Algebra Seminar

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 TBA TBA
October 27 9 4:05 – 4:55 LIT 368 TBA TBA
November 3 9 4:05 – 4:55 LIT 368 Richard Crew TBA
November 10 9 4:05 – 4:55 LIT 368 Richard Crew TBA
November 17 9 4:05 – 4:55 LIT 368 Kevin Keating TBA
November 24 9 4:05 – 4:55 LIT 368 No seminar Thanksgiving
December 1 9 4:05 – 4:55 LIT 368 TBA TBA

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