Title: Brauer's height zero conjecture

Abstract:  We report on the recent solution, in joint work with Gabriel Navarro, Mandi Schaeffer Fry and Pham Huu Tiep, and building on previous work of many authors, of Brauer's 1955 conjecture relating heights of characters in a Brauer block of a finite group to the structure of its defect groups.

Title: Commutators in SL(2,O)

Abstract: Following the proof of the Ore Conjecture by Tiep and his collaborators the challenge as to which elements are commutators in S-arithmetic groups was put forth by Shalev et-al. We discuss progress on this problem which uses recent tools from diophantine analysis. Joint work with A.Ghosh and C.Meiri.

Title: Some recent progress on Thompson's Conjecture

Abstract: I will discuss some recent joint work with Tiep and others on the Thompson's Conjecture: every finite simple group G has a conjugacy class C such that C^2 = G.

Title: Local description of blocks of symmetric groups and their double covers

Abstract: Broue’s Abelian Defect Conjecture predicts a local description, up to derived equivalence, of blocks of finite groups with abelian defect in terms of local data. For the case of symmetric groups the conjecture was proved by Chuang-Rouquier in 2008 using categorical representation theory (as well as the work of Chuang-Kessar on RoCK blocks for symmetric groups). For the case of the double covers of symmetric groups the conjecture was proved by Ebert-Lauda-Vera and Brundan-Kleshchev in 2022 (using again categorical representation theory, as well as the work of Kleshchev-Livesey on RoCK blocks for symmetric groups). In this talk we describe these results and discuss what can be said about the non-abelian defect case. 

Title: G-adequate subgroups

Abstract: In 2012 I defined the notion of adequate subgroup of GL_n(k), where k is a finite field of characteristic. This is a purely group-theoretic condition, formulated for its applications to Galois representations and the Langlands programme, and was shown (by Guralnick—Herzig—Taylor—T) to be equivalent to absolute irreducibility in the tautological representation, provided that p is large enough relative to n. The situation in small characteristics was investigated further by Guralnick, Herzig, and Tiep.

In this talk I will review some of this history and discuss new work of my student Dmitri Whitmore, who has defined what it means for a subgroup of G(k) to be G-adequate (where G now is a connected reductive group) and obtained new applications to problems in the Langlands programme.

Title: Topological generation of algebraic groups

Abstract: Let G be an algebraic group over an algebraically closed field and recall that a subset of G is a topological generating set if it generates a dense subgroup. In this talk, I will report on recent work with Spencer Gerhardt and Bob Guralnick on the topological generation of simple algebraic groups by elements in specified conjugacy classes. I will also present an application concerning the random generation of finite simple groups of Lie type.

Title: The Grothendieck group of unipotent representations: a new basis

Abstract: The Grothendieck group in the title has a new basis B which is related to the standard basis S by an upper triangular matrix with entries natural numbers and with 1 on diagonal. One of the main virtues of B is that B is closer to its Fourier transform FB than S to its Fourier transform FS. We will illustrate this in the case of the symplectic group over F_q.

Title: Galois groups of random polynomials

Abstract:  Let $N_n(G,H)$ denote the number of monic integer polynomials of degree $n$ having Galois group $G\subset S_n$ and coefficients bounded in absolute value by $H$. Then $N_n(S_n,H)\asymp H^n$ by Hilbert irreducibility.  In 1936, van der Waerden conjectured that for all $G\neq S_n$, we should have $N_n(G,H)\ll H^{n-1}$.  In this talk, we describe new methods to bound $N_n(G,H)$ in terms of certain invariants of $G$, which in particular imply van der Waerden's Conjecture.

Title: Rationality of Representations of Finite Reductive Groups

Abstract: Each complex character of a finite group has two natural invariants that have a number theoretic flavour. The first is its character field, obtained from the rationals by adjoining all character values, and the second is its Schur index. This second measures the failure of a character to be afforded by a representation over its character field. In this talk I’ll discuss on-going work to obtain further information on these invariants for finite reductive groups.

Title: On the McKay Conjecture

Abstract: John McKay’s Conjecture (from 1971) predicts that, for any finite group G and prime $\ell$, the number of complex irreducible representations of G with a degree not divisible by  $\ell$ is controlled by the normaliser of a Sylow $\ell$-subgroup of G.  By work of Isaacs, Malle and Navarro this conjecture was reduced to a statement on finite quasi-simple groups and their representation theory. After a series of results by various authors it is sufficient to verify a statement on quasisimple groups of type $D$, namely Spin groups is left open. In the talk I will report on recent progress. In joint work with Marc Cabanes we analyse the representation theory of a new subgroup of those Spin groups. 

Title: Aut(F_n) actions of presentations and representations

Abstract: Let G be a group and F_n the free group on n generations.  The group Aut(F_n) acts naturally on Hom(F_n, G). The study of this action (its orbits and its dynamic) has shown up in several research directions (e.g., presentation theory, computational group theory, ergodic theory, etc.). The results depend also on what is G:  finite (simple), infinite, compact, or algebraic group and sometimes on a  dichotomy between n=2 and n>2. We will describe some classical and old results from this unified perspective; a perspective that suggests some new conjectures. We will mention also a few more recent results and observations. In particular, recent joint work with Will Chen and Pham Tiep shows the failure of the "baby Weigold conjecture" for F_2.  In simple terms: F_2 has infinitely many finite index characteristic subgroups R for which F_2/R  is a finite simple group. Interestingly, the proof is based on the Burau representation of the Braid group B_4. The result also provides counterexamples to some conjectures in the theory of non-congruence subgroups of SL(2,Z). 

Title: On the irreducibility of p-adic Banach space representations

Abstract: Suppose that G is (the group of Q_p-points of) a connected reductive group over Q_p.  Motivated by the Langlands program, it is of interest to understand representations of G on p-adic Banach spaces, i.e. continuous representations of G on a complete normed vector space over Q_p.  We will report on recent joint work with Noriyuki Abe in which we prove many new results concerning the topological irreducibility of p-adic Banach principal series representations. 

Title: Theta correspondence over finite fields and Springer correspondence

Abstract: Theta correspondence relates representations of an orthogonal group and a symplectic group. When the groups involved are finite groups of Lie type, the exact form of the correspondence, which is far from being one-to-one, was conjectured by Aubert-Michel-Rouquier, and proved recently by S-Y. Pan, and then by Ma--Qiu--Zou using a different method. In this talk I will explain yet another proof (and a reformulation) for the theta correspondence between unipotent principal series representations, in terms of the Springer correspondence. I hope it will help clarify the geometry behind the theta correspondence. This is joint work with Jiajun Ma, Congling Qiu and Jialiang Zou.

Title: Exotic representation theory

Abstract:  I will report on an ongoing  project with Gunter Malle and Jason Semeraro which  explores connections  between the p-modular representation theory of finite groups of Lie type (cross-characteristic) and the theory of p-compact groups.

Title: Isometry groups of norms

Abstract: There are many well-known and important examples of norms on R^n, and each has an isometry group that is a compact subgroup of GL(n,R). A classical question, going back to J Lindenstrauss and others, asks which compact linear groups can occur as the isometry group of some norm. I shall present a necessary and sufficient criterion for this which can be stated in a rather simple way in terms of the orbits of the group. One consequence is that every finite subgroup of GL(n,R) is the isometry group of a norm, an old result of Gordon and Loewy. The criterion leads to other interesting results and questions about compact Lie groups and their representations, which I shall discuss.

Title: New directions for modular representation theory of symmetric groups

Abstract: The focus will be on the representations of symmetric groups over the complex numbers and fields of positive characteristic. Over the complex numbers our understanding is very good, however the case of positive characteristic fields has turned out to be more complicated than (I suspect) the pioneers would have ever imagined. Remarkably, there appears to be a way forward which combines ideas which emerged in the Langlands program with techniques from mod p algebraic topology (Smith theory). This leads to a new algorithm for computation of decomposition numbers, which should be orders of magnitude better than existing algorithms. All that remains to do is implement it… 

Title: The rationality and p-rationality of character values of finite groups

Abstract: Studying values of characters is a fundamental problem in the representation theory of finite groups. Within this realm, I will discuss some (both older and recent) results on the rationality and p-rationality of character values. The rationality part focuses on a conjectural link proposed recently by Tiep and myself between the field of values, the conductor, and the degree of an irreducible character. In the second part, I will explain how the p-rationality of p'-degree irreducible characters is encoded in the p-Sylow normalizer through the McKay-Navarro conjecture and present some confirmed consequences on the continuity of p-rationality and bounding characters. The talk is based on joint works with Malle and Maroti, with Schaeffer Fry and Vallejo, and with Tiep.

Title: Seven Ways to Converge

Abstract: I will review recent progress on mixing time estimates for random walks on finite groups. It is useful to delineate various notions of mixing; geometrically ergodic, polynomial time, honest bounds, useful bounds, statistical testing, cutoffs, and shape theorems. Beautiful work of Teyssier (random transpositions) and Zhang (Hecke algebras) mixes character ratio and probabilistic techniques. I hope to trick Tiep and his coauthors to marry these techniques with their heroic work on Finite groups of Lie type.