# Algebra Seminar

Spring 2020

Date
Period
Time
Location
Speaker
Topic
January 17 9 4:05 – 4:55 LIT 368 Tom Wolf An overview of character theory
January  24 9 4:05 – 4:55 LIT 368 Tom Wolf Non vanishing elements of finite solvable groups
February 4 4 10:40 – 11:30 LIT 368 Eric Moorhouse Ovoids from lattices (Join Seminar with Combinatorics)
Abstract
February 14 9 4:05 – 4:55 LIT 368 Richard Crew Valuations in Algebraic Geometry

Abstract: Valuations of height greater than one were first used by Zariski in his work on the desingularization of algebraic surfaces in 3-folds, and have on occasion been used in other investigations where the language of schemes is not so helpful. Huber’s theory of adic spaces uses valuation theory in a nontrivial way, and Peter Scholze’s theory of perfectoid spaces is built on Huber’s theory.

In these talks I will explain the basic ideas and give some simple (and not-so-simple) examples. The setting will be algebraic throughout, but with some asides to the theory of nonarchimedean spaces. In this first talk I will review some basic facts about valuations and valuation rings, define the valuation spectrum of a ring, and if time permits prove Zariski’s theorem that the valuation spectrum of a field is quasi-compact.

February 21 9 4:05 – 4:55 LIT 368 Richard Crew Valuations in Algebraic Geometry, continued

Abstract: Valuations of height greater than one were first used by Zariski in his work on the desingularization of algebraic surfaces in 3-folds, and have on occasion been used in other investigations where the language of schemes is not so helpful. Huber’s theory of adic spaces uses valuation theory in a nontrivial way, and Peter Scholze’s theory of perfectoid spaces is built on Huber’s theory.

In these talks I will explain the basic ideas and give some simple (and not-so-simple) examples. The setting will be algebraic throughout, but with some asides to the theory of nonarchimedean spaces. In this first talk I will review some basic facts about valuations and valuation rings, define the valuation spectrum of a ring, and if time permits prove Zariski’s theorem that the valuation spectrum of a field is quasi-compact.

February 28 9 3:00 – 3:50 LIT 368 Peter Sin Title: Erdős-Ko-Rado properties of permutation groups.

Abstract: The Erdős-Ko-Rado (EKR) theorem is a famous result in extremal combinatorics answering the question:

In a set of size n, how many subsets of size k can there be such that any two have nonempty intersection, and what are the maximum family of subsets? We can assume n≤2k.

The EKR Theorem states that there are at most binom(n-1,k-1) subsets in such a family, and for n>2k any family attaining the maximum is obtained by taking all subsets of size k having a common element. This result turns out to have many analogies when sets are replaced by other combinatorial objects. There is a whole book on this theme by Godsil and Meagher. I will discuss EKR problems in permutation groups and if time allows, I will present recent joint work with Karen Meagher which proves
a conjecture of Meagher, Spiga and Tiep on 2-transitive permutation groups.

March 6 9 4:05 – 4:55 LIT 368 No seminar Spring break
March 13 9 4:05 – 4:55 VIA ZOOM Peter Sin Title: Erdős-Ko-Rado properties of permutation groups (continued).

Abstract: The Erdős-Ko-Rado (EKR) theorem is a famous result in extremal combinatorics answering the question:

In a set of size n, how many subsets of size k can there be such that any two have nonempty intersection, and what are the maximum family of subsets? We can assume n≤2k.

The EKR Theorem states that there are at most binom(n-1,k-1) subsets in such a family, and for n>2k any family attaining the maximum is obtained by taking all subsets of size k having a common element. This result turns out to have many analogies when sets are replaced by other combinatorial objects. There is a whole book on this theme by Godsil and Meagher. I will discuss EKR problems in permutation groups and if time allows, I will present recent joint work with Karen Meagher which proves
a conjecture of Meagher, Spiga and Tiep on 2-transitive permutation groups.

March 20 9 4:05 – 4:55 LIT 368 No Seminar This week
March 27 9 4:05 – 4:55 LIT 368 No Seminar This week
April 3 9 4:05 – 4:55 LIT 368 No Seminar This week
April 10 9 4:05 – 4:55 LIT 368 No Seminar This week
April 17 9 4:05 – 4:55 VIA ZOOM Kevin Keating The Hasse-Arf Theorem and Nonabelian Extensions

Fall 2019

Date
Period
Time
Location
Speaker
Topic
September 20 9 4:05 – 4:55 LIT 305 Peter Sin Title: The critical group of a graph
Abstract: Given a (simple) finite graph, its critical group is a finite abelian group defined by its Laplacian matrix. The group arises in several contexts (Chip-firing games, abelian sandpile model in physics, graph jacobians, arithmetic geometry of curves). I’ll explain the basic definitions and the connection with the Matrix-Tree Theorem and Chip-firing games. Interesting research topics include trying to compute the critical groups for various families of graphs. This talk is an introduction to this topic, with few prerequisites.
September 27 9 4:05 – 4:55 LIT 305 Alexandre Turull p-basic groups
Abstract: The rationality of representations of finite groups is best measured as an invariant in the group Q/Z of rational numbers modulo integers under addition. The talk will discuss an effective method to calculate these invariants. The method consists of a series of reductions that eventually reduce the problem to the case when the finite groups are actually $p$-basic groups. We will discuss the definition of these $p$-basic groups, and the invariants associated with their irreducible characters.
October 4 9 4:05 – 4:55 LIT 305 No seminar Homecoming
October 11 9 4:05 – 4:55 LIT 305 Felix Gotti Title: The rational-infinite elasticity property and some related conjectures

Abstract: The elasticity is an algebraic statistic introduced by Valenza in 1990 to measure how much a ring of integers deviates from being a UFD. Since then, the elasticity has been used in more general classes of domains and in commutative monoids. While the elasticity of a domain/monoid belongs to the set $\{x \in \mathbb{R} : x \ge 1\} \cup \{ \infty \}$ by definition, for certain important classes of domains/monoids it turns out to be either rational or infinite, in which case we say that such classes satisfy the rational-infinite elasticity property. We will see various classes of monoids and integral domains satisfying the rational-infinite elasticity property as well as classes where such a property is conjectured to hold.

October 18 9 4:05 – 4:55 LIT 305 Qing Xiang Fourier analysis on finite abelian groups and uncertainty principles
Abstract
October 25 9 4:05 – 4:55 LIT 305 Zachary Hamaker Grobner degeneration of matrix Schubert varieties

Abstract: Matrix Schubert varieties are a vast generalization of determinantal varieties, which have been studied since the 19th century in the context of degeneracy loci. I will explain how to use Grobner degeneration, a powerful technique in combinatorial commutative algebra, to compute properties of these varieties. Grobner degenerations are determined by term orders. Historically, people used diagonal term orders, but a major breakthrough by Knutson and Miller showed that antidiagonal terms orders are more natural, and used them to describe the Hilbert series of these varieties combinatorially. Deficiencies of diagonal term orders were discussed in work by Knutson, Miller and Yong. I will outline this history and discuss forthcoming work with Oliver Pechenik and Anna Weigandt where we make new progress with diagonal term orders, including a conjectural way forward in this setting.

November 1 9 4:05 – 4:55 LIT 305 No seminar AMS meeting
November 8 9 4:05 – 4:55 LIT 305 Felix Gotti
Atomic properties of monoid algebras
Abstract: From any commutative ring $R$ with identity, one can construct the ring of polynomials $R[x]$. If, in addition, $M$ is a commutative monoid, one can construct the monoid algebra $R[x;M]$ over $R$ consisting of all polynomial expressions with coefficients in $R$ and exponents in $M$. It is well known that many algebraic properties such as being an integral domain, being a UFD, and being a Noetherian domain transfer from $R$ to $R[x]$. We will discuss some algebraic and atomic properties that transfer from $R$ (or from $M$) to $R[x;M]$. The question of whether the property of being atomic transfers from $R$ and $M$ to $R[x;M]$ was posed by Robert Gilmer back in the 1980s. It has been recently proved that for any integral domain $R$ the property of being atomic does not transfer from $M$ to $R[x;M]$. We will conclude the presentation talking about such a recent progress.
November 15 9 4:05 – 4:55 LIT 305 Ashleigh Thomas Modules over polynomial rings with real exponents, as motivated by persistent homology

Abstract: The study of modules over polynomial rings is extended to modules over polynomial rings with real number exponents (these can also be thought of as modules over the group rings k[N^n] and k[R_{\geq0}^n], respectively, where k is a field). We discuss some of the differences between modules in these two settings. For example, Noetherinity does not apply for modules over k[R_{\geq0}^n], but there are alternative finiteness conditions.

This exploration is motivated by a data analysis technique called persistent homology, which analyses a system or data set across all values in a parameter space. Persistent homology computations produce modules graded by an input parameter space, which is frequently Z, R, Z^n, or R^n. We give a brief introduction to the basic algebraic constructions used for persistent homology.

November 22 9 4:05 – 4:55 LIT 305 Kevin Keating Generic p-extensions

Abstract: Let p be prime and let G be a group of order p^n. In this talk I will discuss the existence of polynomials D_i in F_p[X_0,…,X_{i-1}] with the following property: For every field K_0 of characteristic p and every Galois extension K_n/K_0 with Gal(L/K)=G there are b_i in K_0 such that K_n is constructed recursively by K_{i+1}=K_i(a_i), with a_i^p-a_i=D_i(a_0,..,a_{i-1})+b_i. In addition, under mild assumptions on K_0 and b_i, every extension K_n/K_0 constructed this way is Galois with Gal(K_n/K_0)=G.

November 29 9 4:05 – 4:55 LIT 305 No seminar Thanksgiving

Spring 2019

Date
Period
Time
Location
Speaker
Topic
February 15 9 4:05 – 4:55 LIT 368 Peter Sin Spreads, Ovoids, Opposites and Irreducible Group Representations

Abstract:
A spread in a polar space is set of disjoint generators (maximal totally isotropic subspaces) that cover the set of points. Dually, an ovoid is set of points such that each generator contains exactly one point from the set. These definitions can be extended to Generalized Polygons, using the concept of oppositeness. I will discuss recent work (with Ihringer and Xiang) on the bounds from representation theory on the size of a partial spreads and partial ovoids. In particular, we show that ovoids cannot exist in the finite Tits octagon.
February 22 9 4:05 – 4:55 LIT 368 Qing Xiang Characterization of Intersecting Families of Maximum Size in PSL(2, q)
(Joint work with Ling Long, Rafael Plaza, and Peter Sin).
March 1 9 4:05 – 4:55 LIT 368 Felix Gotti

A question by Gilmer on the Atomicity of Monoid Domains

ABSTRACT: Let $M$ be a commutative cancellative monoid, and let $R$ be an integral domain. The question of whether the monoid domain $R[x;M]$ is atomic provided that both $M$ and $R$ are atomic dates back to the 1980’s. In 1993, Roitman gave a negative answer to the question for $M = \nn_0$: he constructed an atomic integral domain $R$ such that the polynomial ring $R[x]$ is not atomic. However, the question of whether a monoid algebra $F[x;M]$ over a field $F$ is atomic provided that $M$ is atomic has been open since then. Here I will discuss a negative answer to this question when the field $F$ has finite characteristic.

March 8 9 4:05 – 4:55 LIT 368 No seminar Spring break
March 15 9 4:05 – 4:55 LIT 368 Alexander York A Structure Theorem for Quasi-Gorenstein Modules

Abstract: Let R be a Cohen-Macaulay factorial domain.  An R-module M is quasi-Gorenstein if the projective dimension of M is equal to the grade of M and Ext_R^pd(M)(M,R) is isomorphic to M.  An example of such a module is the critical group of a finite graph.  We will show that this definition allows for us to give a structure theorem for quasi-Gorenstein R-modules of projective dimension 1 related to the diagonalizability of matrices composed of entries in R which is an extension of the structure theorem for modules over a PID.  This is accomplished by defining a certain type of filtration of R-modules by quasi-Gorenstein modules allowing us to prove the splitting of a certain short exact sequence.

March 22 9 4:05 – 4:55 LIT 368 No seminar this week
March 29 9 4:05 – 4:55 LIT 368 No seminar this week
April 5 9 4:05 – 4:55 LIT 368 Marly Cormar Atomicity of cyclic rational semirings

Abstract. Every sub-semiring of the field Q that is not a ring (i.e., a rational semiring) is contained in the nonnegative cone of Q. The rational semiring S r generated by a given positive rational r (i.e., the cyclic rational semiring generated by r) is atomic as an additive monoid unless r = 1/n for some n ∈ N. Thus, “almost all” cyclic rational semirings are atomic monoids. I will present some of the most relevant atomic and factorization properties of the cyclic rational semirings.

April 12 9 4:05 – 4:55 LIT 368 Kevin Keating Heights of power series in characteristic p

Abstract: Let k be a perfect field of characteristic p. The set of all series of the form f(x)=x+a_1x^2+a_2x^3+… with a_i in k forms a group with the operation of composition, known as the Nottingham group of k. In this talk we consider several possible definitions for the height of such a series. These definitions are motivated by the definition of the height of a formal group law.

April 22 9 4:05 – 4:55 LIT 368 Tiep Sylow subgroups and character tables

Abstract. What information about Sylow p-subgroups of a finite group can be read off from its character table? This problem was stated by Richard Brauer in his famous 1963 list, and has also received considerable attention because of its connections to the more recent Galois-McKay conjecture. We will discuss various results on this problem, many of which are joint with Gabriel Navarro.

Fall 2018

Date
Period
Time
Location
Speaker
Topic
October 26 9 4:05 – 4:55 LIT 368 A. Turull An invariant for ordinary characters arising from modular characters
November 2 9 4:05 – 4:55 LIT 368 No seminar Homecoming
November 9 9 4:05 – 4:55 LIT 368 Raghu Pantangi
Critical groups of van Lint-Schrijver cyclotomic Strongly Regular Graphs.
Abstract:
The critical group of a finite connected graph is an abelian group defined by the Smith normal form of its Laplacian. Let $K$ be a finite field and $D$ be subgroup of the multiplicative group. A cyclotomic Strongly Regular Graph is a Cayley graph on $(K,+)$ with ”connection” set $D$ that is Strongly regular. We will describe the critical groups of a family of cyclotomic SRG’s discovered by van Lint and Schrijver.
November 16 9 4:05 – 4:55 LIT 368 Felix Gotti On monoid algebras with rational exponents
November 23 9 4:05 – 4:55 LIT 368 No seminar Thanksgiving
November 30 9 4:05 – 4:55 LIT 368 Hossein Shahrtash The implications of rational class sizes for the structure of a finite group

Spring 2018

Date
Period
Time
Location
Speaker
Topic
January 26 9 4:05 – 4:55 LIT 305 No seminar Colloquium talk
February 2 9 4:05 – 4:55 LIT 305 No seminar Colloquium talk
February 9 9 4:05 – 4:55 LIT 305 No seminar Colloquium talk
February 16 9 4:05 – 4:55 LIT 305 Tom Wolf Automorphism Towers and subnormal series
February 23 9 4:05 – 4:55 LIT 305 Tom Wolf Automorphism Towers and subnormal series, cont
March 2 9 4:05 – 4:55 LIT 305 Tom Wolf Coprime actions and invariant Hall subgroups
March 9 9 4:05 – 4:55 LIT 305 No seminar Spring break
March 16 9 4:05 – 4:55 LIT 305 No seminar This week
March 23 9 4:05 – 4:55 LIT 305 No seminar This week
March 30 9 4:05 – 4:55 LIT 339 The Atrium Hossein Shahrtash Rational class sizes and their implications about the structure of a finite group
April 6 9 4:05 – 4:55 LIT 305 No seminar This week
April 13 9 4:05 – 4:55 LIT 305 Alexandre Turull The invariant of a character
April 20 9 4:05 – 4:55 LIT 305 Kevin Keating The Artin character of a local field extension

Fall 2017

Date
Period
Time
Location
Speaker
Topic
September 1 9 4:05 – 4:55 LIT 368 Alexandre Turull Dade’s conjectures and related conjectures, I
September 8 9 4:05 – 4:55 LIT 368 No seminar Hurricane Irma
September 15 9 4:05 – 4:55 LIT 368 Alexandre Turull Dade’s conjectures and related conjectures, II
September 22 9 4:05 – 4:55 LIT 368 Peter Sin Erdős-Ko-Rado problems for permutation groups
September 29 9 4:05 – 4:55 LIT 368 Peter Sin EKR for PSL(2,q) acting on the projective line
October 6 9 4:05 – 4:55 LIT 368 No seminar Homecoming
October 13 9 4:05 – 4:55 LIT 368 Peter Sin EKR for PSL(2,q) acting on the projective line (continued)
October 20 9 4:05 – 4:55 LIT 368 No seminar This week
October 27 9 4:05 – 4:55 LIT 368 No seminar This week
November 3 9 4:05 – 4:55 LIT 368 Richard Crew F-isocrystals, Weil groups and local class field theory
November 10 9 4:05 – 4:55 LIT 368 No seminar This week
November 17 9 4:05 – 4:55 LIT 368 Richard Crew F-isocrystals, Weil groups and local class field theory, cont.
November 24 9 4:05 – 4:55 LIT 368 No seminar Thanksgiving
December 1 9 4:05 – 4:55 LIT 368 Kevin Keating What are the p-adic numbers?

Spring 2017

Date
Period
Time
Location
Speaker
Topic
February 10 9 4:05 – 4:55 LIT 368 Tom Wolf Regular orbits
February 17 9 4:05 – 4:55 LIT 368 Tom Wolf Regular orbits (continued)
February 24 9 4:05 – 4:55 LIT 368 Pantangi Smith group and Critical group of the Symplectic polar graph.

Abstract: The Smith group and Critical group are interesting invariants of a graph. The Smith group a graph is the co-kernel of it’s adjacency matrix. The critical group a graph is a finite Abelian group whose order is the number of spanning forests of the graph.

In this presentation, we will focus on some elementary linear algebra techniques that give us partial information about the Smith group and Critical group of a Strongly regular graph. We will also apply these techniques and some representation theory to find the Smith group and Critical group of the Symplectic Polar graph.

March 3 9 4:05 – 4:55 LIT 368 Cyr Semipermutability of subgroups in some simple groups
March 10 9 4:05 – 4:55 LIT 368 No seminar Spring Break
March 17 9 4:05 – 4:55 LIT 368 Kevin Keating What is a Hopf Algebra?
March 24 9 4:05 – 4:55 LIT 368 Kevin Keating Affine group schemes
March 31 9 4:05 – 4:55 LIT 368 Kevin Keating Affine Group Schemes II
April 7 9 4:05 – 4:55 LIT 368 Kevin Keating Hopf-Galois Extensions
April 14 9 4:05 – 4:55 LIT 368 Yong Yang On $p$-parts of character degrees of finite groups

Abstract:
Let $G$ be a finite group and $P$ be a Sylow $p$-subgroup of $G$, it is reasonable to expect that the degrees of irreducible characters of $G$ somehow restrict the structure of $P$. The Ito-Michler Theorem proves that every ordinary irreducible character degree is coprime to $p$ if and only if $G$ has a normal abelian Sylow $p$-subgroup. Of course, this implies that $|G:F(G)|_p=1$ where $F(G)$ is the Fitting subgroup of $G$.

Let $G$ be a finite group and $\Irr(G)$ the set of irreducible complex characters of $G$. Let $e_p(G)$ be the largest integer such that $p^{e_p(G)}$ divides $\chi(1)$ for some $\chi \in \Irr(G)$. In this talk, we show that $|G:F(G)|_p \leq p^{K e_p(G)}$ for a universal constant $K$. This settles a conjecture of A. Moreto.

Fall 2016

Date
Period
Time
Location
Speaker
Topic
October 17 9 4:05 – 4:55 LIT 368 Alexandre Turull Maximal Subgroups of Finite Groups

Abstract:
We will discuss elementary properties of maximal subgroups of finite groups with particular emphasis on the maximal subgroups of finite solvable groups. Some recent results and open questions will also be discussed.

October 24 9 4:05 – 4:55 LIT 368 No seminar Because of colloquium talk
October 31 9 4:05 – 4:55 LIT 368 Alexandre Turull Maximal Subgroups of Finite Groups, II
November 7 9 4:05 – 4:55 LIT 368 No seminar Because of colloquium talk
November 14 9 4:05 – 4:55 LIT 368 Alexandre Turull Maximal Subgroups of Finite Groups, III
November 21 9 4:05 – 4:55 LIT 368 No Seminar This week
November 28 9 4:05 – 4:55 LIT 368 Alexandre Turull Small orbits and regular orbits

Abstract:
We will discuss the existence of regular orbits when the size of the field of definition of the module in question is large compared with the size of the orbits of a finite group on it.

December 5 9 4:05 – 4:55 LIT 368 Alexandre Turull Small orbits and regular orbits, II

Spring 2016

Date
Period
Time
Location
Speaker
Topic
February 5 7 1:55 – 2:45 LIT 305 Alexandre Turull Characters of Finite Groups

Abstract:
An introduction to the character theory of finite groups. No previous knowledge of character theory of finite groups will be assumed.

February 12 7 1:55 – 2:45 LIT 305 Alexandre Turull Characters of Finite Groups

Abstract:
In this talk, we will continue our introduction to the character theory of finite groups.

February 19 7 1:55 – 2:45 LIT 305 Tom Wolf More character theory
February 26 7 1:55 – 2:45 LIT 305 Tom Wolf More character theory
March 4 7 1:55 – 2:45 LIT 305 No seminar Spring Break
March 11 7 1:55 – 2:45 LIT 305 Alexandre Turull The Strengthened Dade Projective Conjecture

The Dade Projective Conjecture relates the existences of certain characters for the normalizers of chains of p-subgroups of a finite group. This conjecture has been strengthened to include information about the p’-part of their degrees, the fields of definition and the Schur indices. We will discuss a proof of this strengthened conjecture for the all the finite p-solvable groups.

March 18 7 1:55 – 2:45 LIT 305 Alexandre Turull The Strengthened Dade Projective Conjecture II

This will continue the material discussed the previous week.

March 25 7 1:55 – 2:45 LIT 305 Shahrtash Recognizing direct products from their conjugate type vector
March 28 (Monday) 9 4:05 – 4:55 LIT 368 Qing Xiang A Linear Analogue of Kneser’s Theorem and Related Problems
Abstract
April 8 7 1:55 – 2:45 LIT 305 Kevin Keating Galois Modules and Ramification Theory
April 15 7 1:55 – 2:45 LIT 305 Kevin Keating Galois Modules and Ramification Theory (continued)
April 22 7 1:55 – 2:45 LIT 305 Josh Ducey Critical groups of strongly regular graphs

Abstract:
The critical group of a graph is an interesting isomorphism invariant; it is a finite abelian group whose order is equal to the number of spanning forests of the graph.  Determination of the critical group is equivalent to finding the Smith normal form of the Laplacian matrix for the graph.

An active line of research has been to calculate the critical group for various families of graphs.  In this talk we illustrate how to obtain partial information about the critical group of any strongly regular graph.  Other methods that can be used to gain further insight will be illustrated through several examples.

Spring 2015

Date
Period
Time
Location
Speaker
Topic
January 27 9 4:05 – 4:55 LIT 305 Alexandre Turull Endoisomorphisms

Abstract:
Many properties of characters of finite groups are obtained using character correspondences. These are certain maps from sets of characters in one group to sets of characters in another. These can be uniquely defined via the use of endoisomorphisms. We will define endoisomorphism and explain how the character correspondences are obtained from them. Then we will describe some natural operations on endoisomorphisms.

February 3 9 4:05 – 4:55 LIT 305 Alexandre Turull Endoisomorphisms

Abstract:
In this talk, we will define endoisomorphisms and show how the character bijection is obtained.

February 10 9 4:05 – 4:55 LIT 305 Peter Sin Title: Representations of the alternating group which are irreducible over subgroups.

Abstract:
Suppose an alternating group A_n has a primitive subgroup H isomorphic to A_m, for some m. It is useful to know whether or not an irreducible representation of A_n can remain irreducible when restricted to the subgroup H. We show that this never happens when n>m ≥ 9. (Joint work with A. Kleshchev and P. H. Tiep.)

February 17 9 4:05 – 4:55 LIT 305 Kevin Keating Trace, Norm, Etc.
February 24 9 4:05 – 4:55 LIT 305 Tom Wolf The Glauberman-Isaacs Correspondence
March 3 9 4:05 – 4:55 LIT 305 No seminar Spring Break
March 10 9 4:05 – 4:55 LIT 305 Richard Crew F-Isocrystals and division algebras
March 17 9 4:05 – 4:55 LIT 305 Liz Wiggins Some Weyl modules for simple algebraic groups

Abstract:
Let $G$ be a simple algebraic group over an algebraically closed field of characteristic $p>0$. In this talk, we will examine groups of type $B_4$ and $D_4$, which are the classical groups $SO(9)$ and $SO(8)$, respectively. We will determine the structure of Weyl modules and characters of simple modules for some particular weights. We will also discuss an application in the theory of spherical buildings.

March 24 9 4:05 – 4:55 LIT 305 No seminar this week
March 31 9 4:05 – 4:55 LIT 305 No seminar this week
April 7 9 4:05 – 4:55 LIT 305 Peter Sin The Smith group of the Hypercube graph.

Abstract. The hypercube graph is a basic example, closely related to the Hamming association scheme, which in turn plays an important role in coding theory. This talk is about the recent calculation of the Smith group of the hypercube, or equivalently the Smith Normal Form of its adjacency matrix. (Joint work with D. Chandler and Q. Xiang.)

April 14 9 4:05 – 4:55 LIT 305 No seminar this week

Fall 2014

Date
Period
Time
Location
Speaker
Topic
July 28 8 3:00 – 3:50 LIT 368 Pham Tiep Nilpotent Hall and abelian Hall subgroups
Abstract: To which extent can one generalize the Sylow theorems? One possible direction is to assume the existence of a nilpotent subgroup whose order and index are coprime. We will discuss recent joint work with various collaborators that gives a criterion to detect the existence of such subgroups in any finite group.
September 12 9 4:05 – 4:55 LIT 368 Alexandre Turull Generalizations of Jordan’s Theorem
September 19 9 4:05 – 4:55 LIT 368 Peter Sin Some remarks on Veronese and Grassmann varieties in characteristic 2
September 26 9 4:05 – 4:55 LIT 368 Alexandre Turull The Glauberman-Isaacs character correspondence and its inverse
October 3 9 4:05 – 4:55 LIT 368 Doug Brozovic An introduction to sharp permutation groups
Abstract
October 10 9 4:05 – 4:55 LIT 368 No seminar this week
October 17 9 4:05 – 4:55 LIT 368 No seminar Homecoming
October 24 9 4:05 – 4:55 LIT 368 Christopher Cyr Title: A Theorem of Isaacs on S-Semipermutable Subgroups and Some Consequences

Abstract: The familiar notion of a permutable subgroup can be generalized in many ways, one of which is S-semipermutability. In recent years many authors have explored what can be said about a group when many subgroups satisfy a particular permutability condition. In this talk, we present a recent theorem of Isaacs concerning the normal closure of an S-semipermutable subgroup H of a finite group G. In addition to proving the theorem, we mention some corollaries which result from considering the special cases where H is a Sylow p-subgroup or a Hall π-subgroup of G.

October 31 9 4:05 – 4:55 LIT 368 Alexander Gruber Title: Design and Cryptanalysis of Matsumoto-Imai and its Variants

Abstract: The Matsumoto-Imai (MI) Cryptosystem was proposed in 1988 as a candidate for the national cryptosystem of the Japanese government. The MI scheme exploits the difficulty of determining the hidden structure of a finite field extension, promising similar security to RSA, yet with much faster encryption and decryption speeds. MI was broken in 1995 with an algebraic attack published by Jacques Patarin; however, there are several proposed improvements that offer increased security. In this talk, we outline the MI scheme, discuss its cryptanalysis, and
present several of its variants.

November 7 9 4:05 – 4:55 LIT 368 Venkata Raghu Tej Pantangi Title: Introduction to symmetric functions

Abstract: Let $X$ be a set of variables indexed by the natural numbers. A symmetric function $f$ is an element of $\mathbb{Q}[[X]]$ which is invariant under any permutation of $X$ and the degrees of monomials involved in $f$ are bounded. One can also view them as the elements of the inverse limit of the rings of symmetric polynomials in finite number of variables, considered as graded rings. Any $\mathbb{Q}-$basis of the ring of symmetric polynomials is indexed by partitions of natural numbers. We will see four important bases and the linear relations among these bases. The role played by symmetric functions in the character theory of symmetric groups will also be discussed.

November 14 9 4:05 – 4:55 LIT 368 Yong Yang Title: Orbits of group actions

Abstract: The idea of a group is the mathematical abstraction of the common notion of symmetry. Group actions allow the study of groups via their action on suitable sets (such as vector spaces) which models the ways they arise in the real world. Naturally, the information on the orbits induced by a group action is central to the understanding of the action. As in the case of Sylow’s theorems, a result on the orbits of a group action is often at the core of a seemingly unrelated problem. In this talk, we discuss some recent developments in this area, and address some open problems.

November 21 9 4:05 – 4:55 LIT 368 No seminar this week
November 28 9 4:05 – 4:55 LIT 368 No seminar Thanksgiving
December 5 9 4:05 – 4:55 LIT 368 No seminar this week