Title: The polynomial question in modular invariant theory

Abstract: Let G < GL(V) be a finite group, where V is a finite dimensional vector space over a field F.
G acts as a group of automorphisms on S(V), the symmetric algebra on V.

We shall consider the following:

Q. When is S(V)^G ,the subring of G-invariants, a polynomial ring?

This had been completely settled by Shephard-Todd-Chevalley-Serre if (char F, |G|)=1 ,and is still open otherwise.

We shall describe our solution to this problem when G< SL(V).

An application to the classification problem of isolated quotient singularities in prime characteristic, will be indicated.

Title: The geometries of the Freudenthal-Tits magic square

Abstract: I will give an overview of a long-term programme investigating projective embeddings of (exceptional) geometries, which Hendrik Van Maldeghem and I started in 2010.

Title: Parabolic Kazhdan-Lusztig polynomials and oriented Temperley-Lieb algebras

Abstract: Work by Brundan and Stroppel in the 2010s showed that parabolic Kazhdan-Lusztig polynomials in type A arise from diagrammatic algebras they called extended Khovanov arc algebras. These polynomials appear all over the place in representation theory. I will talk about another place they seem to crop up -- work in progress with Olivier Dudas. And I will discuss how to explain the diagrammatic rule for computing them using an oriented version of the Temperley-Lieb algebra -- work with Chris Bowman, Maud De Visscher, Niamh Farrell, and Amit Hazi.

Title: Generic stabilizers and double coset density

Abstract: We consider faithful actions of simple algebraic groups on self-dual irreducible modules, and on the associated varieties of totally singular subspaces. In all cases where the dimension of the acting group is at least as large as the dimension of the variety, we find a dense open subset where the stabilizer of any point is conjugate (or isomorphic) to a fixed subgroup. We use these results to determine whether there exists a dense orbit. This in turn gets us very close to a complete answer to the following question. Given two maximal connected subgroups of a simple algebraic group, is there a dense double coset?

Title: Simple-minded view of the sporadic Lie groups and Lie algebras

Abstract: The Weyl group of E6 is contained in the geometry of the 27 lines on the cubic surface and the 45 triangles they form. I use this relation to nail down the Lie algebra and Lie group of type E6, and one of its prominent symmetric spaces. The same ideas go up to E7 and E8 (with increasing effort of calculation), and down to the smaller but still interesting cases of D5 and F4.

The talk is intended to be elementary, and most of this is well-known to specialists. However, I have had fun rediscovering it for myself, and in some cases my calculations may simplify the traditional ideas used by algebraists.

Title: Representations of algebraic groups: a journey into the world of contramodules.

Abstract: The ultimate goal of this talk will be to give an explicit construction of projective covers of simple G-modules. However, we are not going to find such covers in the realms of comodules. Therefore, after equipping ourselves with some definitions, examples and constructions we wander into the contramodule world in hopes of constructing our desired covers there.

Title: Invariable generation and totally deranged elements of almost simple groups

Abstract: By a classical theorem of Jordan, every faithful transitive action of a nontrivial finite group admits a derangement (an element with no fixed points). More recently, the existence of derangements with additional properties has attracted much attention, especially for primitive actions of almost simple groups. Surprisingly, there exist almost simple groups with elements that are derangements in every faithful primitive action; we say that these elements are totally deranged. I’ll talk about recent work to classify the totally deranged elements of almost simple groups, and I’ll mention how this solves a question of Garzoni about invariable generating sets for simple groups.

Title: Simple Modules for Algebraic Groups

I will report on current work with David Stewart. In this work we have: a) given a classification of simple modules for algebraic groups over arbitrary fields, which extends well-known high weight classifications when the field is algebraically closed and/or the group is reductive; b) begun to explore the structure of these simple modules, in particular how they look after extensions of the ground field. The key to progress is to understand so-called "pseudo-reductive groups". I will spend most of the talk on an extended example of such a group and its representations, which can be constructed pretty concretely from the simple algebraic group SL₂.

Title: Some composition multiplicities for tensor products of irreducible representations of GL(n).

Abstract: Understanding the composition factors of tensor products is an important question in representation theory. In characteristic 0 the classical Littlewood-Richardson coefficients describe the composition factors of both the tensor products of simple CGLn(C)-modules and the restriction of simple CGLn(C)-modules to some Levi subgroup.

Now let F denote an algebraically closed field of characteristic p > 0. In comparison very little is known about composition factors of tensor products of simple FGLn(F)-modules but it is thought that there may still be a relationship with the restriction of simple FGLn(F)-modules to some Levi subgroup. In this talk explore and explicit relationship of this kind for tensor products of simple FGLn(F)-modules with the wedge square of the dual natural module and see how this might be used to find composition factors.

Title: Representation varieties and rigidity in finite simple groups

Abstract: Building from the now-classical theorem that every non-abelian finite simple group is 2-generated, one can ask much more delicate questions, for instance on the abundance of generating pairs, or on the existence of generating pairs with particular orders, or from particular conjugacy classes.

For groups of Lie type, these questions can be studied using the representation variety Hom(F,G) where F is finitely generated and G is a reductive algebraic group. In work with Ben Martin (Aberdeen), we use the conjugation action of G on Hom(F,G) to interpret generators for groups of Lie type as certain Zariski-closed orbits. This allows us to prove and generalise a 2010 conjecture of C. Marion, and motivates this as an avenue in the wider study of generating sets in groups of Lie type.

Title: Homotopy type of SL2 quotients of simple complex Lie groups

Abstract: We say an element X in a Lie algebra g is nilpotent if ad(X) is a nilpotent operator. It is known that G_{ad}-orbits of nilpotent elements of a complex semisimple Lie algebra g are in 1-1 correspondence with Lie algebra homomorphisms \phi: sl2 -> g, which are in turn in 1-1 correspondence with Lie group homomorphisms SL2 -> G.
Thus, we may denote the homogeneous space obtained by quotienting G by the image of a Lie group homomorphism SL2 -> G by X_v, where v is a nilpotent element in the corresponding G_{ad}-orbit.
In this talk we introduce some algebraic and topological tools that one can use to attempt to classify the homogeneous spaces, X_v, up to homotopy equivalence.

Title: On endotrivial modules

Abstract: Let G be a finite group and k a field of positive characteristic p diving the order of G. An endotrivial kG-module is a finitely generated kG-module which is "invertible" in some suitable sense. Since the late 70s, these modules have been intensely studied in modular representation theory. In this talk, we briefly review the background, before presenting some results (joint with Carlson, Grodal and Nakano) about endotrivial modules for some "very important" finite groups.

Title: Selected topics on the number of conjugacy classes in a finite group

Abstract: The number k(G) of conjugacy classes in a finite group G plays an important role in both group theory and representation theory.

In this talk, we will discuss a few topics on this invariant including lower bounds for k(G), upper bounds for k(G), and the commuting probability.

Title: Primitive amalgams and the Goldschmidt-Sims conjecture

Abstract: The Classification of Finite Simple Groups has led to substantial progress on deriving sharp order bounds in various natural families of finite groups. One of the most well-known instances of this is Sims' conjecture, which states that a point stabiliser in a primitive permutation group has order bounded in terms of its smallest non-trivial orbit length (this was proved by Cameron, Praeger, Saxl and Seitz using the CFSG in 1983). In the meantime, Goldschmidt observed that a generalised version of Sims' conjecture, which we now call the Goldschmidt--Sims conjecture, would lead to important applications in graph theory. In this talk, we will describe the conjecture, and discuss some recent progress. Joint work with L. Pyber.

Title: The Orbital Diameter of Primitive Permutation Groups

Abstract: Let G be a group acting transitively on a finite set Ω. Then G acts on ΩxΩ component wise. Define the orbitals to be the orbits of G on ΩxΩ. The diagonal orbital is the orbital of the form ∆ = {(α, α)|α ∈ Ω}. The others are called non-diagonal orbitals. Let Γ be a non-diagonal orbital. Define an orbital graph to be the non-directed graph with vertex set Ω and edge set (α,β)∈ Γ with α,β∈ Ω. If the action of G on Ω is primitive, then all non-diagonal orbital graphs are connected. The orbital diameter of a primitive permutation group is the supremum of the diameters of its non-diagonal orbital graphs.

There has been a lot of interest in finding bounds on the orbital diameter of primitive permutation groups. In my talk I will outline some important background information and the progress made towards finding explicit bounds on the orbital diameter. In particular, I will discuss some results on the orbital diameter of the groups of simple diagonal type and their connection to the covering number of finite simple groups. I will also discuss some results for affine groups, which provides a nice connection to the representation theory of quasisimple groups.

Title: Some problems on representations of simple algebraic groups

Abstract: Some open questions on the weight structure of tensor-decomposable representations of simple algebraic groups will be discussed.

Title: The Boston--Shalev conjecture for conjugacy classes

Abstract: The Boston--Shalev conjecture (proved by Fulman and Guralnick in 2015) asserts that in any nonabelian simple group G in any nontrivial permutation action the proportion of derangements is at least some absolute constant c > 0. Since the set of derangements is closed under conjugacy it is also natural to ask about the proportion of *conjugacy classes* containing derangements. It is easy to see that this version of the question has a negative answer for alternating groups, but Guralnick and Zalesski asked whether it holds for groups of Lie type. I will outline a proof. We can also (1) extend to the case of almost simple groups, which is not true for the original conjecture, and (2) deduce the original conjecture, which amounts to a simplification of the Fulman--Guralnick proof. The key turns out to be a kind of analytic number theory for palindromic polynomials. This is ongoing work with Daniele Garzoni.

Title: Bounding Character Values at Regular Semisimple Elements

Abstract: If $\chi : G \to \mathbb{C}$ is an irreducible character of a finite group $G$ then one can ask how large the absolute value $|\chi(g)| \in \mathbb{R}$ at a given element $g \in G$ is. Typically one wants to obtain some $C \in \mathbb{R}$ such that the bound $|\chi(g)| < C$ holds as $g$ varies over a suitable subset of elements. In this talk we’ll try to explain why this question is important and discuss recent joint work with P. H. Tiep and M. Larsen obtaining new bounds in the case where $G$ is a finite reductive group and $g$ varies over the regular semisimple elements.

Title: Knutson's Conjecture on the representation ring

Abstract: Donald Knutson proposed the conjecture, later disproven and refined by Savitskii, that for every irreducible character of a finite group, there existed a virtual character such their tensor product was the regular character. In this talk, we disprove both this conjecture and its refinement. We then introduce the Knutson Index as a measure of the failure of Knutson's Conjecture and discuss its algebraic properties.

Title: Comparing $\mathfrak{sl}_2$-subalgebras and $\text{SL}_2$-subgroups of classical algebraic groups

Abstract: Let $G$ be a classical algebraic group, and take $\mathfrak{g}$ to be its Lie algebra. There is a Springer isomorphism from the variety of unipotent elements in $G$ to the unipotent elements in $\mathfrak{g}$. Hence it is natural to consider to what extent results which hold for nilpotent elements in $\mathfrak{g}$ hold for unipotent elements in $G$. In this talk I will discuss when there is a unique $\mathfrak{sl}_2$-subalgebra for a given nilpotent orbit in $\mathfrak{g}$, and compare this to when there is a unique $\text{SL}_2$-subgroup for a given unipotent orbit in $G$.