Please think about giving a talk in the Algebra Seminar, or inviting an outside speaker.

Abstract: The well-known action of $PSL(2,R)$ on hyperbolic 2-space by Moebius transformations $z\to \frac{az+b}{cz+d}$, gives tesselations by hyperbolic triangles when the action is restricted to an arithmetic group such as $PSL(2,Z)$. This is the math behind the famous Escher picture ``Circle Limit IV” when hyperbolic 2-space is viewed as the Poincaré disk. Hyperbolic 3-space can be viewed as the interior of a unit ball, or an upper half-space in the real quaternions, and the group of all isometries is $PSL(2,C)$. I will discuss this material, and whether it might lead to an understanding of how an infinite dimensional hyperbolic Kac-Moody group might act on a hyperbolic $\infty$-space.

Abstract: This talk will give an intro to, and examples of, the profinite topology on an abstract group. After reviewing the example of the p-adic topologies on the integers and rationals, we will define the profinite topology and give Furstenberg's proof of the infinitude of primes. We will use this to define the abstract commensurator of a group and provide some examples. Examples will connect with the theory of profinite groups, Galois groups, and non-Archimedean local fields.

Abstract: We will briefly review the definition of the profinite topology on a group and the definition of the group of germs of automorphisms, and give some examples. We then explain how this topological language naturally appears in statements of 1) Neukirch and Uchida about rigidity in Galois groups of number fields, and 2) Bass–Milnor–Serre, and others, about the congruence subgroup property of certain arithmetic groups.

Abstract: Let $(\mathbb{K},\mathcal{O},\mathbb{F})$ be a $p$-modular system which is large enough for finite groups $G$ and $H$. Let $ A$ be a $p$-block of the group algebra $\mathcal{O}G$, and $B$ be a $p$-block of the group algebra $\mathcal{O}H$. In 1990, Michel Broué introduced the definition of a perfect isometry between the $p$-blocks $A$ and $B$ which is a generalized $\mathbb{K}$-valued character leading to a special bijection between the sets of ordinary irreducible characters of $A$ and $B$. In this talk, we introduce and investigate the notion of stable perfect isometries–a way to consider perfect isometries up to generalized projective characters of the corresponding $p$-blocks. Our interest lies in understanding in which cases a stable perfect isometry can be lifted to a perfect isometry. We will answer this question for the $p$-block $\mathcal{O}P$ where $P$ is an abelian $p$-group. (Joint work with Robert Boltje).

Abstract: A complex irreducible character $\chi$ of a finite group $G$ is said to be rational (or real) if $\chi$ takes only rational (or real) values, that is, $\chi(g)\in\mathbb{Q}$ (or $\chi(g)\in\mathbb{R}$) for all $g\in G.$ An element $g\in G$ is said to be rational if $\chi(g)\in\mathbb{Q}$ for all irreducible characters $\chi$ of $G$. Similarly, an element $g\in G$ is real if $\chi(g)\in \mathbb{R}$ for all irreducible characters $\chi$ of $G$ or equivalently $g$ and $g^{-1}$ are conjugate in $G$. The Itô-Michler theorem for the prime $p=2$ states that if all irreducible characters of $G$ have odd degree, then $G$ has a normal abelian Sylow $2$-subgroup $P$. A real version of this theorem was obtained by Dolfi, Navarro and Tiep in $2008$. It was shown that if all irreducible real characters of $G$ have odd degree, then $G$ has a normal Sylow $2$-subgroup. There is no rational version of this result since all the irreducible rational characters of the non-abelian simple group $\textrm{PSL}_2(3^{f})$, where $f\ge 3$ is an odd integer, have odd degree. In this talk, we discuss a new variant of the above mentioned result for rational characters. This is a joint work with P. H. Tiep.

Abstract: We will give an introduction to the McKay graph and the Mckay correspondence. Namely, the one-to-one correspondence between the Mckay graphs of the finite subgroups G of $SL(2,\mathbb{C})$ and the affine simply laced Dynkin diagrams. We will also discuss the relations between these diagrams and the resolution of singularities of $\mathbb{C}^2$ quotient by G.

Abstract: The influence of the set of character degrees of a finite group $G$ has been well studied. Over the past twenty years, it has also become clear that we can restrict our view to characters which take their values in a particular subfield of the complex numbers while still gaining some knowledge of the group’s structure. During that same time, the so-called codegrees of the characters have been studied. In this talk, I will define codegree, and give some insight into the impact the codegrees of the real-valued characters have on the structure of finite groups.

Abstract: The case when a linear group $G$ acting primitively on the vector space $V$ is of central importance in the theory of representations of solvable groups. In short, such groups have an invariant $e$ that measures their complexity. It is known that if $e > 118$, $G$ has a regular orbit.

I was able to improve this result dramatically by classifying all the cases when the regular orbit exists. In some of my early papers, I gave a coarse classification of the existence of regular orbits for primitive solvable linear groups, and the results have been widely used by other people and myself to study related problems of arithmetic properties of group invariants. A more detailed final classification has been completed in some of my recent work along with several further applications.

Abstract: We consider a finite group $G$ with a normal subgroup $N$ so that all elements of $G \setminus N$ have prime power order. We prove that if there is a prime $p$ so that all the elements in $G \setminus N$ have $p$-power order, then either $G$ is a $p$-group or $G = PN$ where $P$ is a Sylow $p$-subgroup and $(G,P,P \cap N)$ is a Frobenius-Wielandt triple. We also prove that if all the elements of $G \setminus N$ have prime power orders and the orders are divisible by two primes $p$ and $q$, then $G$ is a $\{ p, q \}$-group and $G/N$ is either a Frobenius group or a $2$-Frobenius group. If all the elements of $G \setminus N$ have prime power orders and the orders are divisible by at least three primes, then all elements of $G$ have prime power order and $G/N$ is nonsolvable.

Abstract: The Lie algebra determines many aspects of the structure theory and representation theory of a Lie group. Remarkably, linear algebraic groups defined over algebraically closed fields of non-zero characteristic exhibit many of the same properties. In this expository talk we cover the basics of linear algebraic groups, with a focus on how geometric properties over an algebraically closed field descend to properties of finite subgroups defined over finite fields. The finite groups which arise in this way are called finite groups of Lie type and play a major role in the classification of finite simple groups. We discuss Deligne and Lusztig's classification of irreducible complex representations of finite groups of Lie type, which relies on a cohomology theory originally constructed to solve the Weil conjectures. We also provide some applications.

Abstract: You may be familiar with the Fibonacci (sequence) rabbit problem. Essentially, the growth of a hypothetical rabbit population can be described by an infinite binary tree, whose levels correspond perfectly with the Fibonacci sequence. Formally, we can think of rooted paths down this tree as functions from the natural numbers to a two-element set, subject to some conditions. This talk introduces a slight twist on these conditions and illustrates how Boolean semirings (of all things) come into play!