L4

Let G be a finite Chevalley group, i.e., the group of F_q points of a reductive group over F_q. Virtual representations of G in defining characteristic can be constructed in two ways, either by Brauer-Nesbitt reduction of complex representations, or by restricting an algebraic representation. G. Lusztig conjectured the shape of formulas connecting the two procedures; I will discuss a realization of his proposal related to decomposition of the class of diagonal for G/B coming from summands in the push-forward of the structure sheaf under Frobenius. 

Time permitting I will discuss a different, unrelated at present, way to describe such virtual representations linking it to homology of an affine Springer fiber. This found application in the work of Tony Feng and Viet Bao Le Hung on Breuil-Mezard conjectures. 

Based on joint works with Finkelberg, Kazhdan and Morton-Ferguson and with Boixeda Alvarez, McBreen and Yun respectively.

We describe the solution to two problems concerning indefinite integral ternary quadratic forms. The first about anisotropic forms was popularized by Margulis following his solution of the Oppenheim Conjecture. The second about the density of isotropic forms was raised by Serre. Joint work with A. Gamburd, A. Ghosh and J. Whang.

It is known for a long time, due to a celebrated theorem of Ornstein and Weiss, that (classical/plain) orbit equivalence offers no information about ergodic probability measure preserving actions of amenable groups. On the other hand, conjugacy is too intractable, and effectively hopeless to study in full generality. Quantitative orbit equivalence aims to bridge this gap by adding intermediate layers of rigidity— a strategy that has borne fruit already in the late 1960s but was used as a general framework only semi-recently. In this talk, Spyridon Petrakos will introduce aspects of quantitative orbit equivalence and present a complete picture of it for integer odometers. This is joint work with Petr Naryshkin.

I will discuss some general PDE techniques, for example ``extended substitute kernel'' and ``linearized gluing'', and their application to various gluing constructions in Differential Geometry.

I will review how the large N expansion can be used in the context of the renormalisation group to probe some strongly coupled regimes. In particular, I will discuss a work by Gawedzki and Kupiainen where the authors study the three-dimensional non-Gaussian infrared fixed point of Phi^4 in the case of a hierarchical model of rank-one covariance, and explain how their approach could generalise to more realistic models. 

This is a joint work with Ajay Chandra.  

Frobenius’ theorem in differential geometry asserts that, given a smooth manifold $M,$ every involutive subbundle $E \subset T_M$ determines a decomposition of $M$ into smooth leaves tangent to $E$. I will explain an infinitesimal analogue of this integration phenomenon for suitably nice schemes over coherent base rings, and then discuss an application. This talk is based on joint work with Magidson and Nuiten and ties into the work of Jiaqi Fu.

Given a Fano variety X with smooth anticanonical divisor D, one may consider the enumerative geometry of X, of the pair (X,D) or of D. A-model Quantum Lefschetz, Residue and Serre relate counts of genus 0 curves in X,  (X,D) and D. While the A-model statements are fairly involved, they become standard integral transforms when formulated as B-model correspondences within the Intrinsic Mirror Construction of Gross-Siebert. I will explain how this works. Time permitting, I will explain how for K-polystable del Pezzo surfaces, genus 0 log BPS instanton expansions transform into modular forms.