seminars:alge

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seminars:alge [2019/04/28 23:04]
alex
seminars:alge [2019/10/10 10:47] (current)
tongviet
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-~~META:​title=Algebra Seminar~~+~~META:​title=Fall 2019~~

<WRAP center box 68%> <WRAP center box 68%>
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-=====Spring ​2019=====+=====Fall 2019=====

+   * **August 27**\\ Organizational meeting
+
+   * **September 3**\\ <​html>​ <span style="​color:​blue;​font-size:​120%">​Casey Donoven </​span></​html>​(Binghamton University) \\ **//​Automata acting on Fractal Spaces//** \\    \\  <WRAP center box 90%> **//​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. ​

-   * **January 22**\\ ​ <​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.
</​WRAP>​ </​WRAP>​

+   * **September 10**\\ <​html>​ <span style="​color:​blue;​font-size:​120%">​Matt Evans </​span></​html>​(Binghamton University) \\ **//​BCK-algebras and generalized spectral spaces
+//** \\    \\  <WRAP center box 90%> **//​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.

-   * **January 29**\\ ​ <​html>​ <span style="​color:​blue;​font-size:​120%">​Ben Brewster (Binghamton University) </​span></​html>​ \\      **//The values ​of the Chermak-Delgado measure//** \\    \\  <WRAP center box 90%> **//​Abstract//​**:​ Let $G$ be a finite group. For $H\leq G$$m_G(H) = |H|\ |C_G(H)|. Let m^*(G) = max\{m_G(H)\mid H\leq G\} and CD(G) = \{H\leq G\mid m_G(H)=m^*(G)\}. Then CD(G) is a self-dual modular sublattice of the subgroup lattice of G. +In this talk I will briefly discuss Hochster'​s characterization ​of spectral spaces, and then show - +that the spectrum ​of a commutative BCK-algebra ​is generalized spectral. -It is known that if |G| > 1, then not every subgroup ​of G is member of CD(G), that is, |\{m_G(H)\mid H\leq G\}| > 1. Following some ideas of M. Tarnauceanu,​ we examine possibilities for |\{m_G(H)\mid H\leq G\}|, its form and the distribution of subgroups of same measure.+ </​WRAP>​ </​WRAP>​ - + * **September 17**\\ <​html>​ <span style="​color:​blue;​font-size:​120%">​Jonathan Doane </​span></​html>​(Binghamton University) ​\\ **// Dualizing Kleene Algebras//** \\ \\ <WRAP center box 90%> **//​Abstract//​**: ​It is well-known that the class of Boolean algebras is "​generated"​ by - * **February 5**\\ <​html>​ <span style="​color:​blue;​font-size:​120%">​Alex Feingold (Binghamton University)</​span></​html>​ \\ **//An introduction to Lie algebras//** \\ \\ <WRAP center box 90%> **//​Abstract//​**: ​A Lie algebra ​is a vector space equipped with a bilinear product, denoted by [\cdot,\cdot], such that [x,x]=0and [x,[y,z]] + [y,[z,x]] + [z,[x,y]] = 0 (Jacobi Identity)will give an introduction ​to the basic ideas and examples.+the two element chain F<​T ​equipped with negation ​\neg F:= T$$\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 coursethere is a famous correspondence between Boolean algebras and
+Boolean topological spacesnamed Stone duality;
+this leads us to wonder if we can somehow represent Kleene algebras by
+topological spaces as well.
+In factStone 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.
</​WRAP>​ </​WRAP>​

-   * **February 12**\\  <​html>​ <span style="​color:​blue;​font-size:​120%">​Canceled due to inclement weather ​</​span></​html>​ \\   ​+   * **September 24**\\ <​html>​ <span style="​color:​blue;​font-size:​120%">​Cancelled ​</​span></​html>​ \\ **////** \\    \\  ​

-   * **February 19**\\  <​html>​ <span style="​color:​blue;​font-size:​120%">​Daniel Rossi (Binghamton University)</​span></​html>​ \\      **//The structure of finite groups with exactly three rational-valued +   * **October 1**\\ <​html>​ <span style="​color:​blue;​font-size:​120%">​No Classes ​</​span></​html>​(University) ​\\ **//Title//** \\    \\  <WRAP center box 90%> **//​Abstract//​**: ​Abstract
-irreducible characters//** \\    \\  <WRAP center box 90%> **//​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 $sl(n)$.+
</​WRAP>​ </​WRAP>​

+   * **October 8**\\ <​html>​ <span style="​color:​blue;​font-size:​120%">​Ben Brewster </​span></​html>​(Binghamton University) \\ **//The collection of intersections of Sylow p-subgroups of G//** \\    \\  <WRAP center box 90%> **//​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.

-   * **February 26**\\ ​ <​html>​ <span style="​color:​blue;​font-size:​120%">​Casey Donoven (Binghamton University)</​span></​html>​ \\      **//​Thompson'​s Group $V$ and Finite Permutation Groups//** \\    \\  <WRAP center box 90%> **//​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 $Vcalled$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$pointsthe group$V_n(G)$marries the finite prefix exchanges with iterated permutations from$G$. The primary theorem ​will present states that$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. +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.
-</​WRAP>​+
+During a sabbatical in 1989, went to TübingenGermany where Peter Hauck and 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.

-   * **March 5**\\  <​html>​ <span style="​color:​blue;​font-size:​120%">​Matt Evans (Binghamton University)</​span></​html>​ \\      **//Spectra of cBCK-algebras//​** \\    \\  <WRAP center box 90%> **//​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.
</​WRAP>​ </​WRAP>​

- +   * **October 15**\\ <​html>​ <span style="​color:​blue;​font-size:​120%">​Fikreab Admasu ​</​span></​html>​(Binghamton University) ​\\ **//Generating series for counting finite p-groups ​of class 2//** \\    \\  <WRAP center box 90%> **//​Abstract//​**:​ In 2009C. 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
-   * **March 12**\\  <​html>​ <span style="​color:​blue;​font-size:​120%">​Hung Tong-Viet (Binghamton University)</​span></​html>​ \\      **//Real conjugacy class sizes and orders ​of real elements//** \\    \\  <WRAP center box 90%> **//​Abstract//​**:​ In this talkI 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 +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.
</​WRAP>​ </​WRAP>​

-   * **March 19**\\  <​html>​ <span style="​color:​blue;​font-size:​120%">​Spring Break</​span></​html>​ \\      **//No  Talk//** \\    \\  <WRAP center box 90%> **//​Abstract//​**: ​Text of Abstract+   * **October 22**\\ <​html>​ <span style="​color:​blue;​font-size:​120%">​Eran Crockett ​</​span></​html>​(Binghamton University) ​\\ **//Finitely related clones//** \\    \\  <WRAP center box 90%> **//​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.
</​WRAP>​ </​WRAP>​

-   * **March 26**\\  <​html>​ <span style="​color:​blue;​font-size:​120%">​No Talk</​span></​html>​ \\      **//​Title ​of Talk//** \\    \\  <WRAP center box 90%> **//​Abstract//​**: ​Text of Abstract+   * **October 29**\\ <​html>​ <span style="​color:​blue;​font-size:​120%">​Speaker ​</​span></​html>​(University) ​\\ **//​Title//​** \\    \\  <WRAP center box 90%> **//​Abstract//​**:​ Abstract
</​WRAP>​ </​WRAP>​

- +   * **November 5**\\ <​html>​ <span style="​color:​blue;​font-size:​120%">​Luise Kappe </​span></​html>​(Binghamton University) ​\\ **//A GAP-conjecture ​and its solution: ​ isomorphism classes of capable special $p$-groups of rank 2//** \\    \\  <WRAP center box 90%> **//​Abstract//​**: ​A group is said to be capable if it is central quotient group and a $p$-group is special ​of rank 2 if its center is elementary abelian ​of rank 2
-   * **April 2**\\  <​html>​ <span style="​color:​blue;​font-size:​120%">​John Brown (Binghamton University)</​span></​html>​ \\      **//A small step toward proving a character theory ​conjecture//​** \\    \\  <WRAP center box 90%> **//​Abstract//​**: ​In this talk we'll discuss ​bit of the work done on a conjecture by Isaacs ​and others which states that the degree of any primitive character of finite ​group G divides the size of some conjugacy class of GWe'll focus on the case that G is symmetric or alternatingwith view to showing that the result holds for every irreducible character ​of either groupIf time permits we may discuss ideas for the next steps toward, as well as some of the obstructions toa general result.+and equal to its commutator subgroup In 1990, Heineken showed ​that if $Gis a capable special$p$-group of rank 2then$p^5 \leq |G| \leq p^7$. Over decade ago we asked GAP to determine ​the number of isomorphism classes of capable special$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$. Finallywe were able to show that what +GAP conjectured is true for all primes$p$. </​WRAP>​ </​WRAP>​ - * **April 9**\\ <​html>​ <span style="​color:​blue;​font-size:​120%">​Jonathan Doane (Binghamton University)</​span></​html>​ \\ **//Restriction of Stone Duality to Generalized Cantor Spaces//** \\ \\ <WRAP center box 90%> **//​Abstract//​**: ​Stone duality is a correspondence between Boolean algebras (BAs) and Boolean/​Stone topological spaces. ​ Dualizing the free BA$\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?''​+ * **November 12**\\ <​html>​ <span style="​color:​blue;​font-size:​120%">​Dikran Karagueuzian ​</​span></​html>​(Binghamton University) ​\\ **//Title//** \\ \\ <WRAP center box 90%> **//​Abstract//​**: ​Abstract </​WRAP>​ </​WRAP>​ - + * **November 19**\\ <​html>​ <span style="​color:​blue;​font-size:​120%">​Zach Costanzo ​</​span></​html>​(Binghamton University) ​\\ **//Title//** \\ \\ <WRAP center box 90%> **//​Abstract//​**: ​Abstract - * **April 16**\\ <​html>​ <span style="​color:​blue;​font-size:​120%">​Casey Donoven (Binghamton University)</​span></​html>​ \\ **//Inverse Semigroups//** \\ \\ <WRAP center box 90%> **//​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.+
</​WRAP>​ </​WRAP>​

-   * **April 23**\\  <​html>​ <span style="​color:​blue;​font-size:​120%">​Joseph Cyr (Binghamton University)</​span></​html>​ \\      **//The Structure of Medial Quandles//** \\    \\  <WRAP center box 90%> **//​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.+   * **November 26**\\ <​html>​ <span style="​color:​blue;​font-size:​120%">​Speaker ​</​span></​html>​(University) ​\\ **//Title//** \\    \\  <WRAP center box 90%> **//​Abstract//​**: ​Abstract
</​WRAP>​ </​WRAP>​

-   * **April 30**\\  <​html>​ <span style="​color:​blue;​font-size:​120%">​Dikran Karagueuzian (Binghamton University)</​span></​html> ​\\      **//​Coalescence of Polynomials over Finite Fields//** \\    \\  <WRAP center box 90%> **//​Abstract//​**:​ A polynomial over a finite field may be compared to a random map from +   * **December 3 **\\ <​html>​ <span style="​color:​blue;​font-size:​120%">​Speaker ​</​span></​html>​(University) \\ **//Title//** \\    \\  <WRAP center box 90%> **//​Abstract//​**: ​Abstract
-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.  +
-</​WRAP>​ +
- +
-   * **May 7**\\  <​html>​ <span style="​color:​blue;​font-size:​120%">​Joshua Carey (Binghamton ​University)</​span></​html> ​\\      **//Representation Theory of Affine Kac-Moody Lie Algebras (Candidacy Exam, Part 1)//** \\    \\  <WRAP center box 90%> **//​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. ​+
</​WRAP>​ </​WRAP>​

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* [[seminars:​alge:​alge-fall2016]]  ​   * [[seminars:​alge:​alge-fall2016]]  ​
* [[seminars:​alge:​alge-Spring2017|Spring 2017]]   * [[seminars:​alge:​alge-Spring2017|Spring 2017]]
-  * [[seminars:​alge:​alge-Fall2017|Fall 2017]]+  * [[seminars:​alge:​alge-fall2017]]
* [[seminars:​alge:​alge-Spring2018|Spring 2018]]   * [[seminars:​alge:​alge-Spring2018|Spring 2018]]
-  * [[seminars:​alge:​alge-Fall2017|Fall 2018]]

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seminars/alge.1556507047.txt · Last modified: 2019/04/28 23:04 by alex

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