Abstract: In this talk, we will discuss several problems related to the set of irreducible character degrees of a finite group. We will see how to use orbit structure of group actions to tackle these problems.

Abstract: The usual local fields that are used in algebraic number theory, such as the p-adic fields Q_p, can be viewed as local fields of dimension 1. In this expository talk I will define local fields of dimension n, and explain why they are of interest.

Abstract: An integral domain is atomic if every nonzero nonunit factors into irreducibles. Back in 1968, P. Cohn conjectured that every atomic integral domain satisfies the ascending chain condition on principal ideals (ACCP). In 1974, A. Grams disproved Cohn’s conjecture by constructing the first example of an atomic domain that does not satisfy the ACCP. Only a few classes of atomic domains without the ACCP have been found since then, even though both the atomic and the ACCP properties have been systematically investigated. In this talk, we will discuss Grams’s construction as well as some recent progress in this direction.

Abstract: Working in characteristic p, N.P. Byott (University of Exeter) and G.G. Elder (University of Nebraska Omaha) gave sufficient conditions for the totally ramified degree $p^2$ extensions of local fields, L/K, to have a Galois scaffold. They then used the scaffold to say exactly when the ring of integers $\mathfrak{O}_L$ is free over its associated order $\mathfrak{A}_{L/K} $. We will translate their work into characteristic 0 whilst trying to preserve as much generality as possible.

Abstract:

I will briefly survey the origin of topological techniques in algebraic geometry, and then discuss the Weil conjectures and their cohomological formulation. I will then describe Grothendieck’s approach to the problem, emphasizing the difference between the l-adic etale cohomology and the various p-adic theories. Finally I will describe recent joint work with Tomoyuki Abe showing the nonexistence of certain “integral” p-adic theories with reasonable finiteness properties.

Abstract: Dedekind domains have been systematically studied for many years given their relevance in both commutative ring theory and algebraic number theory. As a result, several generalizations of Dedekind domains have been introduced and investigated during the last century, including almost Dedekind domains, Prüfer domains, and Krull domains. After briefly reviewing some known aspects on the atomicity (factorization into irreducibles) of Dedekind domains, we will discuss various results and recent progress on the atomicity of some of their most standard generalizations.

Abstract: The Lazard Correspondence gives a method for classifying certain
p-groups in terms of Lie theory. In this expository talk I will
describe the Lazard correspondence and consider several examples.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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.

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

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

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.

This will continue the material discussed the previous week.

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.

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.

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

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.)

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.

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.)

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.

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.

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.

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.