Department of Mathematics and Statistics, Binghamton University

The Combinatorics Seminar

2020-01-29T14:03:07-04:00

The Combinatorics Seminar

FALL 2008

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Directions to the department.

Organizers: Emanuele Delucchi, and Thomas Zaslavsky.

The usual day, time, and place are:

Tuesdays, 1:15 - 2:15, in

Room LN-2205,

with coffee, tea, and cookies at 3:45 in the Anderson Room, LN-2207.

Some meetings will be at other times, e.g., when joint with other seminars.

This semester we will have several talks on non-crossing partitions. Here is the link to a short bibliography, including papers that will be presented. Here are links to a first list and second list of relevant papers with reviews (there is overlap with the short bibliography).

Tuesday, September 2
Organizational Meeting (all should come)
+ chat with
Speaker: Thomas Zaslavsky (Binghamton)
Title: Fun at Summer Conferences
Time: 1:15 - 2:15
Room: LN-2205
Tuesday, September 9
Speaker: Emanuele Delucchi (Binghamton)
Title: Finite Reflection Groups, Non-Crossing Partitions, and a Theorem of Deligne
Time: 1:15 - 2:15
Room: LN-2205
Tuesday, September 16
Speaker: Thomas Zaslavsky (Binghamton)
Title: Quasigroups via Graphs
Time: 1:15 - 2:15
Room: LN-2205
Tuesday, September 23
Speaker: Ed Swartz (Cornell)
Title: Three Complexes
Time: 1:15 - 2:15
Room: LN-2205
Tuesday, September 30
Holiday; no meeting.
Tuesday, October 7
Speaker: Thomas Zaslavsky (Binghamton)
Title: Tutte Functions of Matroids
Time: 1:15 - 2:15
Room: LN-2205
Tuesday, October 14
Speaker: Garry Bowlin (Binghamton)
Title: Non-Crossing Partitions, I
Time: 1:15 - 2:15
Room: LN-2205
Tuesday, October 21
Speaker: Garry Bowlin (Binghamton)
Title: Non-Crossing Partitions, II
Time: 1:15 - 2:15
Room: LN-2205
Wednesday, October 29 (Special day)
Speaker: Laura Anderson (Binghamton)
Title: Representation of Matroids by Homotopy Spheres
Time: 2:20 - 3:20 (Special time)
Room: LN-2205
Tuesday, November 4
No meeting today – Election day.
Tuesday, November 11
Speaker: Nate Reff (Binghamton)
Title: The Lattice of Non-Crossing Partitions
Time: 1:15 - 2:15
Room: LN-2205
Thursday, November 13 (Colloquium)
Speaker: Joanna Ellis-Monaghan (St. Michael's College)
Title: The Tutte Polynomial and Potts Model in Statistical Mechanics
Time: 4:30 - 5:30
Room: LN-2205
Friday, November 14 (Special day)
Speaker: Joanna Ellis-Monaghan (St. Michael's College)
Title: Multivariable Tutte and Transition Polynomials
Time: 2:20 - 3:20 (Special time)
Room: LN-2205
Tuesday, November 18
Speaker: Jackie Kaminski (Binghamton)
Title: Regular Non-Crossing Partitions
Time: 1:15 - 2:15
Room: LN-2205
Tuesday, November 25 (joint with the Algebra Seminar)
Speaker: Thomas Zaslavsky (Binghamton)
Title: Graphic Matrices Over a Group
Time: 1:15 - 2:15
Room: LN-2205
Tuesday, December 2 (joint with the Algebra Seminar)
Speaker: Simon Joyce (Binghamton)
Title: The Symmetric Group and Non-Crossing Partitions
Time: 1:15 - 2:15
Room: LN-2205

I will define a poset relation on the symmetric group S_n, which gives a natural order-preserving function from S_n to the lattice of partitions. If we restrict our attention to elements in S_n under a particular n-cycle, we have a lattice which is isomorphic to the lattice of non-crossing partitions. If time permits I'll talk about some of the implications. This work is based on a paper by Thomas Brady.
Tuesday, December 9
Speaker: Lucas Rusnak (Binghamton)
Title: A Multidirected Hypergraph Representation of Matrices with 0, 1, −1 Entries, Part I
Time: 1:15 - 2:15
Room: LN-2205

A multi-directed hypergraph is a combinatorial representation of {0, +1, −1}-matrices that extends the concepts of signed graphs to hypergraphic analogs. I will discuss their discovery and development from hypergraphic matrices and the problems in extending the signed-graphic treatment of the classification of column dependencies.
Tuesday, December 16
Speaker: Lucas Rusnak (Binghamton)
Title: A Multidirected Hypergraph Representation of Matrices with 0, 1, −1 Entries, Part II
Time: 2:50 - 3:50 (Special time)
Room: SW-231 (Special room)

A multi-directed hypergraph is a combinatorial representation of {0, +1, −1}-matrices that extends the concepts of signed graphs to hypergraphic analogs. I will discuss their discovery and development from hypergraphic matrices and the problems in extending the signed-graphic treatment of the classification of column dependencies.

Departmental home page.

Algebra Seminar

2025-05-16T13:08:18-04:00

The Algebra Seminar

The seminar will meet in-person on Tuesdays in room WH-100E at 2:50 p.m. There should be refreshments served at 4:00 in room WH-102. Masks are optional.

Anyone wishing to give a talk in the Algebra Seminar this semester is requested to contact the organizers at least one week ahead of time, to provide a title and abstract. If a speaker prefers to give a zoom talk, the organizers will need to be notified at least one week ahead of time, and a link will be posted on this page.

If needed, the following link would be used for a zoom meeting (Meeting ID: 93487611842) of the Algebra Seminar:

Algebra Seminar Zoom Meeting Link

Organizers: Alex Feingold, Daniel Studenmund and Hung Tong-Viet

To receive announcements of seminar talks by email, please email one of the organizers with your name, email address and reason for joining this list if you are external to Binghamton University.

Spring 2025

January 21
Organizational Meeting

Please think about giving a talk in the Algebra Seminar, or inviting an outside speaker.

January 28
Daniel Studenmund (Binghamton University)
Piecewise isometry groups arising from Weyl groups

Abstract: Here’s a fun way to build a group by cutting and pasting: Start with a Euclidean, spherical, or hyperbolic model geometry $X$ carrying a collection $\mathcal{H}$ of totally geodesic codimension-1 submanifolds determining a regular tessellation $\Delta$ of $X$. A piecewise isometry of $\Delta$ is defined by cutting out finitely many subspaces $S_1,\dotsc, S_k \in \mathcal{H}$ and isometrically mapping the components of what remains to the components obtained by cutting out another finite collection of subspaces $T_1,\dotsc, T_k \in \mathcal{H}$. The collection of all piecewise isometries is a group $PI(\Delta)$. When $\Delta$ is a tessellation of $\mathbb{R}$ by isometric line segments, $PI(\Delta)$ is an extension of Houghton’s group $H_2$. When $\Delta$ is a tessellation of the hyperbolic plane by ideal triangles, $PI(\Delta)$ naturally extends Thompson’s group $V$. Bieri and Sach studied $PI(\mathbb{Z}^n)$, where $\mathbb{Z}^n$ is the standard tessellation of Euclidean space by isometric cubes, obtaining lower bounds on their finiteness lengths and presenting a careful analysis of their normal subgroup structure.

Our story will start with the piecewise isometry group of the tessellation of the Euclidean plane by equilateral triangles, and generalize to piecewise isometry groups of Euclidean tessellations associated with affine Weyl groups of type $A_n$. Pictures will be drawn and preliminary results on algebraic structure and finiteness properties will be discussed. Time permitting, we will connect our discussion to the tessellation of hyperbolic 3-space by regular ideal tetrahedra. This talk covers work in progress with Robert Bieri and Alex Feingold.

February 4
Dikran Karagueuzian (Binghamton University)
You Need a Yoneda

Abstract: The Yoneda Lemma is widely regarded as the most-commonly-quoted result of category theory. This (expository) talk will discuss instances of the lemma appearing in the undergraduate mathematics curriculum, particularly linear algebra.

February 11
Hung Tong-Viet (Binghamton University)
Orders of commutators in finite groups

Abstract: In this talk, I will discuss some problems concerning the orders of some commutators in finite groups and how they affect the structure of the group.

February 18
No Speaker, No Meeting

February 25
No Speaker, No Meeting

March 4
Thu Quan (Binghamton University)
Squaring a conjugacy class in a finite group

Abstract: Let $G$ be a finite group and $K$ be a conjugacy class of $G$. Then $K^2$ consists of the products of any two elements in $K$. In this talk, we consider some equivalent conditions for $K^2$ to be a conjugacy class of $G$. This talk is based on the paper by Guralnick and Navarro in 2015.

March 11
No Meeting, Spring Break

March 18
James Hyde (Binghamton University)
Small generating sets for groups of homeomorphisms of the Cantor set

Abstract: I will give the definition of Chabauty's space of marked groups and use it to give a nicer proof of a result from my thesis. I will then discuss joint work with Collin Bleak, Casey Donoven, Scott Harper on stronger notions of small generating sets for groups of homeomorphisms of the Cantor set.

March 25
Chris Schroeder (Binghamton University)
Finite groups whose maximal subgroups have almost odd index

Abstract: A recurring theme in finite group theory is understanding how the structure of a finite group is determined by the arithmetic properties of group invariants. There are results in the literature determining the structure of finite groups whose irreducible character degrees, conjugacy class sizes or indices of maximal subgroups are odd. These results have been extended to include those finite groups whose character degrees or conjugacy class sizes are not divisible by 4. In this paper, we determine the structure of finite groups whose maximal subgroups have index not divisible by 4. As a consequence, we obtain some new 2-nilpotency criteria. This is joint work with Prof. Hung Tong-Viet.

April 1
Andrew Velasquez-Berroteran (Binghamton University)
Coverings of Groups and Rings

Abstract: Given a group G, a covering of G is a collection of proper subgroups of G whose set-theoretic union is G. The first part of my talk will be dedicated to some history of coverings of groups and providing some results on which finite groups have an equal covering, which is a type of covering where each subgroup is of the same order. The second part of my talk will be dedicated to extending the notion of coverings of groups to that of rings. One result of this extension is determining necessary conditions for a ring $R$ so that the ring of polynomials R[X] has a special type of covering.

April 8
Edgar A Bering IV (San Jose State University)
Two-generator subgroups of free-by-cyclic groups

Abstract: In general, the classification of finitely generated subgroups of a given group is intractable. Even restricting to two-generator subgroups is not enough. However, in a geometric setting classification is possible. For example, a two-generator subgroup of a right-angled Artin group is either free or free abelian. Jaco and Shalen proved that a two-generator subgroup of the fundamental group of an orientable atoroidal irreducible 3-manifold is either free, free-abelian, or finite-index. In this talk I will present recent work proving a similar classification theorem for two generator mapping-torus groups of free group endomorphisms: every two generator subgroup is either free or conjugate to a sub-mapping-torus group. As an application we obtain an analog of the Jaco-Shalen result for free-by-cyclic groups with fully irreducible atoroidal monodromy. While the statement is algebraic, the proof technique uses the topology of finite graphs, a la Stallings. This is joint work with Naomi Andrew, Ilya Kapovich, and Stefano Vidussi.

April 15
No Algebra Seminar

April 22
No Algebra Seminar - Monday Classes Meet

April 29
Hanlim Jang (Binghamton University)
Dehn function and the van Kampen diagram

Abstract: Historically, the word problem leads to the definition of the Dehn function which measures the difficulty of solving the word problem. We will discuss how questions concerning Dehn functions turn into questions concerning the geometry of certain planar 2-complexes called van Kampen diagrams. This translation also explains a link between Riemannian filling problems and word problems. Also, we will discuss the lower bounds on Dehn functions for semi-direct products of Z^n and Z. These results are classical and our approach is based on the work of Bridson and Gersten.

May 6
No Algebra Seminar

Have a good summer! Talks will resume in the fall.

Algebra Seminar

2025-12-13T09:09:06-04:00

The Algebra Seminar

The seminar will meet in-person on Tuesdays in room WH-100E at 2:45 p.m. There should be refreshments served at 3:45 in our new lounge/coffee room, WH-104. Masks are optional.

If needed, the following link would be used for a zoom meeting (Meeting ID: 948 2031 8435, Passcode: 053702) of the Algebra Seminar:

Algebra Seminar Zoom Meeting Link

Organizers: Alex Feingold, Daniel Studenmund and Hung Tong-Viet

To receive announcements of seminar talks by email, please email one of the organizers with your name, email address and reason for joining this list if you are external to Binghamton University.

Fall 2025

August 19
Organizational Meeting

Please think about giving a talk in the Algebra Seminar, or inviting an outside speaker.

August 26
Ryan McCulloch (Binghamton University)
The commuting graph and the centralizer graph of a group

Abstract: Let $G$ be a group. The commuting graph $\mathfrak{C}(G)$ for $G$ is the graph whose vertices are $G-Z(G)$ and if $a, b \in G-Z(G)$, $a \neq b$, then there is an edge between $a$ and $b$ if $ab = ba$. A close cousin of $\mathfrak{C}(G)$ is the centralizer graph, which we define. When a connected component of $\mathfrak{C}(G)$ is a complete graph, the corresponding component in the centralizer graph is an isolated vertex, and we call such a component trivial. Otherwise, the natural bijection between the commuting graph and the centralizer graph preserves the diameter of connected components.

One sees that if $G$ is a Frobenius group with a nonabelian kernel and a nonabelian complement where the complement has nontrivial center, then the centralizer graph of $G$ has more than one nontrivial component. Can this happen in a $p$-group? The answer is yes! In fact, for any specified number $k$ of nontrivial components and any diameter sizes $n_1,\dots, n_k$, one can construct a $p$-group of nilpotency class 2 whose centralizer graph has these specs. This is joint work with Mark Lewis.

September 2
No Meeting (Monday classes meet)

September 9
Chris Schroeder (Binghamton University)
A topological quantum field theory and invariants of finite groups

Abstract: In this talk, we will discuss the properties of finite groups that are witnessed by the group invariants arising in the context of Dijkgraaf-Witten theory, a topological quantum field theory, as invariants of surfaces. Assuming the theory is derived from the complex group algebra of a finite group, these invariants are generalizations of the commuting probability, an invariant that has been well studied in the literature. The main goal of this talk is to construct these invariants from scratch, assuming no previous knowledge of quantum mechanics.

September 16
Alex Feingold (Binghamton University)
Lie Algebras, Representations, Roots, Weights, Weyl groups and Cliﬀord Algebras

Abstract: Lie algebras and their representations have been well-studied and have applications in mathematics and physics. The classification of finite dimensional Lie algebras over C by Killing and Cartan inspired the classification of finite simple groups. Geometry and combinatorics are both involved through root and weight systems of representations, with the Weyl group of symmetries playing a vital role. Infinite dimensional Kac-Moody Lie algebras have deeply enriched the subject and connected with string theory and conformal field theory. In a collaboration with Robert Bieri and Daniel Studenmund, we have been studying tessellations of Euclidean and hyperbolic spaces which arise from the action of affine and hyperbolic Weyl groups. Our goal has been to define and study piecewise isometry groups acting on such tessellations.

Today I will present background material on Lie algebras, representations and examples which show the essential structures. I will present a construction of representations of the orthogonal Lie algebras, $so(2n,F)$, of type $D_n$ as matrices and also using Clifford algebras to get spinor representations.

September 23
No Algebra Seminar

September 30
Thu Quan (Binghamton University)
A generalization of Camina pairs and orders of elements in cosets

Abstract: Let $G$ be a finite group with a nontrivial proper subgroup $H$. If $H$ is normal in $G$ and for every element $x\in G\setminus H$, $x$ is conjugate to $xh$ for all $h\in H$, then the pair $(G,H)$ is called a Camina pair. In 1992, Kuisch and van der Waall proved that $(G,H)$ is a Camina pair if and only if every nontrivial irreducible character of $H$ induces homogeneously to $G$. In this talk, we discuss the equivalence of these two conditions on the pair $(G,H)$ without assuming that $H$ is normal in $G$. Furthermore, we determine the structure of $H$ under the hypothesis that, for every element $x\in G\setminus H$ of odd order, all elements in the coset $xH$ also have odd order.

October 7
No Algebra Seminar

October 14
Hung Tong-Viet (Binghamton University)
Orders of commutators and Products of conjugacy classes in finite groups

Abstract: Let $G$ be a finite group, $x\in G$, and let $p$ be a prime. In this talk, we explore conditions that forces $x$ to lie in certain characteristic subgroups of $G$. In particular, we prove that the commutator $[x,g]$ is a $p$-element for all $g\in G$ if and only if $x$ is central modulo $O_p(G)$, the largest normal $p$-subgroup of $G$. This result unifies and generalizes aspects of both the Baer-Suzuki theorem and Glauberman's $Z_p^*$-theorem. Additionally, we show that if $x\in G$ is a $p$-element and there exists an integer $m\ge 1$ such that for every $g\in G$, the commutator $[x,g]$ is either trivial or has order $m$, then the subgroup generated by the conjugacy class of $x$ is solvable. As an application, we confirm a conjecture of Beltran, Felipe, and Melchor: if $K$ is a conjugacy class in $G$ such that the product $K^{-1}K=1\cup D\cup D^{-1}$ for some conjugacy class $D$, then the subgroup generated by $K$ is solvable.

October 21
Inna Sysoeva (Binghamton University)
Welded braid groups, their (irreducible) representations and linearity questions

Abstract: Welded braid group $WB_n$ is a generalization of a classical braid group $B_n,$ for which some crossings are allowed to be “welded”. From the group-theoretical point of view, $WB_n$ is finitely presented by braid-like and permutation-like generators and relations.

In this talk I am going to define welded braid groups and describe how they are related to the classical braid groups, the groups of automorphisms of free groups and some other interesting groups. I will present my recent results on the classification of the irreducible representations of $WB_n$ of dimension $\leq n$ (https://arxiv.org/abs/2412.21133)

No prior knowledge of the aforementioned groups and their representations is expected.

October 28
Daniel Studenmund (Binghamton University)
Non-simplicity of commensurators of free groups

Abstract: The abstract commensurator of a group is the group of ``germs of automorphisms,'' equivalence classes of isomorphisms between finite-index subgroups. The abstract commensurator of a finitely-generated free group is a fundamental object in mathematics that has a rich structure. We will discuss some results exploring the structure of the commensurator joint with Khalid Bou-Rabee and Edgar Bering, and discuss a recent preprint of Barnea, Ershov, Le Boudec, Reid, Vannacci, and Weigel that proves that the commensurator has no simple finite-index subgroup.

November 4
James Hyde (Binghamton University)
On the Permutational Boone-Higman Conjecture

Abstract: The Boone-Higman Conjecture asserts that a finitely generated group has solvable word problem exactly if it embeds into some finitely presented simple group. I will survey the work that has been done on the Boone-Higman Conjecture and describe work of Jim Belk, Francesco Fournier-Facio, Matt Zaremsky and myself relating it to its permutational variant.

November 11
Tae Young Lee (Binghamton University)
Relations between character values of symmetric groups

Abstract: I will discuss some polynomial relations between the values of complex irreducible characters of finite symmetric groups, and their consequences. In particular, I will show that no irreducible character can vanish on certain sets of conjugacy classes, and use these sets to prove that if n satisfies certain conditions, then it is impossible to cover S_n\{1} with the zero sets of three irreducible characters.

November 18
Nguyen N. Hung (University of Akron)
The McKay theorem with degree inequality

Abstract: The McKay theorem, recently proved by Cabane and Späth, states that for a finite group $G$ and a prime $p$, there exists a bijection $\tau$ between the irreducible $p'$-degree characters of $G$ and those of a $p$-Sylow normalizer. Giannelli proposed that such a bijection should also satisfy the degree inequality $\tau(\chi)(1) \le \chi(1)$ for every irreducible $p'$-degree character $\chi$ of $G$. Proving this strengthened version requires establishing the “inductive McKay condition” with the additional degree inequality, which in turn requires our understanding of the smallest $p'$-degree of $G$ and the largest $p'$-degree of its normalizer. In this talk, I will discuss the details and outline the proof in the case $p = 2$. This is joint work with J. Miquel Martinez and G. Navarro.

November 25
Pratik Misra (Binghamton University)
Directed Gaussian graphical models with toric vanishing ideals

Abstract: Directed Gaussian graphical models use directed acyclic graphs (DAGs) to encode conditional independence relations among jointly Gaussian random variables. Beyond their statistical interpretation, these DAGs also provide a combinatorial parametrization of the covariance matrices that lie in the model, allowing us to view them as algebraic varieties. Understanding the vanishing ideals, that is, the defining polynomial equations of these varieties offers valuable insights into fundamental statistical problems such as model identifiability and causal discovery.

In this talk, I will discuss recent progress toward characterizing those DAGs whose vanishing ideals are toric ideals. In particular, I will present some combinatorial criteria for constructing such DAGs from smaller ones that already have toric vanishing ideals. A key ingredient in this characterization is a monomial map known as the shortest trek map, which plays a central role in describing toric Gaussian DAG models. These results provide a generalized solution to a conjecture of Sturmfels and Uhler, originally posed for undirected graphical models.

December 2
No Algebra Seminar

Algebra Seminar

2024-12-05T15:33:31-04:00

The Algebra Seminar

The seminar will meet in-person on Tuesdays in room WH-100E at 2:50 p.m. There should be refreshments served at 4:00 in room WH-102. Masks are optional.

If needed, the following link would be used for a zoom meeting (Meeting ID: 93487611842) of the Algebra Seminar:

Algebra Seminar Zoom Meeting Link

Organizers: Alex Feingold, Daniel Studenmund and Hung Tong-Viet

To receive announcements of seminar talks by email, please email one of the organizers with your name, email address and reason for joining this list if you are external to Binghamton University.

Fall 2024

August 20
Organizational Meeting

Please think about giving a talk in the Algebra Seminar, or inviting an outside speaker.

August 27
Alex Feingold (Binghamton University)
Tessellations from Affine and Hyperbolic Weyl groups

Abstract: Weyl groups are Coxeter groups generated by reflections determined by a Cartan matrix associated with a Kac-Moody Lie algebra. We will discuss examples where the Weyl group acts on a Euclidean space (affine case) as well as examples where it acts on a hyperbolic space (hyperbolic case). In either case, the action is properly discontinuous so it has a fundamental domain whose reflections tessellate the space. In joint work with Robert Bieri and Daniel Studenmund, we are are investigating the geometry of such tessellations in order to define and obtain generators for groups of piecewise isometries of the tessellations which generalize the Thompson group PPSL(2,Z).

September 3
Dikran Karagueuzian (Binghamton University)
Elliptic Curves for Dummies

Abstract: A theorem on the random variable of inverse image sizes for a polynomial over a finite field of order q computes the moments of the random variable of inverse images sizes up to an error term. This error term decreases with the inverse of the square root of q. Standard results in the theory of elliptic curves will be used to show that this error term cannot generally be improved. No familiarity with the extensive theory of elliptic curves will be assumed.

September 10
Fernando Guzman (Binghamton University)
The Isomorphism Theorems for g-digroups

Abstract: Digroups, and generalized digroups, g-digroups for short, have been considered as a generalization of continuous groups whose tangent space is a Leibniz algebra. This structure has been seen as a generalization of groups, therefore, efforts have been done to study properties and results that come from group theory, to explore if they hold in this new setting. In this talk, we'll discuss the isomorphism theorems for g-digroups, and show that the results for groups do extend to g-digroups. This is joint work with Olga Patricia Salazar-Diaz.

September 17
Ryan McCulloch (Binghamton University)
Centralizers of a group

Abstract: Given a group $G$, define an equivalence relation on the elements of $G$ by $x\sim y$ iff $C_G(x) = C_G(y)$, and let $X$ denote a fixed set of representative elements. The set $X$ can be used to define any centralizer in $G$, since for every centralizer $H$, there is a unique maximal subset $S$ of $X$ so that $H = C_G(S)$. We observe a Galois connection between the lattice of centralizer subgroups of $G$ and this poset of maximal subsets of $X$ corresponding to centralizers. We show other interesting properties related to this relationship.

This is joint work with Wil Cocke and Mark Lewis.

September 24
No Algebra Seminar

October 1
No Algebra Seminar (Recess 1 PM).

October 8
No Algebra Seminar (Friday Classes Meet).

October 15
HanLim Jang (Binghamton University)
Lefschetz properties through a topological lens

Abstract: The topic of this lecture is the algebraic Lefschetz properties, which are abstractions of the important Hard Lefschetz theorem from geometry. I will introduce what are Lefschetz properties and show the relation between Lefschetz properties and commutative algebra.

October 22
No Algebra Seminar

October 29
Marcin Mazur (Binghamton University)
Applications of Zsigmondy's theorem to some results in algebra

Abstract: This is an expository talk which should be accessible to anyone with just basic knowledge of rings and groups. We will state a result from elementary number theory, called Zsigmondy's theorem, and show how to use it to get rather straightforward proofs of some basic results from algebra like the celebrated theorem of Wedderburn that any finite division algebra is a field.

October 29, 4:15 Arithmetic Seminar
Jaiung Jun (SUNY New Paltz, IAS)
Schemes over the natural numbers

Abstract: In this talk, I will first explain how a notion of positivity in algebraic geometry / number theory could be captured in terms of semirings by providing an example of the narrow class group of a number field as a reflexive Picard group. Then, I will introduce a notion of equivariant vector bundles over the natural numbers, and prove a version of Klyachko classification theorem of toric vector bundles in this setting.

November 5
Tae Young Lee (Binghamton University)
Realizing quotients of étale fundamental groups as monodromy groups

Abstract: The étale fundamental groups are analogues of the fundamental groups, in the sense that it classifies the finite étale covering just like how the fundamental groups classify the covering spaces. It is a profinite group, and a characterization of its finite quotient groups is known. However, the proof of this doesn't tell us how to actually construct these finite groups as homomorphic images of the étale fundamental group. In this talk, I will discuss a project of Katz, Rojas Léon, Tiep and me, which classifies the finite quotients of the étale fundamental group of the multiplicative group $\mathbb{G}_m$ over the algebraic closure of finite fields that can be realized as monodromy groups of certain local systems called hypergeometric sheaves.

November 12
Chris Schroeder (Binghamton University)
An introduction to permutation representations of finite groups

Abstract: One often first encounters the representation theory of groups through their actions on vector spaces. It is also natural to consider group actions on sets with no additional structure, so-called permutation representations. Concretely, whereas a linear representation is a group homomorphism to a general linear group, a permutation representation is a group homomorphism to a symmetric group. In this talk, we will introduce the basic definitions and examples of permutation representations of finite groups, and discuss the analogies with linear representations.

November 19
? (University)
Title

Abstract: Text of Abstract

November 26
No Algebra Seminar (Friday Classes Meet).

December 3
Tan Nhat Tran (Binghamton University)
Worpitzky-compatible sets and the freeness of arrangements between Shi and Catalan

Abstract: Given an irreducible root system, the Worpitzky-compatible subsets are defined by a geometric property of the alcoves inside the fundamental parallelepiped of the root system. This concept is motivated and mainly understood through a lattice point counting formula concerning the characteristic and Ehrhart quasi-polynomials. In this talk, we show that the Worpitzky-compatibility has a simple combinatorial characterization in terms of roots. As a byproduct, we obtain a complete characterization by means of Worpitzky-compatibility for the freeness of the arrangements interpolating between the extended Shi and Catalan arrangements. This completes the earlier result by Yoshinaga in 2010 which was done for simply-laced root systems. This is joint work (arXiv:2403.17274) with Takuro Abe (Tokyo).

Algebra Seminar

2024-08-02T11:47:46-04:00

The Algebra Seminar

The seminar will meet in-person on Tuesdays in room WH-100E at 2:50 p.m. There should be refreshments served at 4:00 in room WH-102. Masks are optional.

If needed, the following link would be used for a zoom meeting (Meeting ID: 93487611842) of the Algebra Seminar:

Algebra Seminar Zoom Meeting Link

Organizers: Alex Feingold, Daniel Studenmund and Hung Tong-Viet

To receive announcements of seminar talks by email, please join the seminar's mailing list.

Spring 2024

January 16
No Meeting

January 23
Organizational Meeting

Please think about giving a talk in the Algebra Seminar, or inviting an outside speaker.

January 30
Dikran Karagueuzian (Binghamton University)
You Were Always Studying Cohomology

Abstract: We will discuss the observation that carrying, as taught in grade-school arithmetic, is a cohomology class. This observation is something of a folk theorem. It was surely known to Eilenberg and MacLane, but the earliest written record I can find is an internet post from the 90s by Dolan.

February 6
Daniel Studenmund (Binghamton University)
Finite generation of lattices and Kazhdan's Property (T)

Abstract: Lattices in semisimple Lie groups form an important and rich class of finitely generated infinite groups. But it is not immediately obvious from their definition that lattices are finitely generated. This was first proved for a large class of lattices by Kazhdan using a property now known as (T). In this expository talk I will introduce the definition of Property (T) and discuss its relationship to amenability and finite generation.

February 13
Hung Tong-Viet (Binghamton University)
Conditional characterization of solvability and nilpotency of finite groups

Abstract: As a consequence of the classification of nonsolvable $N$-groups, Thompson proved in $1968$ that a finite group $G$ is solvable if and only if every two-generated subgroup of $G$ is solvable. Various extensions of this theorem have been obtained over the years. In this talk, I will survey some of these results and discuss new characterizations of finite solvable and nilpotent groups using certain restriction on the two-generated subgroups. I will end the talk with applications to the solvable conjugacy class graph of groups.

February 20
No Meeting

February 27
No Meeting

March 5
No Algebra Seminar - Spring Break

March 12
Inna Sysoeva (Binghamton University)
Irreducible $n-$dimensional representations of the group of conjugating automorphisms of a free group on $n$ generators

Abstract: Let $F_n$ be the free group on $n$ generators $x_1, \dots ,x_n.$ The group of conjugating automorphisms $C_n$ is a subgroup of $Aut(F_n)$ consisting of those automorphisms which map every free group generator $x_i$ into a word of the form $W_i^{-1}( x_1 , \dots x_n)x_{\pi(i)} W_i( x_1 , \dots x_n),$ where $W_i( x_1 , \dots x_n)\in F_n$ and $\pi$ is some permutation of indices, $\pi\in S_n.$ The subgroup of $C_n$ that preserves the product $x_1\dots x_n$ is isomorphic to the braid group on $n$ strings, $B_n.$

In this talk I am going to describe my new results on the extensions of the irreducible $n-$dimensional representations of the braid group $B_n$ to the group of conjugating automorphisms $C_n$ of a free group $F_n.$ I will cover all relevant background material on the above-mentioned groups and their representations, so no previous knowledge of the subject is expected.

March 19
No Meeting

March 26
No Meeting

April 2
Omar Saldarriaga (Highpoint University) presented on Zoom
The Lie algebra of the transformation group of certain affine homogeneous manifolds

Abstract: In this talk we will show that, under certain algebraic conditions, a bi-invariant linear connection $\nabla^+$ on a Lie group $G$ induces an invariant linear connection $\nabla$ on a homogeneous space $G/H$ so that the projection $\pi:G\to G/H$ is an affine map. We will also show that if the subgroup $H$ is discrete, there is a method to compute the Lie algebra of the group of affine transformations of $G/H$ preserving the connection $\nabla$. As an application, we will exhibit the Lie algebra of the group of affine transformations of the orientable flat affine surfaces.

April 9
Luna Elliott (Binghamton University)
How semigroup people think about (inverse) semigroups

Abstract: I will give a beginner friendly introduction to semigroups, inverse semigroups and the concepts which are well-known and heavily used by people in these areas. These include green's relations free objects, wagner-preston and special subclasses of these objects. I'm very open to going on tangents if people want me to talk more about anything in particular.

April 16
Rachel Skipper (University of Utah)
Computing Scale in Neretin's group

Abstract: For an automorphism of a totally disconnected, locally compact (tdlc) group, Willis introduced the notion of scale which arose in the development of the general theory of these groups. In this talk, we will discuss the setting where the tdlc group is Neretin's group and where the automorphism comes from conjugation in the group. This is an ongoing joint work with Michal Ferov and George Willis at the University of Newcastle.

April 23
No Algebra Seminar - Passover Break

April 30
Andrew Velasquez-Berroteran (Binghamton University)
Neuroscience: An Algebraic and Topological Viewpoint

Abstract: Neuroscience is the study of the nervous system, and one popular aspect of the nervous system is the brain. Many fields of mathematics have contributed to neuroscience research which include but are not limited to statistics, partial differential equations, dynamics and mathematical physics, etc.

In this talk, I will talk about a brief overview of how algebra and topology has recently been used in the study of the brain. We will primarily be looking at neural coding, and at the end talk about what’s known as the neural ring and neural ideal. I will present under the assumption that attendees will have basic ring theory and topology knowledge but no background knowledge in neuroscience.

May 7
No Algebra Seminar - Finals Week

Algebra Seminar

2026-04-13T10:01:15-04:00

The Algebra Seminar

The seminar will meet in-person on Tuesdays in room WH-100E at 2:45 p.m. There should be refreshments served at 3:45 in our new lounge/coffee room, WH-104. Masks are optional.

If needed, the following link would be used for a zoom meeting (Meeting ID: 948 2031 8435, Passcode: 053702) of the Algebra Seminar:

Algebra Seminar Zoom Meeting Link

Organizers: Alex Feingold, Daniel Studenmund and Hung Tong-Viet

To receive announcements of seminar talks by email, please email one of the organizers with your name, email address and reason for joining this list if you are external to Binghamton University.

Spring 2026

January 20
Organizational Meeting

Please think about giving a talk in the Algebra Seminar, or inviting an outside speaker.

January 27
Alex Feingold (Binghamton University)
Tessellations from hyperplane families: Weyl and non-Weyl cases

Abstract: In collaboration with Robert Bieri and Daniel Studenmund, we have been studying tessellations of Euclidean spaces which arise from families of hyperplanes. A rich class of examples come from a finite type root system and associated finite Weyl group, W, whose affine extension acts on the tessellation. We have also seen examples which do not come from a root system and Weyl group, so we want to understand exactly what geometric properties of the hyperplane families are needed for our project. Our goal has been to define and study piecewise isometry groups acting on such tessellations. In this talk I will discuss the details of some Weyl and some non-Weyl tessellations.

February 3
Tim Riley (Cornell University)
Conjugator length

Abstract: The conjugacy problem for a finitely generated group $G$ asks for an algorithm which, on input a pair of words u and v, declares whether or not they represent conjugate elements of $G$. The conjugator length function $CL$ is its most direct quantification: $CL(n)$ is the minimal $N$ such that if $u$ and $v$ represent conjugate elements of $G$ and the sum of their lengths is at most $n$, then there is a word $w$ of length at most $N$ such that $uw=wv$ in $G$. I will talk about why this function is interesting and how it can behave, and I will highlight some open questions. En route I will talk about results variously with Martin Bridson, Conan Gillis, and Andrew Sale, as well as recent advances by Conan Gillis and Francis Wagner.

February 10
Ryan McCulloch (Binghamton University)
A p-group Classification Related to Density of Centralizer Subgroups

Abstract: If $\mathfrak{P}$ is a property pertaining to subgroups of a $p$-group $G$, and if each subgroup with property $\mathfrak{P}$ contains $Z(G)$, then a group $G$ whose subgroups are dense with respect to property $\mathfrak{P}$ must satisfy the following criteria:

$|Z(G)|= p$ and every subgroup $H$ of order at least $p^2$ contains $Z(G)$.

I will discuss our progress in obtaining a classification of all such $p$-groups. This is joint work with Mark Lewis and Tae Young Lee.

February 17
Tae Young Lee (Binghamton University)
Title: Finite groups with many elements of the same order

Abstract: It is a well-known fact that if more than 3/4 of the elements of a finite group are involutions then the group is abelian. Berkovich proved that if more than 4/15 are involutions then the group must be solvable. Motivated by these results, Deaconescu asked the following question: If at least half of the elements are of the same order, $k$, does the group have to be solvable? In this talk, we prove this when $k = p^a$ for primes $p$ except when $p = 2,3$ and $a > 1$, and give counterexamples for larger powers of 2 and 3 except $k = 4$, and also for several other types of composite numbers. We also show that when $k > 4$, it is always possible to find a non-solvable group such that at least 3/19 of its elements have order $k$. This is a joint work with Ryan McCulloch.

February 24
Lei Chen (Bielefeld University, by Zoom)
Covering a finite group by the conjugates of a coset

Abstract: It is well known that for a finite group G and a proper subgroup A of G, it is impossible to cover G with the conjugates of A. Thus, instead of the conjugates of A, we take the conjugates of the coset Ax in G and check if the union of $(Ax)^g$ covers G-{1} for g in G. Moreover, if $(Ax)^g$ covers G for all Ax in Cos(G:A), we say that (G,A) is CCI. We are aiming to classify all such pairs. It has been proven by Baumeister-Kaplan-Levy that this can be reduced to the case where A is maximal in G, and so that the action of G on Cos(G:A) is primitive, here Cos(G:A) stands for the set of right cosets of A in G. And they showed that (G,A) is CCI if G is 2-transitive. By O'Nan-Scott Theorem and CFSG (classification of finite simple groups), we see that G is either an affine group or almost simple. In the paper by Baumeister-Kaplan-Levy, it is shown that affine CCI groups are 2-transitive. Thus, it remains to consider the almost simple groups. By employing the knowledge of buildings, representation theory, and Aschbacher-Dynkin theorem, we prove that, apart from finitely many small cases, the CCI almost simple groups are 2-transitive.

March 3
Chaitanya Joglekar (Binghamton University)
Lattice basis reduction and the LLL algorithm

Abstract: A lattice L is a subgroup of $\mathbb{R}^n$ isomorphic to $\mathbb{Z}^n$. Finding a vector in L of the shortest length has many applications in number theory, cryptography and optimisation. While finding a vector with the shortest length is an NP hard problem, the LLL algorithm finds a “short enough” vector in Polynomial time. In this talk, we will go over the LLL algorithm and demonstrate one of its applications, finding a Diophantine approximation for a finite set of rational numbers.

March 10
Hanlim Jang (Binghamton University)
Isoperimetric functions of nilpotent groups

Abstract: Isoperimetric functions are a measure of the efficiency of solving the word problem in a finitely presented group. In terms of the length of a word which is generated by generators of a finitely presented group, isoperimetric functions determine how many relators we need to apply at most in order to transform a reduced word to identity. In this talk, we will prove that every finitely generated nilpotent group of class c admits a polynomial isoperimetric function of degree c+1. Our strategy will be using an induction argument on the class c. This talk is based on the paper Isoperimetric inequalities for nilpotent groups written by S.M Gersten, D. F. Holt, and T. R. Riley.

March 17
William Cocke (Carnegie Mellon University)
Determining the Free Spectrum of $A_5$

Abstract: We determine the structure of the group of commutator word maps on the alternating group $A_5$. As a result obtain a formula for the size of the relatively free groups of finite rank in the variety generated by $A_5$.

March 24
No Meeting (No Speaker)

March 31
No Meeting (Spring Break)

April 7
No Meeting (Monday Classes Meet)

April 14
Luna Gal (Binghamton University)
Automatic Groups and their Dehn Functions

Abstract: A group $G$ generated by a finite set $A$ is automatic when it has a regular language of normal forms for its elements that exhibit a geometric property called fellow travelling. We discuss finite automata, a model of computation from the theory of formal languages, and discuss how finite automata can be used for certain computations in automatic groups. Then, we use the theory of finite automata to show that automatic groups have Dehn functions in $O(n^2)$.

April 21
Sean Cleary (City College of New York and Graduate Center of CUNY)
Geometry in and around some wreath products

Abstract: The group Z wr Z and the lamplighter group are not finitely presentable but arise as subgroups of a variety of finitely presented groups. Their geometry is concrete and gives some insight into languages of geodesics, convexity, dead-end elements, and distortion. This includes joint work with Murray Elder, Tim Riley, and Jennifer Taback.

April 28
Thi Hoai Thu Quan (Binghamton University)
The construction of the Chevalley groups and their simplicity

Abstract: Most of the groups appearing in the classification of finite simple groups are the finite simple groups of Lie type. One way to construct these groups is through subgroups of the automorphism groups of the simple Lie algebras over finite fields; groups obtained in this way are called Chevalley groups.

In this talk, we describe this construction and the $(B,N)$-pair structure of a Chevalley group arising from it. We also explain how the $(B,N)$-pair structure can be used to prove the simplicity of certain Chevalley groups.

May 5
Nguyen N. Hung (University of Akron)
The $p$-rationality of Deligne-Lusztig characters

Abstract: Deligne–Lusztig characters are virtual characters of finite reductive groups, constructed via $\ell$-adic cohomology of varieties associated with $F$-stable maximal tori. They provide a systematic framework for classifying, and often explicitly constructing, irreducible complex representations of these groups. In this talk, I will present a recent result showing that if a Deligne–Lusztig character has degree coprime to $p$, then its $p$-rationality coincides with that of the linear character of the maximal torus from which it is induced. I will also discuss evidence suggesting that, more generally, Lusztig induction preserves $p$-rationality for characters of $p'$-degree.

Algebra Seminar

2023-12-18T09:59:50-04:00

The Algebra Seminar

The seminar will meet in-person on Tuesdays in room WH-100E at 2:50 p.m. There should be refreshments served at 4:00 in room WH-102. As of Saturday, March 26, 2022, masks are optional.

If needed, the following link would be used for a zoom meeting (Meeting ID: 93487611842) of the Algebra Seminar:

Algebra Seminar Zoom Meeting Link

Organizers: Alex Feingold, Daniel Studenmund and Hung Tong-Viet

To receive announcements of seminar talks by email, please join the seminar's mailing list.

Fall 2023

August 22
Organizational Meeting

Please think about giving a talk in the Algebra Seminar, or inviting an outside speaker.

August 29
Chris Schroeder (Binghamton University)
Finite groups with many $p$-regular conjugacy classes

Abstract: It is a classical topic in finite group theory to understand a finite group through its simple composition factors. To this end, one would like to construct group invariants that distinguish the nonabelian finite simple groups. One heuristic for simple groups is that they have “few” inequivalent irreducible linear representations, as they have few normal subgroups. In this talk, we will construct invariants from this observation and show how they can be used to determine the structure of finite groups. Our talk aspires to be accessible and interesting to a wide mathematical audience.

September 5
No Algebra Seminar - Monday classes meet

September 12
Dikran Karagueuzian (Binghamton University)
Polynomial and Random Maps of Finite Fields

Abstract: We study the relations between polynomial and random maps by computing the moments of the random variable of inverse image sizes. In the polynomial case, these moments are connected to the Galois Theory of the polynomial over a function field. For random maps, the moments can sometimes be computed using generating function techniques. These computations show both similarities and differences between the two cases.

September 19
Nguyen N. Hung (University of Akron)
A conjectural link between the field of values and degree of an irreducible character

Abstract: For an irreducible character $\chi$ of a finite group $G$, we define $f(\chi)$ as the `cyclotomic deficiency' of $\chi$. This deficiency is the degree of the field extension from the field of values of $\chi$ to its cyclotomic closure. Over thirty years ago, Cram proved that when $G$ is solvable, $f(\chi)$ is always a divisor of the character degree $\chi(1)$. In this talk, I will present strong evidence suggesting that for all finite groups, $f(\chi)$ is bounded above by $\chi(1).$

September 26
Daniel Studenmund (Binghamton University)
Hidden symmetries of lattices in solvable Lie groups

Abstract: The abstract commensurator of a group $G$, the group of isomorphisms between finite-index subgroups modulo equivalence, encodes symmetries of $G$ that may be ``hidden.'' When $G$ is a lattice in a simple Lie group incommensurable with $\operatorname{PSL}(2,\mathbb{R})$, work of Mostow, Prasad, Borel, and Margulis shows that the abstract commensurator of $G$ detects whether the group arises through arithmetic constructions. In this talk, I will discuss results on abstract commensurators of the other significant class of lattice in Lie groups, the solvable groups. This builds on classical work of Malcev for nilpotent groups and more recent rigidity results of Mostow, Morris, and Baues–Grunewald.

October 3
Daniel Studenmund (Binghamton University)
Hidden symmetries of lattices in solvable Lie groups, part II

Abstract: The abstract commensurator of a group $G$, the group of isomorphisms between finite-index subgroups modulo equivalence, encodes symmetries of $G$ that may be `hidden'. When $G$ is a lattice in a simple Lie group incommensurable with $\operatorname{PSL}(2,\mathbb{R})$, work of Mostow, Prasad, Borel, and Margulis shows that the abstract commensurator of $G$ detects whether the group arises through arithmetic constructions. In this talk, I will discuss results on abstract commensurators of the other significant class of lattice in Lie groups, the solvable groups. This builds on classical work of Malcev for nilpotent groups and more recent rigidity results of Mostow, Morris, and Baues–Grunewald.

October 10
Olga Patricia Salazar-Diaz (Binghamton University and National University of Colombia)
An introduction to generalized digroups (g-digroups)

Abstract: In this talk we describe what a g-digroup is, and some of its group-type properties.

October 17
? ( University)
Title

Abstract: Text of Abstract

October 24
Alireza Salahshoori (Binghamton University)
Totally Unimodular Matrices: An Introduction

Abstract: Totally unimodular matrices have very nice properties with respect to solutions of linear equations, linear programming and combinatorial optimization. I will introduce them and some of the reasons they get attention.

October 31
Chris Schroeder (Binghamton University)
Flipping a quantum coin

Abstract: The state of a quantum system is described by a vector in a complex vector space. Therefore, whenever a group of transformations acts on a quantum system, we obtain a complex representation of the group. In this talk, we will work out a concrete example – the representation theory of physical rotations acting on a two-state quantum system – and discover the spookiness of quantum theory.

November 7
Marwa Mosallam (Binghamton University)
Localization with respect to certain periodic homology theories

Abstract: We will define localization and mention some results of Bousfield on localization in the stable homotopy category. If time permits we might explain the difference between localization of spectra and spaces.

November 14
Mithun Veettil (Binghamton University)
Wreath Product of groups and indicatrix polynomial of a group action

Abstract: First, we will define the wreath product of finite groups. Then we will define a polynomial called `indicatrix of a group' that captures the fixed points of the action of the group on some set. It turns out that the indicatrix behaves `nicely' upon taking the wreath product. If time permits, we shall go through specific examples; we will compute the indicatrix of the symmetric group on k letters, S_k, acting naturally on {1,2,…,k}.

November 21
No Algebra Seminar - Friday classes meet

November 28
Hao Ye (Binghamton University)
Recovering Objects with Morphisms-elementary introduction to category theory

Abstract: It is about the theme of “Recovering Objects with Morphisms.” It will begin by introducing basic concepts of category and related functors, as well as natural transformations. Following that, I will use universal properties and the Yoneda Lemma to illustrate the theme, and I will outline the general approach for proving the Yoneda Lemma. During the talk, I will use some examples from the book “Abstract Algebra” by Dummit and other relevant sources, such as “Category Theory: A Gentle Introduction” by Peter Smith and “The Rising Sea: Foundations of Algebraic Geometry” by Ravi Vakil.

December 5
Andrew Manuel Velasquez-Berroteran (Binghamton University)
Introduction to Group Rings

Abstract: Given a group G and a ring R we can formulate a new ring, called the group ring RG, whereby elements in this ring can be thought of as an R-linear combination of elements of G. We will look at examples of groups rings, theorems and propositions of groups rings as well as look at some basic module theory.

This talk will be accessible to students who have taken a first semester undergraduate course in modern algebra.

Algebra Seminar

2022-12-19T22:53:31-04:00

The Algebra Seminar

The seminar will meet in-person on Tuesdays in room WH-100E at 2:50 p.m. There should be refreshments served at 4:00 in room WH-102. As of Saturday, March 26, 2022, masks are optional.

If needed, the following link would be used for a zoom meeting (Meeting ID: 981 8719 2351) of the Algebra Seminar:

Algebra Seminar Zoom Meeting Link

Organizers: Alex Feingold, Daniel Studenmund and Hung Tong-Viet

To receive announcements of seminar talks by email, please join the seminar's mailing list.

Fall 2022

August 23
Organizational Meeting

Please think about giving a talk in the Algebra Seminar, or inviting an outside speaker.

August 30
Alex Feingold (Binghamton University)
Representations of Kac-Moody Lie Algebras

Abstract: We will first review basic definitions and examples of finite dimensional Lie algebras and their representations. The infinite dimensional Heisenberg Lie algebra will be shown to have a representation on the space of polynomials in infinitely many variables. That space, $V$, is $\bf Z$-graded into subspaces $V_n$ of total degree $n$ with $\dim(V_n) = p(n)$, the classical partition function.

Then we will give a brief introduction to the infinite dimensional Kac-Moody (KM) Lie algebras defined by generators and relations from a generalized Cartan matrix, $A$. We will discuss in detail the cases when $A$ is $2\times 2$, which give either affine or hyperbolic KM algebras. The affine case is related to the Heisenberg algebra, and we will present the root diagram and a weight diagram for an irreducible representation. We will discuss one hyperbolic example, $Fib$, and study its root system, and some irreducible representations (highest weight and non-standard).

September 6
No Algebra Seminar - Monday classes meet

September 13
Dikran Karagueuzian (Binghamton University)
Moments of Polynomials over Finite Fields

Abstract: A polynomial over a finite field can be regarded as a map of the finite field to itself. The variance of the inverse image sizes has been studied in connection with the question of whether such maps are a good substitute for random maps. We are able to show that polynomial maps are not random in the sense that all moments, not just the variance, must be integers, in an asymptotic sense as the size of the finite field becomes large. This is based on joint work with Per Kurlberg.

September 20
Daniel Studenmund (Binghamton University)
Introduction to automorphism towers

Abstract: The automorphism tower of a centerless group $G$ is the increasing sequence $G < Aut(G) < Aut(Aut(G)) <$ etc., which is then extended to be ordinal-indexed at limit ordinals via unions. A group $G$ is called complete if this sequence stabilizes at the second term — or, equivalently, if $Out(G)$ is trivial. Wielandt proved in the 1930s that this sequence stabilizes at some finite stage for any finite, centerless group $G$. Nearly 50 years later, Thomas proved that the sequence stabilizes without the “finite” hypothesis, at the expense of allowing the tower to stabilize at a possibly infinite ordinal. We will review basic definitions and constructions, and if time permits, compare in form to the Neukirch-Uchida theorem.

September 27
No Algebra Seminar - No classes

October 4
No Algebra Seminar - 1 PM Recess

October 11
Luke Elliot (Binghamton University)
A connection between Thomson's Groups and (unindexed) two-sided subshifts of finite type

Abstract: Two-sided subshifts of finite type are a class of topological dynamical system (or equivalently topological unary algebras) which have been studied for over 50 years. Most famously the problem of determining when two subshifts are isomorphic remains unsolved. Thompsons group V has been around for a similarly long amount of time and (together with T) was the first known example of a finitely presented infinite simple group. I plan to introduce both subshifts of finite type and V, and breifly discuss recently discoved connections between them.

October 18
Nham Ngo (University of North Georgia - Gainesville)
Cohomology of $\mathrm{SL}_2$

Abstract: Let $k$ be an algebraically closed field of prime characteristic and $SL_2$ the rank 1 simple algebraic group defined over $k$. Cohomology of $SL_2$ is still unknown for many cases. In this talk, we will give a brief overview of representation and cohomology of this group. Some new calculations on bounding the dimension of $SL_2$-cohomology with coefficients in Weyl modules will be introduced. (This is joint work with Khang Pham.)

October 25
Hung Tong-Viet (Binghamton University)
Variations of Baer-Suzuki theorem and applications

Abstract: The Baer-Suzuki theorem, a classical result in finite group theory, states that if $C$ is a conjugacy class of a finite group $G$ and if every two elements in $C$ generate a nilpotent subgroup, then $C$ generates a nilpotent normal subgroup of $G$. This gives a nice characterization of the Fitting subgroup of $G$, that is, the largest nilpotent normal subgroup of $G$. This theorem was originally proved by Baer and later by M. Suzuki in 1965. A more direct and elementary proof was obtained by Alperin and Lyons in 1971. Many generalizations of this theorem have been proposed and studied over the years. In this talk, I will discuss the proofs of this theorem and some of its variants and will give an application of these results to the character theory of finite groups.

November 1
Yash Madanha (University of Pretoria)
Average number of zeros of characters of finite groups

Abstract: There has been some interest on how the average character degree affects the structure of a finite group. In this talk we will discuss an analogue of the average character degree: denoted by $ \mathrm{anz}(G) $, the average number of zeros of characters of a finite group $ G $ is defined as the number of zeros in the character table of $ G $ divided by the number of irreducible characters of $ G $. We show that if $ \mathrm{anz}(G) < 1 $, then the group $ G $ is solvable and also that if $ \mathrm{anz}(G) < \frac{1}{2} $, then $ G $ is supersolvable. We characterise abelian groups by showing that $ \mathrm{anz}(G) < \frac{1}{3} $ if and only if $ G $ is abelian. We shall also discuss some work by Moreto and some very recent work of Qian on this invariant.

November 8
Luise-Charlotte Kappe (Binghamton University, retired)
A generalization of the Chermak-Delgado lattice to words in two variables

Abstract: The Chermak-Delgado measure of a subgroup $H$ of a finite group $G$ is defined as the product of the order of $H$ with the order of the centralizer of $H$ in $G$, i.e. $m_G(H) = |H|\,|C_G(H)|$, and the set of all subgroups with maximal Chermak-Delgado measure forms a lattice in $G$.

Let $f(x,y)$ be a word in the alphabet $\{x,y,x^{-1},y^{-1}\}$ and $H$ a subgroup of a group $G$. The following sets are subgroups of $G$: \begin{align*} F_1^{\ell}(G,H) &= \{a\in G\mid f(ag,h)=f(g,h)\ \forall g\in G,\forall h\in H\}\\ F_1^r(G,H) &= \{a\in G\mid f(ga,h) = f(g,h)\ \forall g\in G,\forall h\in H\}\\ F_2^{\ell}(G,H) &= \{a\in G\mid f(h,ag) = f(h,g)\ \forall g\in G,\forall h\in H\}\\ F_2^r(G,H) &= \{a\in G\mid f(h,ga) = f(h,g)\ \forall g\in G, \forall h\in H\} \end{align*} For the commutator word $f(x,y) = [x,y]$, we have \[ F_1^{\ell}(H) = F_2^{\ell}(H) = C_G(H)\text{ and } F_1^r(H)=F_2^r(H)= C_G(H^G).\]

The Chermak-Delgado measure associated with the subgroup $F_i^t(G,H)$, with $i=1,2$ and $t=\ell,r$, is $m_G(H) = |H|\,|F_i^t(G,H)|$. The question arises for which words $f(x,y)$ the subgroups having maximal Chermak-Delgado measure form a lattice. We discuss the obstacles that the generalization encounters, and some situations in which they can be surmounted.

November 15, Combinatorics Seminar, 1:15-2:15
Christian Gaetz (Cornell University)
$1$-skeleton posets of Bruhat interval polytopes

Abstract: Bruhat interval polytopes are combinatorially interesting polytopes arising from total positivity and from certain toric varieties. I study the $1$-skeleta of these polytopes, viewed as posets interpolating between weak order and Bruhat order. Interestingly, these posets turn out to be lattices and the polytopes, despite not necessarily being simple, have interesting $h$-vectors. I will give a criterion for determining when these polytopes are simple, or equivalently when generic torus orbit closures in Schubert varieties are smooth, solving a conjecture of Lee–Masuda.

November 15
Christian Gaetz (Cornell University)
Stable characters from permutation patterns

Abstract: We study the expected value (and higher moments) of the number of occurrences of a fixed permutation pattern on conjugacy classes of the symmetric group $S_n$. We prove that this virtual character stabilizes as $n$ grows, so that there is a single polynomial computing these moments on any conjugacy class of any symmetric group. Our proof appears to be the first application of partition algebras to the study of permutation patterns. I’ll also discuss partial progress towards a conjecture on when these virtual characters are genuine characters. This is joint work with Christopher Ryba and Laura Pierson.

November 22
Ryan McCulloch (Elmira College)
On groups with few subgroups not in the Chermak-Delgado lattice

Abstract: The Chermak-Delgado lattice of a finite group $G$, denoted $CD(G)$, is a modular sublattice of the lattice of subgroups of $G$. $CD(G)$ has nice properties, for example it is self-dual (and so if you turn the lattice upside down, you arrive at the same lattice). $CD(G)$ has been studied extensively in recent years. A recent question of Fasolă & Tărnăuceanu asks about subgroups not in $CD(G)$. Let $\delta(G) = |L(G)| - |CD(G)|$ where $L(G)$ denotes the lattice of all of the subgroups of a finite group $G$. Fasolă & Tărnăuceanu classify all finite groups $G$ with $\delta(G) < 3$. We extend the classification for all finite groups $G$ with $\delta(G) < 5$. We also obtain a classification for the non-nilpotent case when $\delta(G) = 5$. The non-nilpotent cases make extensive use of Sylow theory, and the $p$-group cases blend the computational with the theoretical. This is joint work with David Burrell and William Cocke.

November 29
Andrew Velasquez-Berroteran (Binghamton University)
Equal Coverability of Finite Groups

Abstract: A finite group $G$ has an equal covering if it is a union of a collection of proper subgroups of $G$ that are all of the same order. This talk has two parts. The first part will present the work done in my undergraduate thesis at Adelphi University aimed at finding which finite groups under order 61, and which simple groups under order 100,000, have an equal covering. The second part will discuss the work done by my undergraduate advisor, Tuval Foguel, and his colleagues Alireza Moghaddamfar and Jack Smith. Their work this past summer expanded on my thesis, including a proof of the following claim I made in my thesis: No simple finite group has an equal covering.

December 6
Eilidh McKemmie (Rutgers University)
Galois groups of random additive polynomials

Abstract: The Galois group of an additive polynomial over a finite field is contained in a finite general linear group. We will discuss three different probability distributions on these polynomials, and estimate the probability that a random additive polynomial has a “large” Galois group. Our computations use a trick that gives us characteristic polynomials of elements of the Galois group, so we may use our knowledge of the maximal subgroups of GL(n,q). This is joint work with Lior Bary-Soroker and Alexei Entin.

Algebra Seminar

2023-05-15T12:30:54-04:00

The Algebra Seminar

The seminar will usually meet in-person on Tuesdays in room WH-100E at 2:50 p.m. There should be refreshments served at 4:00 in room WH-102. Masks are optional.

If needed, the following link would be used for a zoom meeting (Meeting ID: 981 8719 2351) of the Algebra Seminar:

Algebra Seminar Zoom Meeting Link

Organizers: Alex Feingold, Daniel Studenmund and Hung Tong-Viet

To receive announcements of seminar talks by email, please contact the organizers and explain your background and why you wish to subscribe. The usual subscription method has led to many spam requests.

Spring 2023

January 17
Organizational Meeting

Please think about giving a talk in the Algebra Seminar, or inviting an outside speaker.

January 24
Alex Feingold (Binghamton University)
Representations of Kac-Moody Lie Algebras (Part 1)

Abstract: We will first review basic definitions and examples of finite dimensional Lie algebras and their representations. The infinite dimensional Heisenberg Lie algebra will be shown to have a representation on the space of polynomials in infinitely many variables. That space, $V$, is $\bf Z$-graded into subspaces $V_n$ of total degree $n$ with $\dim(V_n) = p(n)$, the classical partition function.

Then we will give a brief introduction to the infinite dimensional Kac-Moody (KM) Lie algebras defined by generators and relations from a generalized Cartan matrix, $A$. We will discuss in detail the cases when $A$ is $2\times 2$, which give either affine or hyperbolic KM algebras. The affine case is related to the Heisenberg algebra, and we will present the root diagram and a weight diagram for an irreducible representation. We will discuss one hyperbolic example, $Fib$, and study its root system, and some irreducible representations (highest weight and non-standard).

January 31
Alex Feingold (Binghamton University)

Representations of Kac-Moody Lie Algebras (Continued)

February 7
Chris Schroeder (Binghamton University)

Understanding invariants of finite groups

Abstract: In this talk, we will consider the properties of finite groups that are preserved under isomorphism, which we call group invariants. The richness of finite group theory is reflected in the zoo of invariants that have been defined and studied. These invariants typically stem from the abstract group structure or the representation theory. We will discuss how group invariants can shed light on the classification of finite simple groups. This talk aims to be accessible and appealing to a broad mathematical audience.

February 14
Daniel Studenmund (Binghamton University)

Lie groups and pro-$p$ groups

Abstract: We will discuss some results on pro-$p$ groups, with a focus on cases in which pro-$p$ groups behave like Lie groups in some sense. In particular, we will discuss the notion of an analytic structure on certain $p$-pro groups, and ways that this structure is analogous to the analytic structure on Lie groups. This talk will cover no original work, but rather focus on presenting work of Lazard, Serre, and others.

February 21
Robert Bieri (Binghamton University)

Groups of tile-permutations defined by finitary rearrangements of tessellations

Abstract: Let $M$ be the Euclidean or hyperbolic $n$-space with a regular tessellation. By a finitary rearrangement of $M$ (with respect to decompositions of $M$ into finitely many pieces of a pre-specified shape) we mean the process of cutting $M$ along tile-boundaries into finitely many tessellated pieces $X_i$ each of which is either a single tile or isometric to one of the specified infinite shapes, and then mapping each piece by a tessellation-respecting isometric embedding $f_i : X_i \to M$, to a new position, with the property that the images $f_i(X_i)$ cover $M$ and have pairwise disjoint interiors. The union of the maps $f_i$ is well defined on all interior points of the tiles and induces a permutation of the set $\Omega$ of all tile-centers which we call a finitary piecewise isometric tile-permutation of $M$.

It is interesting to replace $M$ by the abstract planar tree $T$ with all its vertices of degree $3$, view it as tessellated by its edges, and consider a corresponding construction. For if one restricts attention to finitary rearrangements of $T$ with respect to decompositions of $T$ whose infinite pieces specified to be rooted dyadic trees (with root of degree $2$ and all other vertices of degree $3$), then one recovers Richard Thompson's key tool to the structure of his groups: It exhibits the the elements of Thompson's group – up to finite permutations – as finitary piecewise planar-tree-isometric edge-permutations of $T$.

Moreover, the tree $T$ occurs as the dual of the tessellation of the hyperbolic plane $H^2$ given by an ideal triangle and its iterated reflections over the sides; and the close connection between the isometries of $H^2$ and the planar-tree-isomorphisms of $T$ allows one to show that Thompson's groups can also be interpreted as groups of piecewise hyperbolic-isometric tile-permutations of $H^2$.

The observation that Thompson's groups are part of the game could nurture hopes that other basic regular tessellations might lead to similar gem-stones. The Euclidean space $E^n$ with its standard unit-cube tessellation is certainly the most natural down-to-earth case; and it is also natural to handle it by decomposing $E^n$ into finitely many orthants: i.e. intersections of tessellated half-spaces. In this situation it is convenient to describe the set of all tile centers directly, identifying them with the lattice $\Omega = Z^n$. The decomposition of $E^n$ into orthants thus decomposes $Z^n$ into a finite pairwise disjoint union of single points, discrete rays, discrete quadrants, … , discrete rank-$n$ orthants; and the elements of our piecewise Euclidean-isometric permutation group $G = pei(Z^n)$ have to rearrange them to a new disjoint-union-decomposition of $Z^n$. On the face of it this looks rather complicated, but those who nurture hope for gem-stones should be prepared to dig deep.

However, it turned out to be doable. In my talk I will indicate how a $G$-equivariant germs-of-orthants structure at infinity of $Z^n$ with a corank-$1$ flow helps to get information on the group structure of $G$ to show:

(1) The group of all piecewise Euclidean isometric permutations of $Z^n$ is the fundamental group of a CW-complex with finite $(2n – 1)$-skeleton. This is joint work with Heike Sach, ArXiv:2016. (Recall that Thompson's groups are of type $F_\infty$).

(2) For each $0\leq k\leq n$ the subgroup $G_k$ consisting of all elements of $G$ supported on the $k$-skeleton of $E^n$ is normal in $G$. In my pandemic related sabbatical from BU I added: Every normal subgroup $N$ of rank $k$ in $G_k$ contains the unique subgroup of index $2$ of $G_{k-1}$. See J. London Math. Soc. 2022. (Recall that Thompson's groups $V$ and $T$ are simple and every normal subgroup of $F$ contains $F'$).

February 28
Banafsheh Akbari (Cornell University)

The Structure of Finite Groups affected by Vertex Neighborhoods of their Solubility Graphs

Abstract: The solubility graph associated with a finite group $G$ is a simple graph whose vertices are the elements of $G$, and there is an edge between two distinct vertices if and only if they generate a soluble subgroup. Properties of this graph can have dramatic consequences on the structure of $G$. For example, it has recently been proved that if some non-identity element has prime degree then $G$ is an abelian simple group. So in this talk, we focus our attention on the set of neighbors of a vertex $x$ which we call it the solubilizer of $x$ in $G$, $Sol_G(x)$. We investigate both arithmetic and structural properties of this set and discuss how restrictions on the structure of this set affect the structure of $G$.

March 7
Alex Feingold (Binghamton University)

Fusion Rules for Affine Kac-Moody Algebras

Abstract: This is an expository introduction to fusion algebras for affine Kac-Moody algebras, with major focus on the algorithmic aspects of their computation and the relationship with tensor product decompositions. Explicit examples are included illustrating the rank 2 cases. Previous work of the author and collaborators on a different approach to fusion rules from elementary group theory is also explained.

March 14
Lucia Morotti (Heinrich-Heine-Universität Düsseldorf)

Decomposition matrices of spin representations of symmetric groups

Abstract: Not much is known about decomposition numbers of spin representations of symmetric groups. For example it is not even known whether in general the decomposition matrices are triangular. I will show how certain parts of the decomposition matrices look like and in particular focus on rows corresponding to spin representations labeled by partitions with at most 2 parts.

March 21
(? University)

Title

Abstract: Text of Abstract

March 28
(? University)

Title

Abstract: Text of Abstract

April 4
No Seminar (Spring Break)

April 11
Duc-Khanh Nguyen (SUNY Albany)

A generalization of the Murnaghan-Nakayama rule for $K$-$k$-Schur and $k$-Schur functions

Abstract: We introduce a generalization of $K$-$k$-Schur functions and $k$-Schur functions via the Pieri rule. Then we obtain the Murnaghan-Nakayama rule for the generalized functions. The rule are described explicitly in the cases of $K$-$k$-Schur functions and $k$-Schur functions, with concrete descriptions and algorithms for coefficients. Our work recovers the result of Bandlow, Schilling, and Zabrocki for $k$-Schur functions, and explains it as a degeneration of the rule for $K$-$k$-Schur functions. In particular, many other special cases promise to be detailed in the future.

April 18
Nicholas Packauskas (SUNY Cortland)

Growth of Betti Sequences over Commutative Rings

Abstract: One method of studying a local commutative ring is to study its category of finitely generated modules. An invariant of particular interest is a module’s Betti sequence. This sequence contains various information about the module, but in particular it measures the growth of the minimal free resolution of the module. Studying such sequences can also provide information about the ring. In particular, it is known that if every finitely generated module has a minimal free resolution that grows on the order of a polynomial, then the ring must be a complete intersection. One can say much more in this case; each Betti sequence will be governed by a quasipolynomial of period 2. We will explore such sequences, and establish a bound on the discrepancy between the polynomials which govern their even and odd terms. The bound is computed using an invariant of the ring called its quadratic codimension. (Joint work with Lucho Avramov and Mark Walker.)

April 25
No Seminar

May 2
No Seminar (Friday Classes Meet)

Algebra Seminar

2021-12-19T11:43:35-04:00

The Algebra Seminar

If we have a normal semester, the seminar will meet in-person on Tuesdays in room WH-100E at 2:50 p.m. There should be refreshments served at 4:00 in room WH-102. If the COVID Delta variant resurgence causes us to change the plan, we would go back to having only zoom talks. Currently we are required to use masks for all indoor campus events, so all speakers and attendees must wear masks in the Algebra Seminar.

If needed, the following link would be used for a zoom meeting (Meeting ID: 981 8719 2351) of the Algebra Seminar:

Algebra Seminar Zoom Meeting Link

Organizers: Alex Feingold and Hung Tong-Viet

To receive announcements of seminar talks by email, please join the seminar's mailing list.

Fall 2021

August 24
Organizational Meeting

Please think about giving a talk in the Algebra Seminar, or inviting an outside speaker.

August 31
Fikreab Admasu (Binghamton University)
Plane partitions, Schur functions, and representation theory of $S_n$

Abstract: This is an expository talk on how the theory of symmetric functions, such as Schur functions, ties in with the theory of plane partitions and the representation theory of $S_n$. We will conclude with some open problems that have connections with number theory and discuss current attempts at extensions of the above relations.

September 7
No Seminar

September 14
Fikreab Admasu (Binghamton University)
Plane partitions, Schur functions, and representation theory of $S_n$

Abstract: This is a continuation of the first talk on Aug. 31.

September 21
Alex Feingold (Binghamton University)
Combinatorics related to representations of Clifford and Lie algebras

Abstract: I will give an introduction to Clifford algebras and Lie algebras, and give examples showing how their representation theory is connected in certain cases to combinatorics. In particular, I will discuss the bosonic representation of an infinite dimensional Heisenberg Lie algebra whose weight spaces have dimensions given by the classical partition function. In contrast, Clifford algebras have fermionic representations whose weight spaces have dimensions given by certain restricted partitions (into distinct parts). These provide simple models of two kinds of fundamental particles in physics, bosons and fermions.

September 28
Anthony Ercolano (Binghamton University)
Cellular Automata, The Garden of Eden Theorem and Surjunctive Groups

Abstract: In this talk we will introduce the theory of cellular automata, beginning with a simple explanation of the most famous cellular automaton, John Conway's Game of Life. We will then generalize the Game of Life, putting the theory on more rigorous mathematical footing so we can discuss important classes of groups related to the theory, namely surjunctive groups, residually finite groups, and Hopfian groups. Our talk will conclude with a discussion of some of the open problems related to them.

October 5
Daniel Studenmund (Binghamton University)
Finite-index subgroups of infinite nilpotent groups

Abstract: Finitely generated infinite nilpotent groups, such as the discrete Heisenberg group, possess rich geometric structure while having algebraic structure amenable to a variety of computational techniques. We will review some established results about word growth and subgroup growth in nilpotent groups due to Milnor, Bass, Guivarc'h, Gromov, Grunewald, du Sautoy, and others. We then introduce a metric space structure on the collection of finite-index subgroups, and state results on growth of metric balls in this space. This covers work joint with Khalid Bou-Rabee of CCNY.

October 12
Sailun Zhan (Binghamton University)
Hilbert schemes of points of algebraic surfaces

Abstract: Moduli spaces parametrizing objects associated with a given space are a rich source of spaces with interesting structures. A Hilbert scheme of n points of a projective/quasi-projective variety is the moduli space parametrizing 0-dimensional subschemes of length n. It parametrizes “n points” on the space and is highly related to the n-th symmetric product. But it contains more information than that. The local structure is also related to partition functions and Young diagrams. We will use concrete examples (e.g. the space being $\mathbb{C}$ or $\mathbb{C}^{2}$) and give an expository talk on the topic.

October 19
Jonathan Doane (Binghamton University)
Varieties with constants

Abstract: Rings and groups are among the many classes of (universal) algebras, called varieties, which contain constant(s) as part of their structure. In fact, the ring with unity $\textbf{R}=\langle\{0,1\};+,\cdot,0,1\rangle$ whose only elements are constants, generates a variety interesting in its own right, namely Boolean rings. Notably, $\textbf{R}$ is 0-generated as it is required to contain both 0 and 1. This talk aims to provide a small, general theory for those varieties which are generated by a single 0-generated algebra.

October 26
Zach Costanzo (Binghamton University)
Real-valued character degree patterns of finite groups

Abstract: Given a finite group $G$, we can consider the influence the set of character degrees has on the structure of the group. In this talk, we will restrict our view to the characters which take their values over $\mathbb{R}$, and examine groups whose real-valued character degrees satisfy certain patterns.

November 2
Hung Tong-Viet (Binghamton University)
Character degrees and conjugacy class sizes

Abstract: There is a strong but mysterious connection between the complex character degrees and conjugacy class sizes of finite groups. Many important results on character degrees admit a dual version for conjugacy class sizes and vice versa. Both of these numerical invariants have strong influence on the structure of the groups. In this expository talk, I will survey some known results and open problems concerning these two invariants.

November 9
Christopher Schroeder (Binghamton University)
The structure of finite groups via relative character degrees

Abstract: A fruitful line of research in the representation theory of finite groups is to relate group structure and the degrees of ordinary irreducible characters (that is, the dimensions of the irreducible complex representations). For example, the celebrated It${\hat o}$-Michler theorem says that a finite group has normal, abelian Sylow p-subgroups if and only if all its ordinary irreducible character degrees are coprime to p. In this talk, we discuss a more general setting in which only the relative degrees of characters over a normal subgroup are considered.

November 16
Tung T. Nguyen (Western University/Nature Claim)
Join of circulant matrices

Abstract: Circulants matrices provide a nontrivial, beautiful, and simple set of objects in matrix theory. They appear quite naturally in many problems in network theory and non-linear dynamics. The circulant diagonalization theorem describes the eigenspectrum and eigenspaces of a circulant matrix explicitly via the fast Fourier transform. Consequently, many problems involving circulant matrices have closed-form or analytical solutions. In this talk, we generalize the circulant diagonalization theorem to the join of several circulant matrices. We then discuss some applications of our theorems to computational neuroscience and spectral graph theory.

A panopto recording of the talk by Tung T. Nguyen is available through the following link: Algebra Seminar talk by Tung T. Nguyen

November 23
No Seminar

November 30
Yiyong Yan (Binghamton University)
Is there a classification of finite nilpotent groups?

Abstract: We define $f(n)$ to be the number of groups of order $n$ up to isomorphism. As $n$ increases, the problem of classifying groups of order $n$ becomes hard. Charles Sims [86] proved in 1965 that there is a upper bound for the number of groups of order $n$ up to isomorphism. This talk will show that the error term in the Sims bound for the number of $p$-groups of order $p^m$ may be improved if we restrict our enumeration to $p$-groups of nilpotency class at most $3$. Indeed, our aim is to show that the number of such groups is at most $p^{(2m^3/27+O(m^2))}$.

December 7
No Seminar

Algebra Seminar

2018-08-05T16:33:40-04:00

The Algebra Seminar

Unless stated otherwise, the seminar meets Tuesdays in room WH-100E at 2:50 p.m. There will be refreshments served at 4:00 in room WH-102.

Organizers: Alex Feingold and Marcin Mazur

To receive announcements of seminar talks by email, please join the seminar's mailing list.

Spring 2018

January 16
Jonas Deré (KU Leuven Kulak)
Which manifolds admit expanding maps

Abstract: In 1981, M. Gromov completed the proof that every manifold admitting an expanding map is, up to finite cover, homeomorphic to a nilmanifold. Since then it was an open question to give an algebraic characterization of the nilmanifolds admitting an expanding map. During my talk, I will start by introducing the basic notions of expanding maps and nilmanifolds. Then I explain how the existence of such an expanding map only depends on the covering Lie group and on the existence of certain gradings on the corresponding Lie algebra. One of the applications is the construction of a nilmanifold admitting an Anosov diffeomorphism but no expanding map, which is the first example of this type.

January 23
Organizational Meeting
Title of Talk

Abstract: Please come or contact the organizers if you are interested in giving a talk this semester or want to invite someone.

January 30
Casey Donoven (Binghamton University)
Subgroups and Supergroups of Thompson's Group V

Abstract: Thompson's group V is a group of homeomorphisms of Cantor space. It acts by finite prefix exchanges on infinite sequences over a finite alphabet. These finite prefix exchanges can be restricted or augmented to form subgroups and supergroups of V. I will describe these groups and when they are isomorphic (conjugate) to V.

February 6
Jaiung Jun (Binghamton University)
Algebraic geometry in characteristic one

Abstract: Algebraic geometry in characteristic one (also known as Geometry over the ``field with one element'') is a recent area of mathematics emerging from certain heuristic ideas relating combinatorics, algebraic geometry, and number theory. Two main driving forces of the theory are to understand combinatorial objects in an algebraic geometric way and to translate the proof of Weil conjectures to the case of Riemann hypothesis. In this talk, we outline the meaning of working in characteristic one and highlight two particular perspectives of the theory (tropical geometry and algebraic geometry over hyperrings).

February 13
Ben Brewster (Binghamton University)
Sylow Intersections and the Chermak-Delgado Lattice

Abstract: In a first graduate course focussed on Group Theory, being able to count non- identity elements when distinct Sylow subgroups intersect trivially, is a memorable experi- ence. There are numerous etensions and intensive examination of these Sylow Intersections. One of these is known as Brodkey’s Theorem. It is closely tied to the existence of a regular orbit under a suitable group action.

In the seminar, I will give full definitions and exhibit some of the points mentioned above.

Recently R. McCulloch and M. Tarnauceanu have, using suggestions from I.M. Isaacs, been able to find application of these ideas to the Chermak-Delgado Lattice of a finite group.

Recall if $G$ is finite and $H \leq G$, one defines $m_G(H) = |H| |C_G (H)|$, $m^{*}(G) = \max\{m_G (H): H\leq G\}$, and $CD(G) = \{H: m_G (H) = m^{*}(G)\}$. It is known that $CD(G)$ is a modular self- dual sublattice of subgroup lattice of $G$. McCulloch and Tarnauceanu seek a classification of groups where $|CD(G)| = 1$. Ideas from above can be organized to prove a nice result.

Theorem (Isaacs, McCulloch, Tarnauceanu 2017). If $G = AB$ where $A$ and $B$ are abelian, $A$ is normal in $G$, and $(|A|, |B|) = 1$, then $CD(G) = \{AC_B(A)\}$.

February 20
Victor Protsak (Cornell University)
Invariants of representations

Abstract: One of the most fundamental questions in any field of mathematics is the Classification Problem: Determine all objects of a given kind up to some natural notion of equivalence. Next, comes the Recognition Problem: Given an object, identify its place in the classification. A general approach to these problems is via invariants, entities which are preserved under equivalences and can be explicitly or effectively computed. In Lie representation theory, we are interested in the classification of irreducible admissible representations of reductive Lie groups and of certain classes of modules over Lie algebras. Familiar invariants include the infinitesimal character, K-types, highest weight and the Weyl character. I will focus on invariants related to ideals in enveloping algebras, especially rank, the minimal polynomial and quantized elementary divisors. This approach has applications to description of certain noncommutative rings by generators and relations and to Howe duality.

February 27
Bruno Duchesne (L’Institut Élie Cartan de Lorraine )
Kaleidoscopic groups

Abstract: Inspired by Burger-Mozes construction of universal groups acting a regular tree, we construct universal groups with a prescribed local action on dendrites. Dendrites are compacta that can be thought as continuous analogs of trees. The groups we construct are Polish groups with nice properties that reflect the higher transitivity properties of a given (infinite) permutation group. No knowledge of dendrites or Polish groups is required. This is joint work with Nicolas Monod and Phillip Wesolek.

March 6
No Classes
Title of Talk

Abstract: No meeting

March 13
Hung Tong-Viet (Binghamton University)
Brauer characters and normal Sylow $p$-subgroups

Abstract: The celebrated $It\hat{o}-Michler$ theorem for Brauer characters states that if a prime $p$ does not divide the degrees of any irreducible $p$-Brauer characters of a finite group $G$, then $G$ has a normal Sylow $p$-subgroup. In this talk, I will discuss several generalizations of this theorem using various inequalities involving $p$-parts and $p'$-parts of the $p$-Brauer character degrees.

March 13 (Cross Listing from Combinatorics Seminar. Time: 1:15-2:15. WH-100E)
Victor Reiner (Minnesota)
Finite General Linear Groups and Symmetric Groups

Abstract:In recent years we have seen surprising counting phenomena related to the symmetric group, with striking analogues for general linear groups over finite fields. Often the explanations come from invariant theory. I will give examples and pose some intriguing conjectures that come from pursuing the analogy further.

March 20
Joe Cyr (Binghamton University)
Quandles and the “Abelian iff Quasi-Affine” Problem

Abstract: The “abelian iff quasi-affine” problem in universal algebra asks what is the relationship between a generalized notion of abeliannes and the representability of an algebra by modules. I will introduce the concept of abeliannes for general algebras (which at first does not all seem to resemble the definition for groups) and present a recently published proof that indeed a quandle is abelian if and only it is quasi-affine. No prior knowledge about quandles or general algebras will be assumed.

March 27
Matt Evans (Binghamton University)
Bounded commutative BCK-algebras do not form a discriminator variety

Abstract: In this talk I will explain what it means for an algebra (in the sense of universal algebra) to be a discriminator algebra, and for a variety of algebras to be a discriminator variety. Discriminator varieties are well-studied and have many nice properties. For example, in a discriminator variety, the notions of simple, subdirectly irreducible, and directly indecomposable all coincide. I will show that the variety of bounded commutative BCK-algebras is not a discriminator variety.

March 27 (Cross Listing from Combinatorics Seminar. Time: 1:15-2:15. WH-100E)
Farbod Shokrieh (Cornell University)
Effective Divisor Classes on Graphs

Abstract: Graphs can be viewed as (non-Archimedean or tropical) analogues of Riemann surfaces. For example, there are well-behaved (and useful) notions of divisors, Riemann-Roch, and Abel-Jacobi theory on graphs. I introduce the notion of semibreak divisors, which provide nice representatives for effective divisor classes on graphs. I then discuss a few applications about the generic behavior of effective divisor classes, analogous to some classical results on Riemann surfaces. Proofs in this tropical setting are more subtle.

This is joint work with Andreas Gross (Imperial College) and Lilla Tóthmérész (Cornell).

April 3
Spring Break
Title of Talk

Abstract: No meeting

April 10
Eric Bucher (Michigan State University)
Introducing cluster algebras and their applications

Abstract: Cluster algebras were first invented by Fomin and Zelevinsky in 2003 to study total positivity of canonical bases. Since their inception, these mathematical objects have popped up in a large variety of seemingly unrelated areas including: Teichmuller theory, Calabi-Yao categories, integrable systems, and the study of high energy particle physics. In this talk we will lay the basic groundwork for working with cluster algebras as well as discuss a few of their applications to the above areas. This talk is intended to be introductory so no background or definitions will be assumed. The intention is to have everyone walk away having learned about this new and fascinating algebraic object.

April 17
Rachel Skipper (Binghamton University)
The congruence subgroup problem for a family of branch groups

Abstract: A group, G, acting on a regular rooted tree has the congruence subgroup property (CSP) if every subgroup of finite index contains the stabilizer of a level of the tree. When the subgroup structure of G resembles that of the full automorphism group of the tree, additional tools are available for determining if G has the CSP.

In this talk, we look at the Hanoi towers group which fails to have the CSP in a particular way. Then we will generalize this construction to a new family of groups and discuss the CSP for them.

This talk is part of Rachel Skipper's doctoral dissertation defense. The examining committee consists of Benjamin Brewster, Matthew Brin, Patrick Madden, and Marcin Mazur (chair).

April 24
Gabriel Conant (Notre Dame University)
VC-dimension, pseudofinite groups, and arithmetic regularity

Abstract: I will discuss recent work on the structure of VC-sets in groups, i.e. subsets whose family of left translates has absolutely bounded VC-dimension. In joint work with A. Pillay, we show that VC-set A in a pseudofinite group G is “generically dominated” by a certain compact group of the form G/H, where H is a canonical normal subgroup of G associated to A. Informally, this implies that almost all cosets of H are either almost contained in A or almost disjoint from A. More formally, if C is the set of cosets of H, which intersect both A and its complement in large sets with respect to the pseudofinite measure on G, then the Haar measure of C in G/H is zero.

In joint work with A. Pillay and C. Terry, we use this generic domination to prove arithmetic regularity lemmas for VC-sets in finite groups. These results are motivated by Green's arithmetic regularity lemma for abelian groups, and generalize (without effective bounds), work of Alon-Fox-Zhao and Terry-Wolf on improved arithmetic regularity for VC-sets.

May 1
David Biddle (Binghamton University)
Generating systems for finitely generated groups and their relation to special classes of characteristic subgroups of free groups

Abstract: Neumann & Neumann introduced two strengthenings of the concept of characteristic subgroup in the early 1950's. In particular, given a group $\Gamma$ and quotient group $Q$, they were interested in studying the set of all $N$ so that $\Gamma/N\cong Q$. We show how these extended definitions of characteristic subgroups give rise to sequences of groups with extremely strong universal mapping properties with $\Gamma=F_n$ a free group and $Q=\cap_{\phi}\, \ker\phi$ where the intersection is over all members of the set $Epi(F_n,G)$ for a fixed group $G$

Algebra Seminar

2021-05-20T16:41:49-04:00

The Algebra Seminar

In a normal semester, the seminar would meet on Tuesdays in room WH-100E at 2:50 p.m. There would be refreshments served at 4:00 in room WH-102. But under the current COVID-19 pandemic, in-person seminar talks do not seem safe, and no refreshments or socializing in person will be allowed in Fall 2020 and Spring 2021.

We therefore propose that all speakers at the Algebra Seminar present their talks using Zoom. Anyone wishing to give a talk in the Algebra Seminar this semester is requested to contact the organizers at least one week ahead of time, to provide a title and abstract. The Zoom meeting link is below.

Organizers: Alex Feingold and Hung Tong-Viet

To receive announcements of seminar talks by email, please join the seminar's mailing list.

You may wish to download and import the following iCalendar (.ics) files to your calendar system. Weekly: https://binghamton.zoom.us/meeting/tJEtdOmuqzwpEtGNr3pf7HnHS0ysG2xzxC99/ics?icsToken=98tyKuCtrjgqHNGSsxGCRowMAojCc-7wmGZej7d6sg22EyYESg3eBbJbIZUtCMLI

You may also use the following link to join each Algebra Seminar Zoom Meeting: https://binghamton.zoom.us/j/95030657385

Please note that we have now implemented a Passcode required to join the zoom meeting. That passcode is the number obtained by adding up all the positive integers from 1 to 100, 1+2+…+100. That should keep the zoom bombers out!

Spring 2021

February 16
Organizational Meeting

Join Zoom Meeting using this link. The Meeting ID: 950 3065 7385. Please think about giving a talk in the Algebra Seminar, or inviting an outside speaker.

February 23 at 4:15 PM in the Arithmetic Seminar
Fikreab Admasu (Binghamton University)
Bhargava's composition law for binary cubic forms

Abstract: The Delone-Fadeev correspondence shows that binary cubic forms with integer coefficients parametrize orders in cubic fields. With this result in mind, Bhargava constructs a binary cubic form 2x3x3 boxes of integers and proves that there is a natural composition law for the boxes of integers. The group resulting from this law is then shown to be isomorphic to the class group of a corresponding cubic order. This is a cubic analogue of Gauss's theory of composition for binary quadratic forms and its relation to ideal classes of quadratic orders. The talk is based on Bhargava's “Higher composition laws II: On cubic analogues of Gauss composition.”

March 2
Jonathan Doane (Binghamton University)
Affine duality, indeed

Abstract: Rings with unity and bounded lattices have two underlying monoid structures - one with 0 and another with 1. There are exactly four twice monoid structures for which $\{0,1\}$ can be equipped; two of these actually resemble Boolean rings (BRs), one is a bounded distributive lattice (BDL), and the last is an elementary Abelian 2-group with 1, called $\textbf{A}$. Since BRs and BDLs are dualizable to Boolean spaces and Priestley spaces, respectively, it is natural to ask whether or not $\textbf{A}$ is dualizable as well. The aim of this talk is to answer this question in the affirmative.

March 9
Heiko Dietrich (Monash University)
An update on group isomorphism

Abstract: In this talk I will present some results of an ongoing project with James Wilson (Colorado State University) on the group isomorphism problem (which asks whether two given groups are isomorphic). For example, we have shown that group isomorphism is “easy” for almost all group orders. While the latter result is for the Cayley table model, I will also report on some more practical results: if time permits, then I will comment on cubefree groups, C-groups, and some groups of 'small order type'. The latter is joint work with my former Honours student Darren Low and current MPhil student Eileen Pan.

March 16 at 4:15 PM in the Arithmetic Seminar
Fikreab Admasu (Binghamton University)
Bhargava's composition law for binary cubic forms, part 2

Abstract: This is a continuation of the talk from February 23 with a focus on illustrative examples and some questions.

March 23
No Seminar

March 30
Zekeriya (Yalcin) Karatas (University of Cincinnati Blue Ash College)
Structure of groups whose proper subgroups are of a certain type

Abstract: One of the attractive general problems in the area of group theory is determining the structure of groups whose proper subgroups are of a certain type. The history of these types of problems goes back to Dedekind and Baer. The first known example is the structure of the groups whose all proper subgroups normal which was given by Dedekind and Baer. The types of conditions used for the subgroups were generalized more and increased by time which made these problems attractive and challenging. In this talk, I will give the history of these type of problems including the most significant results and some possible open problems. I will also present our recent results in this area.

Slides of the talk.

April 6
Justin Lynd (University of Louisiana at Lafayette)
Punctured groups for exotic fusion systems

Abstract: The fusion system of a finite group $G$ at a prime $p$ is a category whose objects are the subgroups of a fixed Sylow $p$-subgroup $S$, and where the morphisms are the conjugation homomorphisms induced by the elements of $G$. The notion of a saturated fusion system is abstracted from this standard example, and provides a coarse representation of what is meant by the $p$-local structure of a finite group. Once the group $G$ is abstracted away, there appear many exotic fusion systems, namely fusion systems which do not arise in the above fashion. Given an exotic fusion system, one might like to ask: just how “group-like” is it? I'll present one answer to this question, look at examples such as the Benson-Solomon exotic fusion systems at the prime 2, and mention an application to the topology of classifying spaces.

Slides of the talk.

April 13
Martino Garonzi (Universidade de Brasília)
Elements pairwise generating the symmetric groups of even degree

Abstract: In this talk I will present a recent work joint with F. Fumagalli and A. Maróti where we compute the maximal size of a subset S of the symmetric group G=Sym(n) (of even degree n at least 26) with the property that any two elements of S generate G. In particular, we show that it equals the covering number of G. Arxiv link

Slides of the talk. Recording

April 20
No Seminar

April 27
Mandi A. Schaeffer Fry (Metropolitan State University of Denver)
The McKay–Navarro Conjecture: The Conjecture That Keeps on Giving!

Abstract: The McKay conjecture is one of the main open conjectures in the realm of the local-global philosophy in character theory. It posits a bijection between the set of irreducible characters of a group with p’-degree and the corresponding set for the normalizer of a Sylow p-subgroup. In this talk, I’ll give an overview of a refinement of the McKay conjecture due to Gabriel Navarro, which brings the action of Galois automorphisms into the picture. A lot of recent work has been done on this conjecture, but possibly even more interesting is the amount of information it yields about the character table of a finite group. I’ll discuss some recent results on the McKay—Navarro conjecture, as well as some of the implications the conjecture has had for other interesting character-theoretic problems.

Slides of the talk. Recording

May 4
Bettina Eick (Technische Universitat Braunschweig)
Groups and their integral group rings (Joint work with Tobias Moede)

Abstract: The integral group ring ${\mathbf Z}G$ of a group $G$ plays an important role in the theory of integral representations. This talk gives a brief introduction to this topic and then shows how such group rings can be investigated using computational tools. In particular, the quotients $I^n(G)/I^{n+1}(G)$, where $I^n(G)$ is the $n$-th power ideal of the augmentation ideal $I(G)$, are an interesting invariant of the group ring ${\mathbf Z}G$ and we show how to determine them for given $n$ and given finitely presented $G$. We then exhibit a variety of example applications for finite and infinite groups $G$.

May 11
Lucas Gagnon (University of Colorado Boulder)
Gluing Supercharacter Theories and pre-GGG Characters

Abstract: Understanding the representation theory of the maximal unipotent subgroups in a given type of finite classical group is essentially impossible. Since these subgroups control a significant amount of the representation theory of said classical group, it is desirable to approximate these representation theories in some way. Supercharacter theory is a framework for doing this, but the ``how to'' of building a supercharacter theory to any meaningful specification remains somewhat mysterious. In this talk I will show how to construct a supercharacter theory from a lattice of normal subgroups and certain data on subintervals of this lattice, much like the gluing lemma in topology. By varying the lattice and interval data, it is possible to customize the resulting supercharacter theory as needed. To complete the story, I will show how the right choice of “gluing supercharacter theory” offers a new perspective on the Generalized Gelfand-Graev (GGG) representations of a finite classical group.

May 18
Hatice Mutlu-Akaturk (‎University of California, Santa Cruz)
Monomial posets and their Lefschetz invariants

Abstract: The Euler-Poincare characteristic of a given poset $X$ is defined as the alternating sum of the orders of the sets of chains $\mathrm{Sd}_n(X)$ with cardinality $n+1$ over the natural numbers $n$. Given a finite gorup $G$, Thevenaz extended this definition to $G$-posets and defined the Lefschetz invariant of a $G$-poset $X$ as the alternating sum of the $G$-sets of chains $\mathrm{Sd}_n(X)$ with cardinality $n+1$ over the natural numbers $n$ which is an element of Burnside ring $B(G)$. Let $A$ be an abelian group. We will introduce the notions of $A$-monomial $G$-posets and $A$-monomial $G$-sets, and state some of their categorical properties. The category of $A$-monomial $G$-sets gives a new description of the $A$-monomial Burnside ring $B_A(G)$. We will also introduce Lefschetz invariants of $A$-monomial $G$-posets, which are elements of $B_A(G)$. An application of the Lefschetz invariants of $A$-monomial $G$-posets is the $A$-monomial tensor induction. Another application is a work in progress that aims to give a reformulation of the canonical induction formula for ordinary characters via $A$-monomial $G$-posets and their Lefschetz invariants. For this reformulation we will introduce $A$-monomial $G$-simplicial complexes and utilize the smooth $G$-manifolds and complex $G$-equivariant line bundles on them.

Algebra Seminar

2020-06-01T11:55:36-04:00

The Algebra Seminar

Unless stated otherwise, the seminar meets Tuesdays in room WH-100E at 2:50 p.m. There will be refreshments served at 4:00 in room WH-102.

Organizers: Alex Feingold and Hung Tong-Viet

To receive announcements of seminar talks by email, please join the seminar's mailing list.

Spring 2020

January 21
No Algebra Seminar Meeting

Please think about giving a talk in the Algebra Seminar, or inviting an outside speaker.

January 28
Organizational Meeting

February 4
Casey Donoven (Binghamton University)
Thompson's Group V is 3/2-Generated

Abstract: Every finite simple group can be generated by two elements and furthermore, every nontrivial element is contained in a generating pair. Groups with this property are said to be 3/2-generated. Thompson’s group V, a finitely presented infinite simple group, is one of a small number of examples of infinite noncyclic 3/2-generated groups. I will present a constructive proof of this fact and mention extensions of this theorem to generalizations of V.

February 11
Cancelled

February 18
Eran Crockett (Binghamton University)
Universal algebra and constraint satisfaction problems

Abstract: Constraint satisfaction problems (CSPs) form a class of combinatorial decision problems generalizing graph colorability and Boolean satisfiability. In this expository talk, I will explain how ideas from universal algebra have been instrumental in classifying the computational complexity of CSPs.

February 25
Fikreab Solomon Admasu (Binghamton University)
Subgroups of the integer lattice $\mathbb{Z}^d$ and the higher rank discrete Heisenberg groups

Abstract: A sublattice $L$ of the integer lattice $\mathbb{Z}^d$ is called co-cyclic when the quotient $\mathbb{Z}^d/L$ is a cyclic group. Approximately $85\%$ of sublattices of finite index in $\mathbb{Z}^d$ are co-cyclic. This can be proven by either counting solutions to linear congruence equations or using zeta function methods. We show a similar result holds for subgroups of the discrete Heisenberg groups $H_{2d+1}.$

March 3
Matt Evans (Binghamton University)
Some recent results for spectra of commutative BCK-algebras

Abstract: BCK-algebras are the algebraic semantics of a non-classical logic. Like for commutative rings, there is a notion of a prime ideal in these algebras, and the set of prime ideals is a topological space called the spectrum. By work of Stone (and later, Priestley), there is a close connection between these spectra and distributive lattices with 0.

In this talk I will discuss some recent results on the interplay between commutative BCK-algebras, their spectra, and distributive lattices.

March 10
Aparna Upadhyay (University at Buffalo)
The Benson-Symonds Invariant

Abstract: Let $M$ be a finite dimensional $kG$-module for a finite group $G$ over a field $k$ of characteristic $p$. In a recent paper Dave Benson and Peter Symonds defined a new invariant $\gamma_G(M).$ This invariant measures the non-projective proportion of the module $M$. In this talk, we will see some interesting properties of this invariant. We will then determine this invariant for permutation modules of the symmetric group corresponding to two-part partitions and present a combinatorial formula for the same using tools from representation theory and combinatorics.

March 17
Cancelled

March 24
Cancelled

March 31
Cancelled

April 7
Spring vacation

April 14
Cancelled

April 21
Cancelled

April 28
Cancelled

May 5
Cancelled

Algebra Seminar

2022-05-18T10:51:36-04:00

The Algebra Seminar

The seminar will meet in-person on Tuesdays in room WH-100E at 2:50 p.m. There should be refreshments served at 4:00 in room WH-102. As of Saturday, March 26, masks are optional.

If needed, the following link would be used for a zoom meeting (Meeting ID: 981 8719 2351) of the Algebra Seminar:

Algebra Seminar Zoom Meeting Link

Organizers: Alex Feingold and Hung Tong-Viet

To receive announcements of seminar talks by email, please join the seminar's mailing list.

Spring 2022

January 25
Organizational Meeting

Please think about giving a talk in the Algebra Seminar, or inviting an outside speaker.

February 1
Alex Feingold (Binghamton University)
Groups acting on hyperbolic spaces

Abstract: The well-known action of $PSL(2,R)$ on hyperbolic 2-space by Moebius transformations $z\to \frac{az+b}{cz+d}$, gives tesselations by hyperbolic triangles when the action is restricted to an arithmetic group such as $PSL(2,Z)$. This is the math behind the famous Escher picture ``Circle Limit IV” when hyperbolic 2-space is viewed as the Poincaré disk. Hyperbolic 3-space can be viewed as the interior of a unit ball, or an upper half-space in the real quaternions, and the group of all isometries is $PSL(2,C)$. I will discuss this material, and whether it might lead to an understanding of how an infinite dimensional hyperbolic Kac-Moody group might act on a hyperbolic $\infty$-space.

February 8
Daniel Studenmund (Binghamton University)
The profinite topology and related constructions

Abstract: This talk will give an intro to, and examples of, the profinite topology on an abstract group. After reviewing the example of the p-adic topologies on the integers and rationals, we will define the profinite topology and give Furstenberg's proof of the infinitude of primes. We will use this to define the abstract commensurator of a group and provide some examples. Examples will connect with the theory of profinite groups, Galois groups, and non-Archimedean local fields.

February 15
No speaker

February 22
Daniel Studenmund (Binghamton University)
The profinite topology and related constructions, part 2

Abstract: We will briefly review the definition of the profinite topology on a group and the definition of the group of germs of automorphisms, and give some examples. We then explain how this topological language naturally appears in statements of 1) Neukirch and Uchida about rigidity in Galois groups of number fields, and 2) Bass–Milnor–Serre, and others, about the congruence subgroup property of certain arithmetic groups.

March 1
Cisil Karaguzel (University of California, Santa Cruz)
Stable perfect isometries of blocks of finite groups

Abstract: Let $(\mathbb{K},\mathcal{O},\mathbb{F})$ be a $p$-modular system which is large enough for finite groups $G$ and $H$. Let $ A$ be a $p$-block of the group algebra $\mathcal{O}G$, and $B$ be a $p$-block of the group algebra $\mathcal{O}H$. In 1990, Michel Broué introduced the definition of a perfect isometry between the $p$-blocks $A$ and $B$ which is a generalized $\mathbb{K}$-valued character leading to a special bijection between the sets of ordinary irreducible characters of $A$ and $B$. In this talk, we introduce and investigate the notion of stable perfect isometries–a way to consider perfect isometries up to generalized projective characters of the corresponding $p$-blocks. Our interest lies in understanding in which cases a stable perfect isometry can be lifted to a perfect isometry. We will answer this question for the $p$-block $\mathcal{O}P$ where $P$ is an abelian $p$-group. (Joint work with Robert Boltje).

March 8
Hung Tong-Viet (Binghamton University)
Odd degree rational irreducible characters

Abstract: A complex irreducible character $\chi$ of a finite group $G$ is said to be rational (or real) if $\chi$ takes only rational (or real) values, that is, $\chi(g)\in\mathbb{Q}$ (or $\chi(g)\in\mathbb{R}$) for all $g\in G.$ An element $g\in G$ is said to be rational if $\chi(g)\in\mathbb{Q}$ for all irreducible characters $\chi$ of $G$. Similarly, an element $g\in G$ is real if $\chi(g)\in \mathbb{R}$ for all irreducible characters $\chi$ of $G$ or equivalently $g$ and $g^{-1}$ are conjugate in $G$. The Itô-Michler theorem for the prime $p=2$ states that if all irreducible characters of $G$ have odd degree, then $G$ has a normal abelian Sylow $2$-subgroup $P$. A real version of this theorem was obtained by Dolfi, Navarro and Tiep in $2008$. It was shown that if all irreducible real characters of $G$ have odd degree, then $G$ has a normal Sylow $2$-subgroup. There is no rational version of this result since all the irreducible rational characters of the non-abelian simple group $\textrm{PSL}_2(3^{f})$, where $f\ge 3$ is an odd integer, have odd degree. In this talk, we discuss a new variant of the above mentioned result for rational characters. This is a joint work with P. H. Tiep.

March 15
No Seminar
Spring Break

March 22
Sailun Zhan (Binghamton University)
McKay graphs of the finite subgroups of $SL(2,\mathbb{C})$

Abstract: We will give an introduction to the McKay graph and the Mckay correspondence. Namely, the one-to-one correspondence between the Mckay graphs of the finite subgroups G of $SL(2,\mathbb{C})$ and the affine simply laced Dynkin diagrams. We will also discuss the relations between these diagrams and the resolution of singularities of $\mathbb{C}^2$ quotient by G.

March 29
Zach Costanzo (Binghamton University)
Codegrees of the real-valued characters of a finite group

Abstract: The influence of the set of character degrees of a finite group $G$ has been well studied. Over the past twenty years, it has also become clear that we can restrict our view to characters which take their values in a particular subfield of the complex numbers while still gaining some knowledge of the group’s structure. During that same time, the so-called codegrees of the characters have been studied. In this talk, I will define codegree, and give some insight into the impact the codegrees of the real-valued characters have on the structure of finite groups.

April 5
Yong Yang (Texas State University)
Regular orbits of finite primitive solvable groups

Abstract: The case when a linear group $G$ acting primitively on the vector space $V$ is of central importance in the theory of representations of solvable groups. In short, such groups have an invariant $e$ that measures their complexity. It is known that if $e > 118$, $G$ has a regular orbit.

I was able to improve this result dramatically by classifying all the cases when the regular orbit exists. In some of my early papers, I gave a coarse classification of the existence of regular orbits for primitive solvable linear groups, and the results have been widely used by other people and myself to study related problems of arithmetic properties of group invariants. A more detailed final classification has been completed in some of my recent work along with several further applications.

April 12
Mark L. Lewis (Kent State University)
Groups having all elements off a normal subgroup with prime power order

Abstract: We consider a finite group $G$ with a normal subgroup $N$ so that all elements of $G \setminus N$ have prime power order. We prove that if there is a prime $p$ so that all the elements in $G \setminus N$ have $p$-power order, then either $G$ is a $p$-group or $G = PN$ where $P$ is a Sylow $p$-subgroup and $(G,P,P \cap N)$ is a Frobenius-Wielandt triple. We also prove that if all the elements of $G \setminus N$ have prime power orders and the orders are divisible by two primes $p$ and $q$, then $G$ is a $\{ p, q \}$-group and $G/N$ is either a Frobenius group or a $2$-Frobenius group. If all the elements of $G \setminus N$ have prime power orders and the orders are divisible by at least three primes, then all elements of $G$ have prime power order and $G/N$ is nonsolvable.

April 19
No Seminar (Following Monday's schedule)

April 26
Chris Schroeder (Binghamton University)
A guided tour of the representation theory of finite groups of Lie type

Abstract: The Lie algebra determines many aspects of the structure theory and representation theory of a Lie group. Remarkably, linear algebraic groups defined over algebraically closed fields of non-zero characteristic exhibit many of the same properties. In this expository talk we cover the basics of linear algebraic groups, with a focus on how geometric properties over an algebraically closed field descend to properties of finite subgroups defined over finite fields. The finite groups which arise in this way are called finite groups of Lie type and play a major role in the classification of finite simple groups. We discuss Deligne and Lusztig's classification of irreducible complex representations of finite groups of Lie type, which relies on a cohomology theory originally constructed to solve the Weil conjectures. We also provide some applications.

May 3
Jonathan Doane (Binghamton University)
An infinite Fibonacci-like tree

Abstract: You may be familiar with the Fibonacci (sequence) rabbit problem. Essentially, the growth of a hypothetical rabbit population can be described by an infinite binary tree, whose levels correspond perfectly with the Fibonacci sequence. Formally, we can think of rooted paths down this tree as functions from the natural numbers to a two-element set, subject to some conditions. This talk introduces a slight twist on these conditions and illustrates how Boolean semirings (of all things) come into play!

Algebra Seminar

2019-08-20T11:18:55-04:00

The Algebra Seminar

Unless stated otherwise, the seminar meets Tuesdays in room WH-100E at 2:50 p.m. There will be refreshments served at 4:00 in room WH-102.

Organizers: Alex Feingold and Hung Tong-Viet

To receive announcements of seminar talks by email, please join the seminar's mailing list.

Spring 2019

January 22
Organizational Meeting
Title of Talk

Abstract: Please come or contact the organizers if you are interested in giving a talk this semester or want to invite someone.

January 29
Ben Brewster (Binghamton University)
The values of the Chermak-Delgado measure

Abstract: Let $G$ be a finite group. For $H\leq G$, $m_G(H) = |H|\ |C_G(H)|$. Let $m^*(G) = max\{m_G(H)\mid H\leq G\}$ and $CD(G) = \{H\leq G\mid m_G(H)=m^*(G)\}$. Then $CD(G)$ is a self-dual modular sublattice of the subgroup lattice of $G$.

It is known that if $|G| > 1$, then not every subgroup of $G$ is a member of $CD(G)$, that is, $|\{m_G(H)\mid H\leq G\}| > 1$. Following some ideas of M. Tarnauceanu, we examine possibilities for $|\{m_G(H)\mid H\leq G\}|$, its form and the distribution of subgroups of same measure.

February 5
Alex Feingold (Binghamton University)
An introduction to Lie algebras

Abstract: A Lie algebra is a vector space equipped with a bilinear product, denoted by $[\cdot,\cdot]$, such that $[x,x]=0$ and $[x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0$ (Jacobi Identity). I will give an introduction to the basic ideas and examples.

February 12
Canceled due to inclement weather

February 19
Daniel Rossi (Binghamton University)
The structure of finite groups with exactly three rational-valued irreducible characters

Abstract: Many results in the character theory of finite groups are motivated from the question: to what extent do the irreducible characters of a group $G$ control the structure of $G$ itself? Recently, it has been observed that certain results along these lines can be obtained when one looks not at the set of all irreducible characters of $G$, but only the subset of those characters taking values in some appropriate field. In this talk, I'll characterize the structure of finite groups which have exactly three rational-valued irreducible characters (for solvable groups, this characterization is due to J. Tent). I will attempt to give some of the flavor of the proof – which at one point includes a surprise cameo by the complex Lie algebra $sl(n)$.

February 26
Casey Donoven (Binghamton University)
Thompson's Group $V$ and Finite Permutation Groups

Abstract: Thompson's group $V$ is group of homeomorphisms of Cantor space. It acts by exchanging finite prefixes in infinite strings over a two-letter alphabet. Generalizations of $V$ called $V_n$ act on n-letter alphabets. I will present more generalizations that add the action of finite permutation groups to the finite prefix exchanges. For a finite permutation group $G$ on $n$ points, the group $V_n(G)$ marries the finite prefix exchanges with iterated permutations from $G$. The primary theorem I will present states that $V_n$ is isomorphic to $V_n(G)$ if and only if $G$ is semiregular (i.e. $G$ acts freely). The proof involves the use of automata and orbit dynamics.

March 5
Matt Evans (Binghamton University)
Spectra of cBCK-algebras

Abstract: BCK-algebras are algebraic structures that come from a non-classical logic. Mimicking a well-known construction for commutative rings, we can put a topology on the set of prime ideals of a commutative BCK-algebra; the resulting space is called the spectrum. I will discuss some results/properties of the spectrum of such algebras. A particularly interesting spectrum occurs when the underlying algebra is a so-called BCK-union of a specific algebra. In this case, the spectrum is a spectral space, meaning it is homeomorphic to the spectrum of a commutative ring.

March 12
Hung Tong-Viet (Binghamton University)
Real conjugacy class sizes and orders of real elements

Abstract: In this talk, I will present some recent results concerning the structure of finite groups with restriction on the real conjugacy classes or on the orders of real elements.

March 19
Spring Break
No Talk

Abstract: Text of Abstract

March 26
No Talk
Title of Talk

Abstract: Text of Abstract

April 2
John Brown (Binghamton University)
A small step toward proving a character theory conjecture

Abstract: In this talk we'll discuss a bit of the work done on a conjecture by Isaacs and others which states that the degree of any primitive character of a finite group G divides the size of some conjugacy class of G. We'll focus on the case that G is symmetric or alternating, with a view to showing that the result holds for every irreducible character of either group. If time permits we may discuss ideas for the next steps toward, as well as some of the obstructions to, a general result.

April 9
Jonathan Doane (Binghamton University)
Restriction of Stone Duality to Generalized Cantor Spaces

Abstract: Stone duality is a correspondence between Boolean algebras (BAs) and Boolean/Stone topological spaces. Dualizing the free BA $\textbf{F}(S)$ on set $S$ yields a product space $2^S$, where $2=\{0,1\}$ is discrete. We call $2^S$ a generalized binary Cantor space (GCS$_2$), and similarly define the spaces GCS$_n$ with $n\ge 2$. This talk introduces Stone duality and then answers the question ``what is dual to the class of GCS's?''

April 16
Casey Donoven (Binghamton University)
Inverse Semigroups

Abstract: The inverse of an element $a$ of a semigroup $S$ is an element $b$ such that $aba=a$ and $bab=b$. We define an inverse semigroup to be a semigroup where each element has a unique inverse. I will discuss some introductory inverse semigroup theory, such as equivalent definitions, showing that the idempotents form a semilattice, and the Wagner-Preston Representation Theorem (analogous to Cayley's Theorem). Time permitting, I will present a theorem describing the minimum number of proper inverse subsemigroups needed to cover a finite inverse semigroup.

April 23
Joseph Cyr (Binghamton University)
The Structure of Medial Quandles

Abstract: A medial quandle is a left semigroup in which every polynomial is a multivariable endomorphism. In this talk I will explore a useful structure theorem which shows that any medial quandle can be written as a collection of smaller, easier to understand quandles tied together in what is called an “affine mesh.” This mesh provides a user-friendly way to describe the subdirectly irreducible algebras of the variety.

April 30
Dikran Karagueuzian (Binghamton University)
Coalescence of Polynomials over Finite Fields

Abstract: A polynomial over a finite field may be compared to a random map from the finite field to itself. One way to match random maps to polynomials is to match certain invariants of the maps. One of these invariants is the coalescence, or variance of inverse image sizes. We generalize a coalescence result of Martins and Panario from the case where a Galois group is the symmetric group to an arbitrary Galois group.

This is joint work with Per Kurlberg.

May 7
Joshua Carey (Binghamton University)
Representation Theory of Affine Kac-Moody Lie Algebras (Candidacy Exam, Part 1)

Abstract: Affine Kac-Moody Algebras are infinite dimensional Lie Algebras that have significance in many areas of math as well as theoretical physics. Although they nicely generalize many properties of finite dimensional simple Lie Algebras, it is not so easy to find faithful representations. In this talk I will give some basic definitions and properties of Affine Kac-Moody Algebras and begin to discuss a nice representation using vertex operators.

Algebra Seminar

2020-12-21T15:21:52-04:00

The Algebra Seminar

Organizers: Alex Feingold and Hung Tong-Viet

To receive announcements of seminar talks by email, please join the seminar's mailing list.

Zoom link: https://binghamton.zoom.us/j/95069915454

Fall 2020

September 1
Organizational Meeting

Join Zoom Meeting using this link. The Meeting ID: 821 340 7154. Please think about giving a talk in the Algebra Seminar, or inviting an outside speaker.

September 8
Fikreab Solomon Admasu (Binghamton University)
Composition laws from Gauss to Bhargava

Abstract: One of the fundamental objects of interest in number theory is the composition law of binary quadratic forms. In this talk, we will start with a review of the original complicated formulation due to Gauss and then discuss subsequent simplifications by Dirichlet, et al. Another approach involves identifying equivalence classes of binary quadratic forms with ideal classes in a quadratic ring and using the natural group structure of the ideal class group to formulate the composition law of binary quadratic forms. After about 200 years, M. Bhargava gave a new perspective on Gauss' composition law using so-called 2x2x2 cubes of integers and derived more composition laws. We will discuss the main theorems in 'Higher Composition Laws I' by Bhargava and mention the applications if time permits.

September 15
Fikreab Solomon Admasu (Binghamton University)
Composition laws from Gauss to Bhargava

Abstract: Continuation of the talk from September 8, with focus on the newer Bhargava approach.

September 22
Nguyen Ngoc Hung (University of Akron)
Fields of values of group characters

Abstract : Representation theory and character theory have been developed as tools to study structure of finite groups. I will present some results, both old and new, on fields of character values. In particular, I will present an extension of results of Burnside and Navaro-Tiep on the existence of real/rational irreducible characters in even-order groups.

September 30, 7:30 PM
Burkard Polster (Monash University, Melbourne, Australia)
Mathologer on YouTube

Abstract: Burkard Polster (aka the Mathologer) will talk about his mathematical YouTube channel and discuss interesting questions submitted to him by the audience. This zoom talk will be live from Melbourne, Australia, so in order to accommodate the large time zone difference between Australia and New York, it will take place on Wednesday evening at 7:30 PM (Binghamton time zone).

October 6
Per Kurlberg (KTH Stockholm)
Class numbers and class groups for definite binary quadratic forms

Abstract: Gauss made the remarkable discovery that the set of integral binary quadratic forms of fixed discriminant carries a composition law, i.e., two forms can be “glued together” into a third form. Moreover, as two quadratic forms related to each other via an integral linear change of variables can be viewed as equivalent, it is natural to consider equivalence classes of quadratic forms. Amazingly, Gauss' composition law makes these equivalence classes into a finite abelian group - in a sense it is the first abstract group “found in nature”.

Extensive calculations led Gauss and others to conjecture that the number h(d) of equivalence classes of such forms of negative discriminant d tends to infinity with |d|, and that the class number is h(d) = 1 in exactly 13 cases: d is in { -3, -4, -7, -8, -11, -12, -16, -19, -27, -28, -43, -67, -163 }. While this was known assuming the Generalized Rieman Hypothesis, it was only in the 1960's that the problem was solved by Alan Baker and by Harold Stark.

We will outline the resolution of Gauss' class number one problem and survey some known results regarding the growth of h(d). We will also consider some recent conjectures regarding how often a fixed abelian group occur as a class group, and how often an integer occurs as a class number. In particular: do all abelian groups occur, or are there “missing” class groups?

October 13
Mark Lewis (Kent State University)
Finite, solvable, tidy groups (or how I spent my summer vacation.)

Abstract: Let G be a group and let x be an element of G. We define ${\rm Cyc}_G (x) = \{ g \in G \mid \langle g, x \rangle {\rm ~is~ cyclic} \}$. It is easy to see that ${\rm Cyc}_G (x)$ is not necessarily a subgroup of $G$. We say that a group $G$ is tidy if ${\rm Cyc}_G (x)$ is a subgroup for all $x \in G$. We will find a classification of finite, tidy $p$-groups and finite, tidy $\{ p, q \}$-groups for primes $p$ and $q$. We will see how these can be used to characterize finite, solvable, tidy groups. This includes work that was done during the 2020 Summer REU at Kent State University. Students involved were Nicholas F. Beike, Colin Heath, Kaiwen Lu, and Jamie D. Pearce. The graduate assistants were Rachel Carleton and David Costanzo.

October 20
No seminar

October 27
Hung Tong-Viet (Binghamton University)
Proportions of vanishing elements in finite group

Abstract: An element $g$ of a finite group $G$ is called a \emph{vanishing element} of $G$ if there exists an irreducible complex character $\chi$ of $G$ such that $\chi(g)=0$. In this case, the element $g$ is also called a zero of character. Zeros of characters play an important role in block theory as well as in representation theory.

In this talk, I will discuss the influence of vanishing elements on the structure of finite groups and will focus on bounding the proportion of vanishing elements. In particular, I will show that the proportion of vanishing elements of every finite non-abelian group is bounded below by $1/2$ and classify all finite groups whose proportions of vanishing elements attain this bound. Moreover, if this proportion is less than or equal to $2/3$, then the group must be solvable. (This is joint work with Lucia Morotti).

November 3
Samantha Wyler (Kent State University)
Special, Extra Special, Semi Extra Special, and Ultra Special Groups

Abstract: A non-abelian finite p-group is special if the commutator subgroup, the center, and the Frattini subgroup are equal. In this talk, we will discuss some properties of Special groups, Extra Special groups, and Semi Extra Special groups. Semi Extra Special Groups can be thought of as a generalization of Extra Special groups. Additionally, every Semi Extra Special group is a Special group, and Semi Extra Special groups are also a generalization of Ultra Special groups. This talk will largely be based on Professor Mark Lewis’s paper “Semi-extra special Groups.”

November 10
Zach Costanzo (Binghamton University)
Groups whose real-valued character degrees are all prime powers

Abstract: Let $G$ be a finite group. Dolfi, Pacifici, and Sanus show that if all of the real-valued irreducible characters of $G$ have prime degree, then $G$ is solvable and the real-valued characters are contained in a subset of $\{1, 2, p\}$ for some odd prime $p$. Here, we attempt to generalize some of these results by assuming instead that all of the real-valued irreducible characters of $G$ have prime power degrees. We classify such groups in the case that $G$ is non-solvable, and show a similar result about set of real-valued character degrees in the case that $G$ is solvable.

November 17
Jonathan Doane (Binghamton University)
Affine clone of Boole

Abstract: A Boolean operation is a function $f:\{0,1\}^n\to \{0,1\}$ ($n\in \mathbb{N}$), and it is called affine if it can be expressed as $f(x_1,\ldots, x_n) = c_0+c_1x_1+\cdots+c_nx_n$ ($\{c_i\}_{i=0}^n \subseteq \{0,1\}$) where addition is taken mod 2. In this talk, we will show how the set of all affine Boolean operations arises naturally (as the ``clone'' of an algebra) when explore a generalization of ring (with unity) theory and bounded lattice theory.

November 24
No seminar

December 1
Dikran Karagueuzian (Binghamton University)
Moments of Polynomials over Finite Fields

Abstract: A polynomial over a finite field can be regarded as a map of the finite field to itself. The variance of the inverse image sizes has been studied in connection with the question of whether such maps are a good substitute for random maps. We are able to show that polynomial maps are not random in the sense that all moments, not just the variance, must be integers, in an asymptotic sense as the size of the finite field becomes large. This is based on joint work with Per Kurlberg.

December 8
No seminar

Fall 2019

2020-01-08T15:12:29-04:00

The Algebra Seminar

Unless stated otherwise, the seminar meets Tuesdays in room WH-100E at 2:50 p.m. There will be refreshments served at 4:00 in room WH-102.

Organizers: Alex Feingold and Hung Tong-Viet

To receive announcements of seminar talks by email, please join the seminar's mailing list.

Fall 2019

August 27
Organizational meeting

September 3
Casey Donoven (Binghamton University)
Automata acting on Fractal Spaces

Abstract: A self-similar set is a set that is a union of scaled copies of itself. Through iterated labeling of the $n$ copies, $n^2$ subcopies, and so on, we create a correspondence between infinite sequences over an n letter alphabet and points in the self-similar set. Automata act naturally on infinite sequence, and I will explore groups of homeomorphisms of semi-similar sets induced by automata. I will focus on two examples, the unit interval and Julia set associated to the map $z^2+i$. An important tool in the construction of the automata is the approximation of these self-similar sets as finite graphs.

September 10
Matt Evans (Binghamton University)
BCK-algebras and generalized spectral spaces

Abstract: Commutative BCK-algebras are the algebraic semantics of a non-classical logic. Mimicing the construction of the spectrum of a commutative ring (or Boolean algebra or distributive lattice), we can construct the spectrum of a commutative BCK-algebra.

A topological space is called *spectral* if it is homeomorphic to the spectrum of some commutative ring, and *generalized spectral* if it is homeomorphic to the spectrum of a distributive lattice with 0.

In this talk I will briefly discuss Hochster's characterization of spectral spaces, and then show that the spectrum of a commutative BCK-algebra is generalized spectral.

September 17
Jonathan Doane (Binghamton University)
Dualizing Kleene Algebras

Abstract: It is well-known that the class of Boolean algebras is “generated” by the two element chain $F<T$ equipped with negation $\neg F:= T$, $\neg T:=F$. When we include an uncertainty element $F<U<T$, along with negation $\neg U: =U$, we generate the class of Kleene algebras. Of course, there is a famous correspondence between Boolean algebras and Boolean topological spaces, named Stone duality; this leads us to wonder if we can somehow represent Kleene algebras by topological spaces as well. In fact, Stone duality is but an application of a more general theory of dual equivalences between categories. In this talk, we will utilize this theory to construct a dual equivalence between the categories of Kleene algebras and certain topological spaces.

September 24
Cancelled

October 1
No Classes (University)
Title

Abstract: Abstract

October 8
Ben Brewster (Binghamton University)
The collection of intersections of Sylow p-subgroups of G

Abstract: Suppose $G$ is a finite group, $p$ is a prime integer which divides $|G|$. Brodkey‘s Theorem appeared in 1963. It says that if a Sylow $p$-subgroup of $G$ is abelian, then the intersection of all Sylow $p$-subgroups is the intersection of a pair of them.

I began to wonder what the nature of the collections of intersections of all subsets of Sylow $p$-subgroups looked like. Clearly this is a meet semilattice under set inclusion. There are examples by Ito to show the minimal elements need not be intersections of two Sylow $p$-subgroups.

During a sabbatical in 1989, I went to Tübingen, Germany where Peter Hauck and I collaborated to uncover some things about this collection of all intersections of subsets of Sylow $p$-subgroups.

I will describe a few of the steps we used and hope to show some of main points on how and why some extra hypothesis is needed for some primes to obtain a resemblance of Brodkey’s Theorem.

October 15
Fikreab Admasu (Binghamton University)
Generating series for counting finite p-groups of class 2

Abstract: In 2009, C. Voll computed the numbers $g(n, 2, 2)$ of nilpotent groups of order $n$, of class at most $2$ generated by at most $2$ generators, by giving an explicit formula for the Dirichlet generating function $\sum_{n=1}^{\infty} g(n, 2, 2)n^{-s}.$ Later in $2012$, Ahmad, Magidin and Morse gave a direct enumeration of such groups building on works of M. Bacon, L. Kappe, et al. We use their enumeration to provide a natural multivariable extension of the generating function counting such groups and as a result rederive Voll’s explicit formula. Similar formulas or enumerations for finite groups of nilpotency class $2$ on more than $2$ generators or of at least class $3$ on $2$ or more generators is currently unknown.

October 22
Eran Crockett (Binghamton University)
Finitely related clones

Abstract: What is a clone? What does it mean for a clone to be finitely related? What are some examples of finitely related clones? What are some examples of non-finitely related clones? We answer these questions and more.

October 29
David Biddle (Binghamton University)
Generating tuples of direct products of finite simple groups

Abstract: For any group $G$ we can define the $n^{th}$ Eulerian function $\phi_n(G)$, to be the number of tuples in $G^n$ that generate $G$ and the rank of $G$ to be the smallest integer $d=d(G)$ so that $G$ has a generating set of size $d$. We will show that if we define the reduced Eulerian function to be $r_n(G):=\phi_n(G)/|\text{Aut}(G)|$, that for $S$ finite simple and $n \geq 2$, $\text{rank}(S^{r_n(S)})=n$ precisely. This has been famously used to show for instance that $\text{rank}((A_5)^{20})=3$ while $\text{rank}((A_5)^{19})=2$.

November 5
Luise Kappe (Binghamton University)
A GAP-conjecture and its solution: isomorphism classes of capable special $p$-groups of rank 2

Abstract: A group is said to be capable if it is a central quotient group and a $p$-group is special of rank 2 if its center is elementary abelian of rank 2 and equal to its commutator subgroup. In 1990, Heineken showed that if $G$ is a capable special $p$-group of rank 2, then $p^5 \leq |G| \leq p^7$. Over a decade ago we asked GAP to determine the number of isomorphism classes of capable special $p$-groups of rank 2 for small primes $p$. GAP told us that in these cases, the number of isomorphism classes of special $p$-groups of rank 2 grows with $p$. However, for the capable among them the number of isomorphism classes is independent of the prime $p$. Finally, we were able to show that what GAP conjectured is true for all primes $p$.

November 12
Dikran Karagueuzian (Binghamton University)
Coalescence of Polynomials over Finite Fields

Abstract: A polynomial over a finite field can be regarded as a map of the finite field to itself. The coalescence, or variance of the inverse image sizes, has been studied in connection with whether such maps are a good substitute for random maps. We are able to show that polynomial maps are not random in the sense that the coalescence must be an integer, in an asymptotic sense as the size of the finite field becomes large. This is based on joint work with Per Kurlberg.

November 19
Cancelled

November 26
Alex Feingold (Binghamton University)
Monstrous Moonshine

Abstract: This will be an introduction to the main ideas involved in the Conway-Norton monstrous moonshine, connections between the largest sporadic simple group, modular functions and vertex operator algebras.

December 3
No Meeting
Title

Abstract: Abstract

Algebra Seminar

2019-01-21T16:11:03-04:00

The Algebra Seminar

Unless stated otherwise, the seminar meets Tuesdays in room WH-100E at 2:50 p.m. There will be refreshments served at 4:00 in room WH-102.

Organizers: Alex Feingold and Hung Tong-Viet

To receive announcements of seminar talks by email, please join the seminar's mailing list.

Fall 2018

August 28
Organizational Meeting
Title of Talk

Abstract: Please come or contact the organizers if you are interested in giving a talk this semester or want to invite someone.

September 4
Casey Donoven (Binghamton University)
Covering Number of Semigroups

Abstract: A semigroup is a set $S$ equipped with an associative operation. The covering number of a semigroup $S$ is the minimum number of proper subsemigroups whose union is $S$. In this talk, I will introduce basic semigroup theory and some fundamental examples while proving the following theorem: If $S$ is a finite semigroup that is not a group nor generated by a single element, then the covering number of $S$ is 2. Similar questions have been studied for groups and loops. (Joint work with Luise-Charlotte Kappe and Marcin Mazur.)

September 11
No Classes (University)
Title

Abstract: Abstract text

September 18
Joe Cyr (Binghamton University)
Semilattice Modes

Abstract: A mode A is an algebra which is idempotent and in which every operation is a homomorphism from the appropriate power of A to A. We will explore some results on a particular class of modes which are constructed from semilattices. In particular, we will look at the question of when is a semilattice mode subdirectly irreducible, both in general and in the particular case of when the mode has a single binary operation.

September 25
Matt Evans (Binghamton University)
Spectral properties of involutory BCK-algebras

Abstract: BCK-algebras are algebraic structures arising from non-classical logic. This talk will focus primarily on the classes of commutative BCK-algebras and commutative involutory BCK-algebras. Particularly, I will discuss some basic ideal theory and spectral properties of such algebras, looking at differences between the bounded and unbounded cases.

October 2
Mark Lewis (Kent State University)
Centers of centralizers and maximal abelian subgroups

Abstract: In this talk we consider the centers of the centralizers of elements in finite groups. We will then obtain a lower bound on the order of a maximal abelian subgroup in terms of the indices of the centralizers of elements and the orders of the centers of the centralizers of elements. We will use this to obtain a lower bound for maximal abelian subgroups of semi-extraspecial groups.

October 9
Fernando Guzman (Binghamton University)
Polynomials Automorphisms of the Regular d-ary Tree

Abstract: We explore the question of when does a polynomial with integer coefficients induce an automorphism of the infinite regular d-ary tree. This is well-known for d=2, and there are some partial results for d prime. We extend the results to the case when d is square-free.

October 16
Luise-Charlotte Kappe (Binghamton University)
A generalization of the Chermak-Delgado lattice to words in two variables

Abstract: The Chermak-Delgado measure of a subgroup $H$ of a finite group $G$ is defined as the product of the order of $H$ with the order of the centralizer of $H$ in $G$, $|H||C_G(H)|$, and the set of all subgroups with maximal Chermak-Delgado measure forms a dual sublattice of the subgroup lattice of $G$. In this talk we step back from centralizers and consider four types of centralizer-like subgroups, defined using general words in the alphabet $\{x, y, x^{-1}, y^{-1} \}$ instead of the specific commutator word. We show that this generalization results in four sublattices of the subgroup lattice of a finite group, some of which may be equal to one another depending on the word. We consider which properties of the Chermak-Delgado lattice generalize to the new lattices, and which properties are specialized in the Chermak-Delgado lattice. (This work is joint with Elizabeth Wilcox.)

October 23
Eran Crockett (Ripon College)
The variety generated by the triangle

Abstract: This talk will consist of a (quick) introduction to universal algebra where we focus on three topics: the definability of principal congruences, classifying subdirectly irreducibles, and determining the clone of term operations. We will attempt to understand these topics by focusing on two examples: the two-element semilattice and the three-element non-transitive tournament (a.k.a. the triangle).

October 30
Dan Rossi (Binghamton University)
Brauer characters and fields of values

Abstract: The complex irreducible characters $\text{Irr}(G)$ of a finite group $G$ contain a lot of information about $G$ itself. Recently, it has been realized that some of this information can still be captured if, instead of considering the entire set $\text{Irr}(G)$, one only considers those irreducible characters taking values in some suitable subfield of $\mathbb{C}$. This motivates the following definitions: $\text{Irr}_\mathbb{F}(G)$ is the subset of irreducible characters taking values in the subfield $\mathbb{F}\subseteq \mathbb{C}$; and $\text{Cl}_\mathbb{F}(G)$ is the set of conjugacy classes of $G$ whose elements, when evaluated at every character of $G$, take values in $\mathbb{F}$. A result of Isaacs $\&$ Navarro says that $|\text{Irr}_\mathbb{F}(G)| = |\text{Irr}_\mathbb{F}(G/N)|$ and $|\text{Cl}_\mathbb{F}(G)| = |\text{Cl}_\mathbb{F}(G/N)|$ whenever $N\unlhd G$ contains no non-trivial $\mathbb{F}$-elements.

For a prime $p$, the $p$-Brauer characters of $G$ arise from representations of $G$ over $\overline{\mathbb{F}}_p$. They provide a link between the representation theory of $G$ in characteristic $0$ and in characteristic $p$. Whenever one has a relationship involving characters and conjugacy classes $G$, it is natural to wonder if there is an analogous relationship between the $p$-Brauer characters and $p$-regular conjugacy classes.

I will give some examples of the sorts of $\mathbb{F}$-generalizations alluded to in the first paragraph; introduce the basic notions of Brauer characters; and discuss a Brauer analogue of the Isaacs-Navarro result mentioned above.

November 6
Casey Donoven (Binghamton University)
Covering Number of Semigroups (cont.)

Abstract: I will continue my exploration of covering numbers of semigroups by considering specific classes of semigroups. A monoid is a semigroup with an identity. An inverse semigroup $I$ is a semigroup such that for each element $a\in I$ there exists a unique element $a^{-1}\in I$ such that $aa^{-1}a=a$ and $a^{-1}aa^{-1}=a$. I will give a complete description of the covering number of monoids and inverse semigroups with respect to submonoids and inverse subsemigroups respectively (modulo the covering numbers of groups and semigroups). I will use Green's relations and other results to describe the structure of such semigroups.

November 13
Casey Donoven (Binghamton University)
Fractal Subgroups of Profinite Groups

Abstract: Often, we think of fractals as subsets of $\mathbb{R}^n$. Many definitions in fractal geometry can be generalized to any metric space, including groups equipped with a metric. In particular, the Hausdorff dimension and Box counting dimension can be defined on any metric space. Profinite groups can be equipped with a natural metric, under which we can discuss fractal properties. This will be an expository talk in which I define fractal dimensions and profinite groups. My goal is to set up the following question: Given two fractal subgroups of the automorphism group of the rooted infinite n-ary tree, what is the dimension of their intersection?

November 20
Hung Tong-Viet (Binghamton University)
Conjugacy classes of $p$-elements and normal $p$-complements

Abstract: The commuting probability $d(G)$ of a finite $G$ (introduced by Erdős and Turán in 1968), is defined to be the probability that two randomly chosen elements of $G$ commute. The commuting probability $d(G)$ is also called the commutativity degree of $G$. Erdős and Turán showed that $d(G)=k(G)/|G|$, where $k(G)$ is number of conjugacy classes of $G$. In 1973, W. H. Gustafson proved that $d(G)\leq 5/8$ for any non-abelian group $G$. Since then, there are numerous results concerning the structure of finite groups using various bounds on the commuting probability. In this talk, I will consider a $p$-local version of the commuting probability. Specifically, for a prime $p$, we define $d_p(G)$ to be the ratio $k_p(G)/|P|$, where $k_p(G)$ is the number of conjugacy classes of $p$-elements of $G$ and $P$ is a Sylow $p$-subgroup of $G$. Using the invariant $d_p(G)$, we obtain some new criteria for the existence of normal $p$-complements in finite groups.

November 27
Nicholas Gardner (Binghamton University)
An Introduction to the Chermak-Delgado Lattice

Abstract: For a subgroup $H$ of a finite group $G,$ the Chermak-Delgado measure of $H$ is defined as $m_{G}(H) := |H||C_{G}(H)|$. The subgroups of $G$ with maximum Chermak-Delgado measure form a dual sublattice of the subgroup lattice of $G$. In this talk I will discuss some properties of such maximum-measure subgroups and calculate the Chermak-Delgado lattice for some classes of finite groups.

December 4
Speaker (University)
Title

Abstract: Abstract text

Fall 2017

2019-04-06T00:10:58-04:00

The Algebra Seminar

Unless stated otherwise, the seminar meets Tuesdays in room WH-100E at 2:50 p.m. There will be refreshments served at 4:00 in room WH-102.

Organizers: Alex Feingold and Marcin Mazur

To receive announcements of seminar talks by email, please join the seminar's mailing list.

Fall 2017

August 29
Organizational meeting

September 5
Dikran Karagueuzian (Binghamton University)
Categories for the Skeptical Mathematician

Abstract: Many find it hard to see the mathematical content in Category Theory. I will discuss ways of understanding this content and how I recently decided that it is relevant to a problem in Commutative Algebra which I have studied.

September 12
Hung Tong-Viet (Binghamton University)
Real class sizes

Abstract: An element $x$ in a finite group $G$ is said to be real if there exists an element $g\in G$ such that $x^g=x^{-1}$. A real conjugacy class of $G$ is a conjugacy class which contains some real element and a real class size is the size of a real conjugacy class. Several arithmetic properties of the real class sizes can conveniently be stated using graph theoretic language. The prime graph on the real class sizes of a finite group $G$, denoted by $\Delta^*(G)$, is a simple graph with vertex set $\rho^*(G)$, the set of primes dividing some real class size of $G$, and there is an edge between two vertices $p$ and $q$ if the product $pq$ divides some real class size. In this talk, I will outline the proof that if the prime graph on the real class sizes of a finite group is disconnected, then the group is solvable.

September 19
Phillip Wesolek (Binghamton University)
Elementary amenability and bounded automata groups

Abstract: (Joint work with Kate Juschenko) The bounded automata groups are a natural family of groups, which are defined by a finite amount of information in the form of a certain type of finite automaton. A remarkable property of bounded automata groups is that they are necessarily amenable. On the other hand, the elementary amenable groups are groups that are amenable for `elementary reasons' and thus are somewhat uninteresting. One is thereby motivated to identify the non-elementary amenable bounded automata groups. We isolate three natural families of bounded automata groups, and in each family, we identify the elementary amenable groups.

September 26
Matt Evans (Binghamton University)
Natural Dualities and BCK-algebras
Abstract: BCK-algebras are generalizations of Boolean algebras. There is a well-known natural duality between Boolean algebras and Stone spaces (totally disconnected, compact, Hausdorff spaces), and one might wish to develop a similar type of duality for the class of BCK-algebras. In this talk I will briefly review some category theory before defining “natural duality,” and then discuss some progress on dualizing bounded commutative BCK-algebras.
- This talk has been canceled due to illness of the speaker.

October 3
Joe Cyr (Binghamton University)
Subdirectly Irreducible Medial Quandles of Set Type

Abstract: This talk will be a continuation of my seminar talk last semester on medial quandles (though no prior knowledge will be assumed). We will explore some important aspects of the structure of this class of algebras and find that the subdirectly irreducibles of set type are quasi-reductive. This then allows one to fully classify this class of subdirectly irreducible medial quandles.

October 10
Luise-Charlotte Kappe
On covering numbers of groups

Abstract: A set of proper subgroups is a cover for a group if its union is the whole group. The minimal number of subgroups needed to cover a group is called its covering number. No group is the union of two proper subgroups. Tomkinson showed that the covering number of a solvable group has the form prime-power-plus-one and for each such integer there exists a solvable group having this integer as a covering number. In addition he showed that 7 is not a covering number. So far it has been shown that the integers < 27, which are not covering numbers, are 2,7,11,19,21,22 and 25. We extend this list by determining all integers < 129 which are covering numbers. This is joint work with Martino Garonzi and Eric Swartz.

October 17
Mark Skandera (Lehigh) (joint with the Combinatorics Seminar, 1:15 - 2:15 PM, WH-100E)
Evaluations of Hecke Algebra Traces at the Wiring Diagram Basis

Abstract: The (type A) Hecke algebra H_n(q) is a certain module over Z[q^1/2,q^-1/2] which is a deformation of the group algebra of the symmetric group. The Z[q^1/2,q^-1/2]-module of its trace functions has rank equal to the number of integer partitions of n, and has bases which are natural deformations of those of the trace module of the symmetric group algebra. While no known closed formulas give the evaluation of these traces at the natural basis elements of H_n(q), or at the Kazhdan-Lusztig basis, I present a combinatorial formula for the evaluation of traces induced by the sign character at a certain wiring diagram basis of H_n(q).

October 24
Hung Tong-Viet (Binghamton University)
2-parts of real class sizes

Abstract: An element $x$ in a finite group G is said to be real if $x$ and $x^{-1}$ are G-conjugate. A conjugacy class of G is said to be real if it contains real elements. A real class size is the size of a real conjugacy class. In this talk, I will discuss some results on the normal structures of groups with restrictions on the 2-parts of the real class sizes.

October 31
Eran Crockett (Binghamton University)
The dualizability problem for nilpotent algebras

Abstract: A finite algebra is dualizable if there is a certain dual representation of a category of algebras in a category of topological relational structures. After giving a careful definition of this concept, we ask which finite algebras are dualizable. In particular, I exhibit a class of dualizable nilpotent algebras and ask whether algebras in a generalization of this class are also dualizable.

November 7
Hung Nguyen (The University of Akron)
On the average of character degrees

Abstract: We use the notion of average character degrees to improve some classical results in the character theory of finite groups such as the Ito-Michler theorem, Thompson’s theorem and Navarro-Tiep’s theorem.

November 14
Victor Protsak (Cornell University)
Representations of $\mathfrak{gl}(n)$: a theme and variations

Abstract: Finite-dimensional representation theory of $\mathfrak{gl}(n)$, the general linear Lie algebra of $n\times n$ matrices, has three well-known aspects: algebraic (highest weight theory and characters), algebro-geometric (flag varieties and standard monomials) and combinatorial (Gelfand-Tsetlin theory and Young tableaux). Perhaps less familiar is the noncommutative linear algebra aspect, rooted in the universal enveloping algebra and involving Capelli determinants and characteristic identities.

We will review these aspects of representation theory of $\mathfrak{gl}(n)$ and explore generalizations in two different directions: infinite-dimensional modules over $\mathfrak{gl}(n)$ and integrable modules over the infinite general linear Lie algebra $\mathfrak{gl}(\infty)$.

November 21
Ian Payne (McMaster University)
Subdirectly Irreducible Algebras: Big and Small

Abstract: Given a finite algebra, there is a class called its “generated variety”. For example, it can be shown that the variety generated by the two-element group is precisely the class of groups of exponent two. In any generated variety, there is a subclass of “subdirectly irreducible” members. This subclass depends only on the generating algebra, and universal algebraists are interested in knowing how large the subclass is for a given algebra. This main goals of this talk are to say what subdirectly irreducible algebras are, explain why they are of interest, and discuss some known results about the number of subdirectly irreducible members in some generated varieties.

November 28
Upstate Descriptive Set Theory and Group Theory Day
Several Talks 1:00 - 6:00

Abstract: Please follow this link for details about the special talks: Website for this meeting

December 5
Matt Evans (Binghamton University)
Natural Dualities and BCK-algebras

Abstract: BCK-algebras are generalizations of Boolean algebras. There is a well-known natural duality between Boolean algebras and Stone spaces (totally disconnected, compact, Hausdorff spaces), and one might wish to develop a similar type of duality for the class of BCK-algebras. In this talk I will briefly review some category theory before defining “natural duality,” and then discuss some progress on dualizing bounded commutative BCK-algebras.

Spring 2017

2018-01-10T19:32:19-04:00

The Algebra Seminar

Unless stated otherwise, the seminar meets Tuesdays in room WH-100E at 2:50 p.m. There will be refreshments served at 4:00 in room WH-102.

Organizers: Ben Brewster and Fernando Guzmán

To receive announcements of seminar talks by email, please join the seminar's mailing list.

Spring 2017

January 24
Ben Brewster (Binghamton University)
The Frattini subgroup
Abstract: The Frattini subgroup of group G (denoted Fr G ) is defined as the intersection of all the maximal subgroups of G. If G has no maximal subgroups, the standard device is that G = Fr G. This talk will be concerned primarily with non-trivial finite groups and so this standard device is not relevant.
- The Frattini subgroup is called by the name of its originator in 1885. It is analogous to the Jacobson radical in ring theory and it can be generalized to various posets.
- I will present some examples and basic results about the Frattini subgroup. It has many different uses in Group theory. I will choose a few and give some examples, but then tackle the question of which groups G can be Fr H for some H.
- Speaking now about finite groups, it turns out the Frattini subgroup is nilpotent, every finite abelian group is isomorphic to the Frattini subgroup of an abelian group, but no non-abelian group of order p^3 is isomorphic to the Frattini subgroup of any group.
- I am unaware of the classification of groups G such that G is isomorphic to Fr H for some H.

January 31
Speaker (School)
Title

Abstract:

February 07
Matt Evans (Binghamton University)
Title A Structure Theorem for BCK-algebras

Abstract:

BCK-algebras were introduced in the 1960's as models for set difference and implicational calculus. In this talk I will define BCK-algebras and provide some examples before I restrict to the variety of bounded commutative BCK-algebras. Then I will classify all finite bounded commutative BCK-algebras (up to isomorphism).

February 14
Speaker (School)
Title

Abstract:

February 21
Dikran Karagueuzian (Binghamton University)
Title : Randomness of Polynomials over Finite Fields

Abstract: A polynomial over a finite field may be compared to a random map from the finite field to itself. The extent to which this comparison is valid comes up in the analysis of some primality testing algorithms. Martins and Panario have results on the validity of the comparison for generic polynomials, and we are able to generalize some of these results, including specifically their result on the coalesence, to non-generic polynomials. This is joint work with Per Kurlberg.:

February 28
Joe Cyr (Binghamton University)
Title A Structure Theorem for Entropic Quandles

Abstract: Entropic Quandles form a large subclass of binary modes. In this talk I will introduce this algebra and develop a representation of them using a blend of affine structures.

March 07

Winter Break

March 14
Adam Allan (C.C.S.U.)
Symmetry of Endomorphism Algebras

Abstract: In this talk we will consider the problem of determining whether the endomorphism algebra of a prescribed module is symmetric. In particular,we will focus on the examples of indecomposable modules for the dihedral 2-groups. :

March 21
Speaker (School)
Title

Abstract:

March 28
Hung Tong-Viet (Kent State University)
Derangements in permutation groups

Abstract: A derangement (or fixed-point-free permutation) is a permutation with no fixed points. One of the oldest theorems in probability, the Montmort limit theorem, states that the proportion of derangements in finite symmetric groups $\textrm{S}_n$ tends to $e^{-1}$ as $n$ approaches infinity. A classical theorem of Jordan implies that every finite transitive permutation group of degree greater than $1$ contains derangements. This result has many applications in number theory, game theory and representation theory. There are several interesting questions on the number and the order of derangements that have attracted much attention in recent years. In this talk, I will discuss some recent results on primitive permutation groups with some restriction on derangements (joint with T.C. Burness).

April 04
Eran Crockett (Binghamton University)
Nilpotence vs supernilpotence

Abstract: In universal algebra there are two different notions of nilpotence that happen to coincide for groups. I will describe the relation between the nilpotence class of an algebra and the maximum supernilpotence class of the algebra's congruences.

April 11

Spring Break

April 18
Phillip Wesolek (Binghamton University)
Elementary Groups

Abstract : In the study of totally disconnected locally compact (tdlc) groups, groups “built by hand” from compact groups and discrete groups frequently arise. In particular, such groups arise as obstructions to general theorems. To isolate these groups, we consider the class of elementary groups: The smallest class of tdlc groups which contains the profinite groups and discrete groups and is closed under the elementary operations.In this talk, we explore this class, showing that it is very robust. We conclude by posing several open questions, including a conjectural connection with elementary amenable groups.

April 28
Dikran Karagueuzian (Binghamton University)
Title : Randomness of Polynomials over Finite Fields (2)

Abstract: This is a continuation of the talk earlier this semester with the same title.

A polynomial over a finite field may be compared to a random map from the finite field to itself. The extent to which this comparison is valid comes up in the analysis of some primality testing algorithms. Martins and Panario have results on the validity of the comparison for generic polynomials, and we are able to generalize some of these results, including specifically their result on the coalesence, to non-generic polynomials. We will discuss the proof of the coalescence formula stated in the previous talk.

This is joint work with Per Kurlberg.

May 2
Ben Steinberg (C.C.N.Y.)
Title : Homological finiteness properties for one-relator monoids and related monoids

Abstract:

In 1932 Magnus proved the word problem was decidable for one-relator groups. Algebraists, particularly in the Soviet Union, then began to study the analogous question for other algebraic structures. Despite intensive work, particularly by Adian and his collaborators, the word problem for one-relator monoids remains open.

In the nineties, Kobayashi asked whether one-relator monoids admit a finite complete rewriting system. This would imply decidability of the word problem, but is much stronger. The Anick-Squier-Groves theorem implies that a monoid with a finite complete rewriting system satisfies the homological finiteness condition $FP_{\infty}$.

With this in mind, Kobayashi asked whether all one-relator monoids are of type $FP_{\infty}$. Lyndon had proved all one-relator groups are of type $FP_{\infty}$. Kobayashi proved all one-relator monoids are of type $FP_3$ and that many are of type $FP_4$.

The first class of one-relator monoids for which Adian solved the word problem is that of special one-relator monoids (those with a one-relator presentation of the form $w=1$); special monoids in general (ones with a finite presentation in which all relations are of the form $w_i=1$) were studied in the sixties by Adian and Makanin. They also form a base case for the results of Kobayashi.

Our main result is that special one-relator monoids are of type $FP_{\infty}$ and, more generally, that the homological finiteness properties of a special monoid are determined by those of its group of units. The techniques are topological in nature and rely on a self-similar tree-like structure in the Cayley graph of special monoids and a generalization of a result of Ken Brown on homological finiteness properties of groups acting on contractible CW complexes in terms of those of the cell stabilizers.

This is joint work with Robert Gray (University of East Anglia).

May 9
John Brown (Binghamton University)
Title : Classifying finite hypergeometric groups, height one balanced integral factorial ratio sequences, and some step functions.

Abstract: In this talk we will discuss some connections between hypergeometric series, factorial ratio sequences, and non-negative bounded integer-valued step functions. We will start with a finiteness criterion for hypergeometric groups by Beukers and Heckman, then show how this leads to the classification by Bober of integral balanced factorial ratio sequences of height one, and thus a proof that a conjectured classification of a certain class of step functions by Vasyunin is complete.

Fall 2016

2017-01-19T13:37:14-04:00

Fall 2016

August 30
Organizational Meeting

September 6
No talk this week (see the Geometry/Topology seminar on September 8 here.)

September 13
Eran Crockett (Binghamton University)
Properties of finite algebras

Abstract: We study various properties of finite algebras and the varieties they generate. In particular, we look for counterexamples to the conjecture that every dualizable algebra is finitely based.

September 20
Name (University)
Title of Talk

Abstract: Abstract for Talk

September 27
Name (University)
Title of Talk

Abstract: Abstract for Talk

October 4
Holiday
Title of Talk

Abstract: Abstract for Talk

October 11
Name (University)
Title of Talk

Abstract: Abstract for Talk

October 18
Luise C. Kappe
On auto commutators in infinite abelian groups

Abstract: Abstract for Talk

October 25
Matt Evans (Binghamton University)
An introduction to BCK-algebras

Abstract: In this talk I will introduce BCK-algebras and discuss some of their universal algebraic properties. In the bounded commutative case, I will develop the beginnings of a Priestley duality for BCK-algebras and discuss some complications.

November 1
Rachel Skipper (Binghamton University)
On some groups generated by finite automata

Abstract: Every invertible automaton with finitely many states produces a group of automorphisms of a regular rooted tree. In this talk, we outline how to obtain a group from an automaton and then discuss a particular family of examples.

November 7
Matthew Moore (McMaster University)
Dualizable algebras omitting types 1 and 5 have a cube term

Abstract: An early result in the theory of Natural Dualities is that an algebra with a near unanimity (NU) term is dualizable. A converse to this is also true: if V(A) is congruence distributive and A is dualizable, then A has an NU term. An important generalization of the NU term for congruence distributive varieties is the cube term for congruence modular (CM) varieties, and it has been thought that a similar characterization of dualizability for algebras in a CM variety would also hold. We prove that if A omits tame congruence types 1 and 5 (all locally finite CM varieties omit these types) and is dualizable, then A has a cube term.

November 8
Colin Reid (University of Newcastle)
Totally disconnected, locally compact groups

Abstract: Totally disconnected, locally compact (t.d.l.c.) groups are a large class of topological groups that arise from a few different sources, for instance as automorphism groups of combinatorial structures, or from the study of isomorphisms between finite index subgroups of a given group. Two analogies are that they are like 'discrete groups combined with compact groups' or 'non-Archimedean Lie groups'. A general theory has begun to emerge in recent years, in which we find that the interaction between small-scale and large-scale structure in t.d.l.c. groups is somewhere between the two extremes that these analogies would suggest. I will give a survey of some ways in which these groups arise and a few recent results in the area.

November 15
Andrew Kelley (Binghamton University)
Maximal subgroup growth: current progress and open questions

Abstract: This is an update on my research on the maximal subgroup growth of certain f.g. groups. The focus is on metabelian groups, virtually abelian groups, and on the Baumslag-Solitar groups.

November 22
Name (University)
Title of Talk

Abstract: Abstract for Talk

November 29
Joseph Cyr (Binghamton University)
Embedding Modes into Semimodules

Abstract: A mode is an algebra which is idempotent and whose basic operations are homomorphisms. The main focus of this talk will be to give a generalization of Jezek and Kepka's embedding theorem for groupoid modes. We will show that a mode is embeddable into a subreduct of a semimodule over a commutative semiring if and only if it satisfies the so called Szendrei identities. Thus the operations on Szendrei modes can be represented in a particularly nice way. This will involve thinking of operations “additively”, that is, taking an n-ary operation and considering it as a sum of n unary operations.

December 6
No talk this week (attend the algebra candidate talk on Friday)

The Combinatorics Seminar

2020-01-29T14:03:07-04:00

The Combinatorics Seminar

FALL 2009

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Directions to the department.

Organizers: Laura Anderson, Emanuele Delucchi, and Thomas Zaslavsky.

Tuesday, September 1
Organizational Meeting
Time: 1:15 - 2:00
Room: LN-2205

Tuesday, September 8
Speaker: Lucas Rusnak (Binghamton)
Title: Oriented Incidence and a Generalization of Hypergraphs: An Introduction
Time: 1:15 - 2:15
Room: LN-2205

Tuesday, September 15
Speaker: Lucas Rusnak (Binghamton)
Title: Oriented Incidence and a Generalization of Hypergraphs: Balance and Dependency
Time: 1:15 - 2:15
Room: LN-2205

Tuesday, September 22
Speaker: Thomas Zaslavsky (Binghamton)
Title: Psycho-Graph Math on Two-Mode Signed Networks
Time: 1:15 - 2:15
Room: LN-2205

Tuesday, September 29
Speaker: Amanda Ruiz (Binghamton)
Title: When Do Two Planted Graphs Have the Same Cotransversal Matroid?
Time: 1:15 - 2:15
Room: LN-2205

Tuesday, October 6 (joint with the Geometry and Topology Seminar)
Speaker: Priyavrat Deshpande (Western Ontario)
Title: Arrangements of Submanifolds and the Tangent-Bundle Complement
Time: 1:15 - 2:15
Room: LN-2205

Tuesday, October 13 (joint with the Geometry and Topology Seminar)
Speaker: Max Wakefield (Annapolis)
Title: Topological Formality of Arrangements of Subspaces Derived from Edge-Colored Hypergraphs
Time: 1:15 - 2:15
Room: LN-2205

Tuesday, October 20
Speaker: Tom Head (Binghamton)
Title: Computing Transparently: The Independent Sets in a Graph
Time: 1:15 - 2:15
Room: LN-2205

Tuesday, October 27
Speaker: Emanuele Delucchi & Laura Anderson (Binghamton)
Title: Complex Matroids
Time: 1:15 - 2:15
Room: LN-2205

Tuesday, November 3
Speaker: Nate Reff (Binghamton)
Title: Extensions of Spectral Graph Theory
Time: 1:15 - 2:15
Room: LN-2205

Tuesday, November 10
Speaker: Caroline Klivans (Chicago and Cornell)
Title: A Geometric Interpretation of the Characteristic Polynomial of a Hyperplane Arrangement
Time: 1:15 - 2:15
Room: LN-2205

Tuesday, November 17 (joint with the Algebra and Number Theory Seminars)
Speaker: Justin Lambright (Lehigh)
Title: A Combinatorial Interpretation for Computations in the Quantum Polynomial Ring
Time: 1:15 - 2:15
Room: LN-2205

Tuesday, November 24 (joint with the Algebra and Number Theory Seminars)
Speaker: Matthias Beck (San Francisco State University)
Title: Symmetrically Constrained Compositions of an Integer
Time: 1:15 - 2:15
Room: LN-2205

Tuesday, December 1
Speaker: Lucas Sabalka (Binghamton)
Title: Graph Braid Groups [Cancelled]
Time: 1:15 - 2:15
Room: LN-2205

Tuesday, December 8
Speaker: Simon Joyce (Binghamton)
Title: The Sum of Squares of Degrees in a Graph
Time: 1:15 - 2:15
Room: LN-2205

Friday, December 18 (Special time)
Speaker: Lucas Rusnak (Binghamton)
Title: Oriented Hypergraphs, Parts I & II
Time: 10:30 - 11:30 and 1:00-2:00
Room: LN-1120 (Note special room.)

These lectures are the Ph.D. thesis defense of Mr. Rusnak. His examining committee is Laura Anderson, Gerard Cornuejols (Carnegie-Mellon), Marcin Mazur, and Thomas Zaslavsky (chair).

Everyone is welcome to attend.

Departmental home page.

The Combinatorics Seminar

2020-01-29T14:03:07-04:00

The Combinatorics Seminar

SPRING 2006

Best Viewed With Any Browser

Directions to the department.

Organizers: Laura Anderson, Christopher Hanusa, and Thomas Zaslavsky.

The usual day, time, and place, this semester, are:

Thursdays, 1:15 - 2:15, in

Room LN-2201,

with coffee, tea, and cookies at 3:45 in the Anderson Room, LN-2207.

Some meetings will be at other times, e.g., when joint with other seminars.

Thursday, February 16

Speaker: Garry Bowlin (Binghamton)
Title: Intrinsic Chirality of 3-Connected Graphs
Time: 1:15 - 2:15

Thursday, February 23

Speaker: Leandro Junes (Binghamton)
Title: The Combinatorial Grassmannian and Extension Spaces: I
Time: 1:15 - 2:15
Room: LN-2205
This talk is the first part of Mr. Junes' candidacy exam. The examining committee is Laura Anderson (chair), Chris Hanusa, and Thomas Zaslavsky. All are invited to attend.

Thursday, March 2

Speaker: Leandro Junes (Binghamton)
Title: The Combinatorial Grassmannian and Extension Spaces: II, III
Time: 1:15 - 2:15 and 2:50 - 3:50
Room: LN-2205
These talks are the second and third parts of Mr. Junes' candidacy exam. The examining committee is Laura Anderson (chair), Chris Hanusa, and Thomas Zaslavsky. All are invited to attend.

Thursday, March 9

Speaker: Megan Owen (Cornell)
Title: Combinatorics in Information Theory
Time: 1:15 - 2:15

Tuesday, March 21

Speaker: Carlene Klivans (Chicago)
Cancelled

Thursday, March 23

Speaker: Bruce Sagan (Michigan State and Rutgers)
Title: Counting Permutations by Congruence Class of Major Index
Time: 1:15 - 2:15
Room: LN-2201

Friday, March 24 (Colloquium) (Note unusual date and time.)

Speaker: Bruce Sagan (Michigan State and Rutgers)
Title: Rational Generating Functions and Compositions
Time: 1:10 - 2:10
Room: LN-2205

Thursday, April 20 (joint with the Algebra Seminar)

Speaker: Stephanie van Willigenburg (Univ. of British Columbia)
Title: Coincidences Amongst Skew Schur Functions: A Pictorial Approach
Time: 1:15 - 2:15

Thursday, April 27

Speaker: Steven Tedford (Franklin and Marshall)
Title: Connectivity in Lifts of Greedoids
Time: 1:15 - 2:15
Room: LN-2205 (Note special room.)

Thursday, May 4 (Colloquium)

Speaker: Henry Cohn (Microsoft Research)
Title: Representation Theory, Combinatorics, and Fast Matrix Multiplication
Time: 1:15 - 2:15
Room: LN-2205

Saturday, May 6

Discrete Mathematics Day at Binghamton
Right here in Room SL-212. See the Web page.

Monday, May 8 (joint with the Geometry and Topology Seminar)

Speaker: Daniel Biss (Chicago)
Title: Annihilators in Cayley-Dickson Algebras
Time: 3:30 - 4:30 (Note unusual date and time.)
Room: LN-2205

Departmental home page.