(Vasudevan talk) In 2010, de Cataldo-Miglorini showed how to compute the perverse filtration on the cohomology of an affine variety with values in a constructible sheaf using generic flags. In this talk, I shall introduce the Brylinski-Radon transformation, discuss its properties and derive consequences for the perverse filtration. Time permitting we shall also discuss some arithmetic applications of our results. This is joint work with Ankit Rai.

(Ure talk) The celebrated Bloch-Kato conjecture implies that the cohomology ring of a field with coefficients in a finite cyclic group is generated in degree one. In this talk, I will present a systematic approach to understanding this phenomenon in finite field extensions. Furthermore, I will explore other groups exhibiting this property and characterize all such groups that are finite. This talk is based on joint work with Sunil Chebolu, Ján Mináč, Cihan Okay, and Andrew Schultz.

(Frei talk) The rationality problem in algebraic geometry asks whether a given algebraic variety can be parameterized by algebraic functions. A natural question in the study of rationality is the following: is rationality a deformation invariant in smooth families? In arithmetic families, this question asks how the rationality of a variety defined over ℚ interacts with the rationality of its modulo p reductions for various primes p. In this talk, I'll discuss work in progress with Asher Auel and Alena Pirutka on algebraic varieties over ℚ that are not rational over the complex numbers but are rational modulo p for infinitely many primes p.

(Haines talk) The theory of local models has been a very successful tool for the study of Shimura varieties with parahoric level structure, and the theory is now very developed in that setting. For level structure which is deeper than Iwahori level, many complications arise, and the subject is in its infancy. I will review the basic theory of local models for Iwahori level, concentrating on the cases related to general linear and symplectic groups. A second goal will be to explain what can be said about local models when the level structure is Γ₁(p), which is slightly deeper than Iwahori level. For PEL Shimura varieties of Siegel type, I will define the local models using a linear algebra incarnation of Oort-Tate generators of finite flat group schemes of order p, and then, if time, I will explain how one uses a variant of Beilinson-Drinfeld Grassmannians and Gaitsgory's central functor adapted to pro-p Iwahori level, to study the nearby cycles on the special fibers. This is based on joint work in progress with Qihang Li and Benoit Stroh.

(Chebolu talk) Inspired by the classical Cauchy functional equation, we study the following problem: given a field F, classify all functions f from F to F that satisfy f((x + y)/(x - y)) = (f(x) + f(y))/(f(x) - f(y)) for all x ≠ y in F. We give a complete solution to this problem and, more generally, classify all maps that are equivariant with respect to a Mobius transformation of the form (ax + by)/(cx + dy) on any field. Our results yield a new characterization of the field with five elements and the introduction of a novel group of transformations on a field that contains the automorphism group as a subgroup. This is joint work with Jonathan Love, Apoorva Khare, Anindya Sen, and Akaki Tikaradze.

(Singh talk) TBA

(Joshua talk) TBA