We investigate equivalence relations for quadratic forms that can be expressed in terms of algebro-geometric properties of their associated quadrics, more precisely, birational, stably birational and motivic equivalence, and isomorphism of quadrics. We provide some examples and counterexamples and highlight some important open problems.

# Past Algebra Seminar

Quiver varieties, as introduced by Nakaijma, play a key role in representation theory. They give a very large class of symplectic singularities and, in many cases, their symplectic resolutions too. However, there seems to be no general criterion in the literature for when a quiver variety admits a symplectic resolution. In this talk I will give necessary and sufficient conditions for a quiver variety to admit a symplectic resolution. This result is based on work of Crawley-Bouvey and of Kaledin, Lehn and Sorger. The talk is based on joint work with T. Schedler.

If $R = F_q[t]$ is the polynomial ring over a finite field

then the group $SL_2(R)$ is not finitely generated. The group $SL_3(R)$ is

finitely generated but not finitely presented, while $SL_4(R)$ is

finitely presented. These examples are facets of a larger picture that

I will talk about.

I will report on recent progress towards understanding the growth of the torsion of the homology of subgroups of finite index in a given residually finite group G.

The cases I will consider are when G is amenable (joint work with P, Kropholler and A. Kar) and when G is right angled (joint work with M. Abert and T. Gelander).

Given a vector space V over a field K, let End(V) denote the algebra of linear endomorphisms of V. If V is finite-dimensional, then it is well-known that the diagonalizable subalgebras of End(V) are characterized by their internal algebraic structure: they are the subalgebras isomorphic to K^n for some natural number n.

In case V is infinite dimensional, the diagonalizable subalgebras of End(V) cannot be characterized purely by their internal algebraic structure: one can find diagonalizable and non-diagonalizable subalgebras that are isomorphic. I will explain how to characterize the diagonalizable subalgebras of End(V) as topological algebras, using a natural topology inherited from End(V). I will also illustrate how this characterization relates to an infinite-dimensional Wedderburn-Artin theorem that characterizes "topologically semisimple" algebras.

Let G be a transitive permutation group. If G is finite, then a classical theorem of Jordan implies the existence of fixed-point-free elements, which we call derangements. This result has some interesting and unexpected applications, and it leads to several natural problems on the abundance and order of derangements that have been the focus of recent research. In this talk, I will discuss some of these related problems, and I will report on recent joint work with Hung Tong-Viet on primitive permutation groups with extremal derangement properties.

We exhibit a construction in noncommutative nonnoetherian algebra that should be understood as a positive characteristic analogue of the Bernstein-Sato polynomial or b-function. Recall that the b-function is a polynomial in one variable attached to an analytic function f. It is well-known to be related to the singularities of f and is useful in continuing a certain type of zeta functions, associated with f. We will briefly recall the complex theory and then emphasize the arithmetic aspects of our construction.

The classification of irreducible representations of pin double covers of Weyl groups was initiated by Schur (1911) for the symmetric group and was completed for the other groups by A. Morris, Read and others about 40 years ago. Recently, a new relation between these projective representations, graded Springer representations, and the geometry of the nilpotent cone has emerged. I will explain these connections and the relation with a Dirac operator for (extended) graded affine Hecke algebras. The talk is partly based on joint work with Xuhua He.