University of California, San Diego.
Academic year: 2015-2016
Room: APM 7421 (unless it is announced otherwise)
Monday 2:15-2:45. A pre-talk to prepare graduate students.
Monday 3:00-4:00. We continue the fun with more details.

Date Speaker Topic
Feb 8 Igor Rapinchuk
Harvard University
Michigan State University
  Abstract homomorphisms of algebraic groups and applications


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.

Feb 22 Dan Rogalski
  Calabi-Yau algebras  


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.

Feb 29 Yago Antolin Pichel
Vanderbilt University
  Formal conjugacy growth and hyperbolicity


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.

Mar 3 Adam Chapman
Michigan State University
  Linkage of p-algebras of prime degree.


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.

Mar 7 Naser T. Sardari
Princeton University
  Optimal strong approximation for quadratic forms


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

  • Fix any compact subspace S ⊂ Rd of the affine quadric F(x1,..., xd)=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,..., λd) mod m.

Then we show that

  • there exists an integral solution x=(x1,...,xd) of F(x)=N such that xi ≡ λi 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.

Mar 14 Yuri Bakhturin
Memorial University of Newfoundland