Abstract:

I will discuss several recent results on abstract homomorphisms between the groups of rational points of algebraic groups. The main focus will be on a conjecture of Borel and Tits formulated in their landmark 1973 paper. Our results settle this conjecture in several cases; the proofs make use of the notion of an algebraic ring. I will conclude by discussing several applications to character varieties of finitely generated groups and group actions.

Our speaker kindly accepted to give a pre-talk. In the pre-talk he will recall some basic concepts from the theory of algebraic groups and outline a general philosophy for the study of rigidity phenomena between the groups of rational points of algebraic groups.

Abstract:

Twisted Calabi-Yau algebras are a class of algebras with nice behavior regarding their Hochschild cohomology. They include many classes of examples of recent interest, for example Artin-Schelter regular algebras. We discuss in particular the theory of twisted Calabi-Yau algebras of low global dimension which are factors of path algebras of quivers Q. For example, we have preliminary results regarding the following question: for which quivers Q does there exists a twisted Calabi-Yau algebra of dimension 3 which is a factor of the path algebra of Q?

In the pre-talk, we will give an introduction to some techniques from homological algebra, in particular Hochschild cohomology, which are relevant to the talk.

  Abstract:

Rivin conjectured that the formal conjugacy growth series of an hyperbolic group is rational if and only if the group is virtually cyclic. In this talk, I will present a proof of Rivin's conjecture and supporting evidence for the analogous statement for acylindrically hyperbolic groups. The class of acylindrically hyperbolic groups is a wide class of groups that contains (among many other examples) the outer automorphism groups of free groups and the mapping class groups of hyperbolic sufaces. This is a joint work with Laura Ciobanu.

  Abstract:

Quaternion algebras contain quadratic field extensions of the center. Given two algebras, a natural question to ask is whether they share a common field extension. This gives us an idea of how closely related those algebras are to one another. If the center is of characteristic 2 then those extensions divide into two types - the separable type and the inseparable type. It is known that if two quaternion algebras share an inseparable field extension then they also share a separable field extension and that the converse is not true. We shall discuss this fact and its generalization to p-algebras of arbitrary prime degree.

Pre-talk will be in APM 7218.

  Abstract:

For a non-degenerate integral quadratic form F(x₁,..., x_d) in 5 (or more) variables, we prove an optimal strong approximation theorem.

• Fix any compact subspace S ⊂ R^d of the affine quadric F(x₁,..., x_d)=1.
• Suppose that we are given a small ball B of radius 0< r < 1 inside S, and an integer m.
• Assume that N is a given integer which satisfies N ≥(r^-1m)^{4+ ε} for any ε > 0.
• Assume that we are given an integral vector (λ ₁,..., λ_d) mod m.

Then we show that

• there exists an integral solution x=(x₁,...,x_d) of F(x)=N such that x_i ≡ λ_i mod m and x/√N∈ B, provided that all the local conditions are satisfied.
• 4 is the best possible exponent.

Moreover, for a non-degenerate integral quadratic form F(x₁,..., x₄) in 4 variables we prove the same result if N ≥(r^-1m)^{6+ ε} and some non-singular local conditions for N are satisfied.

Based on some numerical experiments on the diameter of LPS Ramanujan graphs, we conjecture that the optimal strong approximation theorem holds for any quadratic form F(x) in 4 variables with the optimal exponent 4.

  Abstract:

TBA.