Algebra Seminar [Department of Mathematics and Statistics, Binghamton University]

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[[http://www-history.mcs.st-and.ac.uk/Biographies/Galois.html|{{http://www.win.tue.nl/~aeb/at/mathematicians/galois1.jpg?110*135 |Evariste Galois}}]]   [[ http://www-history.mcs.st-and.ac.uk/Mathematicians/Noether_Emmy.html|{{ http://seminars.math.binghamton.edu/AlgebraSem/emmy_noether.jpg?110*135|Emmy Noether}}]]
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**#####The Algebra Seminar#####**
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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: [[:people:alex:start|Alex Feingold]] and [[:people:mazur:start|Marcin Mazur]]

To receive announcements of seminar talks by email, please join the seminar's [[http://www1.math.binghamton.edu/mailman/listinfo/algsem|mailing list]].


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=====Spring 2018=====



   * **January 16**\\  <html> <span style="color:blue;font-size:120%">Jonas Deré </span></html> (KU Leuven Kulak)\\      **// Which manifolds admit expanding maps //** \\    \\  <WRAP center box 90%> **//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.
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   * **January 23**\\  <html> <span style="color:blue;font-size:120%">Organizational Meeting</span></html> \\      **//Title of Talk//** \\    \\  <WRAP center box 90%> **//Abstract//**: Please come or contact the organizers if you are interested in giving a talk this semester or want to invite someone.
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   * **January 30**\\  <html> <span style="color:blue;font-size:120%">Casey Donoven</span></html> (Binghamton University)\\      **//Subgroups and Supergroups of Thompson's Group V//** \\    \\  <WRAP center box 90%> **//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. 
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   * **February 6**\\  <html> <span style="color:blue;font-size:120%">Jaiung Jun</span></html> (Binghamton University)\\      **//  Algebraic geometry in characteristic one   //** \\    \\  <WRAP center box 90%> **//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).
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   * **February 13**\\  <html> <span style="color:blue;font-size:120%">Ben Brewster</span></html> (Binghamton University)\\      **//  Sylow Intersections and the Chermak-Delgado Lattice  //** \\    \\  <WRAP center box 90%> **//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 $CD(G) = \{AC_B(A)\}$.
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   * **February 20**\\  <html> <span style="color:blue;font-size:120%">Victor Protsak</span></html> (Cornell University)\\      **//Invariants of representations//** \\    \\  <WRAP center box 90%> **//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.
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   * **February 27**\\  <html> <span style="color:blue;font-size:120%">Bruno Duchesne</span></html> (L’Institut Élie Cartan de Lorraine )\\      **// Kaleidoscopic groups  //** \\    \\  <WRAP center box 90%> **//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.
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   * **March 6**\\  <html> <span style="color:blue;font-size:120%">No Classes</span></html> \\      **//Title of Talk//** \\    \\  <WRAP center box 90%> **//Abstract//**: No meeting 
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   * **March 13**\\  <html> <span style="color:blue;font-size:120%">Hung Tong-Viet</span></html> (Binghamton University)\\      **//Brauer characters and normal Sylow $p$-subgroups//** \\    \\  <WRAP center box 90%> **//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. 
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   * **March 13 (Cross Listing from Combinatorics Seminar. Time: 1:15-2:15. WH-100E)**\\  <html> <span style="color:blue;font-size:120%">Victor Reiner </span></html> (Minnesota)\\      **//Finite General Linear Groups and Symmetric Groups//** \\    \\  <WRAP center box 90%> **//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. 
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   * **March 20**\\  <html> <span style="color:blue;font-size:120%">Joe Cyr</span></html> (Binghamton University)\\      **//Quandles and the "Abelian iff Quasi-Affine" Problem//** \\    \\  <WRAP center box 90%> **//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. 
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   * **March 27**\\  <html> <span style="color:blue;font-size:120%">Matt Evans</span></html> (Binghamton University)\\      **//Bounded commutative BCK-algebras do not form a discriminator variety//** \\    \\  <WRAP center box 90%> **//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. 
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   * **March 27 (Cross Listing from Combinatorics Seminar. Time: 1:15-2:15. WH-100E)**\\  <html> <span style="color:blue;font-size:120%">Farbod Shokrieh</span></html> (Cornell  University)\\      **//Effective Divisor Classes on Graphs//** \\    \\  <WRAP center box 90%> **//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).  
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   * **April 3**\\  <html> <span style="color:blue;font-size:120%">Spring Break</span></html>\\      **//Title of Talk//** \\    \\  <WRAP center box 90%> **//Abstract//**: No meeting 
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   * **April 10**\\  <html> <span style="color:blue;font-size:120%">  Eric Bucher </span></html> (Michigan State University)\\      **// Introducing cluster algebras and their applications   //** \\    \\  <WRAP center box 90%> **//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.
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   * **April 17**\\  <html> <span style="color:blue;font-size:120%">Rachel Skipper</span></html> (Binghamton University)\\      **// The congruence subgroup problem for a family of branch groups //** \\    \\  <WRAP center box 90%> **//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).

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   * **April 24**\\  <html> <span style="color:blue;font-size:120%">Gabriel Conant</span></html> (Notre Dame University)\\      **// VC-dimension, pseudofinite groups, and arithmetic regularity  //** \\    \\  <WRAP center box 90%> **//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.
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   * **May 1**\\  <html> <span style="color:blue;font-size:120%">David Biddle</span></html> (Binghamton University)\\      **//Generating systems for finitely generated groups and their relation
to special classes of characteristic subgroups of free groups//** \\    \\  <WRAP center box 90%> **//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 $G$ 
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  * [[http://seminars.math.binghamton.edu/AlgebraSem/index.html|Pre-2014 semesters]]\\
  * [[seminars:alge:fall2014]]
  * [[seminars:alge:spring2015]]
  * [[seminars:alge:alge_fall2015]]
  * [[seminars:alge:alge-spring2016]]
  * [[seminars:alge:alge-fall2016]]  
  * [[seminars:alge:alge-Spring2017|Spring 2017]]
  * [[seminars:alge:alge-Fall2017|Fall 2017]]
  * [[seminars:alge:alge-Spring2018|Spring 2018]]

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