L4

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.

Given a Galois representation attached to a regular algebraic cuspidal automorphic representation, the Hodge--Tate weight of the Galois representation is matched with the weight of the automorphic representation. Serre weight conjectures are mod p analogue of such a correspondence, relating ramification at p of a mod p Galois representation and Serre weights of mod p algebraic automorphic forms. In this talk, I will discuss how to understand Serre weight conjectures and modularity lifting as a relationship between representation theory of finite groups of Lie type (e.g. GSp4(Fp)) and the geometry of p-adic local Galois representations. Then I will explain the proof idea in the case of GSp4. This is based on a joint work with Daniel Le and Bao V. Le Hung.