Abstract: In 1966 Shafarevich introduced the notion of "infinite dimensional algebraic group", shortly "ind-group". His main application was the automorphism group of affine n-space Aⁿ for which he claimed some interesting properties. Recently, jointly with J.-Ph. Furter we showed that the automorphism group of any finitely generated (general) algebra has a natural structure of an ind-group, and we further developed the theory.

It turned out that some properties well-know for algebraic groups carry over to ind-groups, but others do not. E.g. every ind-group has a Lie algebra, but the relation between the group and its Lie algebra still remains unclear. As another by-product of this theory we get new interpretations and a better understanding of some classical results, together with short and transparent proofs.

An interesting "test case" is Aut(A²), the automorphism group of affine 2-space, because this group is the amalgamated product of two closed subgroups which implies a number of remarkable properties. E.g. a conjugacy class of an element g in Aut(A²) is closed if and only if g is semi-simple, a result well-known for algebraic groups. A generalization of this to higher dimensions would have very strong and deep consequences, e.g. for the linearization problem.

For the pre-talk speaker kindly accepted to tell our graduate students What an ind-group is.

Abstract:The classic Linear Preserver Problem asks to determine, for a polynomial function f on a vector space V, the linear transformations g of V such that fg = f. In case f is invariant under a simple algebraic group G acting irreducibly on V , we note that the subgroup of GL(V) stabilizing f often has identity component G and we give applications realizing various groups, including the largest exceptional group E8, as automorphism groups of polynomials and algebras. We show that starting with a simple group G and an irreducible representation V, one can almost always find an f whose stabilizer has identity component G and that no such f exists in the short list of excluded cases. The main results are new even in the special case where the field is the complex numbers. This talk is about joint work with Bob Guralnick.

Abstract: Given a representation V of a group G, it is a classical problem to determine the centralizer of its action on the n-th tensor power of V. If V is the natural module of a classical Lie group, this led to the famous Schur-Weyl and Brauer-Weyl dualities.

In this talk, we solve this problem for the spinor representation S. For even-dimensional Spin groups, the centralizer on the n-th tensor power of S is given by a representation of SO(n), with a similar result also for the odd-dimensional Spin groups. Time permitting, we discuss generalizations of this result to quantum groups and to classification of tensor categories.

Abstract: Let Ω ⊂ GL_n(F_p(t)) be a finite symmetric set containing the identity. Let Γ be the group generated by Ω and let G be the Zariski-closure of Γ. We discuss conditions on which the family of Cayley graphs {Cay(Γ mod Q, Ω)} is a family of Cayley graphs as Q ranges through a certain subset Σ of F_p[t]. This problem is a positive characteristic variation of the work of Bourgain-Gamburd, Varju, Salehi Golsefidy-Varju, and others. We focus on the difficulties that arise in positive characteristic.

Abstract:The first Weyl algebra A = k<x,y>/(xy - yx - 1) is Z-graded with deg x = 1 and deg y = -1. Susan Sierra and Paul Smith studied the category of graded modules over A, showing that this category was equivalent to coherent sheaves on a certain quotient stack. In this talk, we investigate the graded module categories over Z-graded rings called generalized Weyl algebras. We construct commutative rings with equivalent graded module categories.

In the pre-talk, we will discuss some preliminaries on categories and graded rings before giving an overview of noncommutative projective geometry.

Abstract: We present an algebraic method to study four-dimensional toric varieties by lifting matrix equations from the special linear group SL(2,Z) to its preimage in the universal cover of SL(2,R). With this method we recover the classification of two dimensional toric fans, and obtain a description of their semitoric analogue. As an application to symplectic geometry of Hamiltonian systems, we give a concise proof of the connectivity of the moduli space of toric integrable systems in dimension four, recovering a known result, and extend it to the case of semitoric integrable systems with a fixed number of focus-focus points. (joint work with Daniel Kane and Alvaro Pelayo)

Abstract: Let G be a group acting (faithfully) on a set X. A base for this action is a subset Y of X so that no element of G fixes every element of Y. The question of what is the minimal size of a base is a classical subject going back to the early days of finite permutation group theory.

In this talk I will mostly focus on the case that G is simple algebraic group and X is an irreducible variety. A closely related problem is to determine a generic stabilizer (if it exists). Note that base size 1 is the same as saying some stabilizer is trivial (and indeed base size b on X is the same as saying base size 1 on b copies of X).

We will consider the case where X = G/H for some maximal closed subgroup H and for the case that X is an irreducible rational G-module. Even if one is only interested in the case of finite groups, these cases are relevant.

Some of this is joint work with Burness and Saxl, some with Lawther and some with Garibaldi.