Given a reductive group 𝐺 and some auxiliary data, one has the Hitchin fibration associated with the adjoint action of 𝐺 on Lie(𝐺), which is successfully used by B. C. Ngô to prove the endoscopic fundamental lemma for Lie algebras. Following the same idea, there is a group analogue called the multiplicative Hitchin fibration by replacing the Lie algebra with reductive monoids, and one can hope to directly prove the fundamental lemma at group level. This project is almost complete and we report the results so far. There are many new features that are not present in the additive case, among which is a pleasant surprise that there might be some strata in the support theorem that are not explained by endoscopy.

We start by recalling gauge theory and some of its applications in low-dimensional topology. We briefly discuss Donaldson-Thomas program to extend the methods of gauge theory to study higher-dimensional manifolds, specially Calabi-Yau 3-folds and G2-manifolds. Finally, we will see that the study of gauge theory in higher dimensions motivates new ideas and questions in low-dimensional topology.

Let $F$ be a number field with ring of adeles $\mathbb{A}_F$. Let $r_1,r_2,r_3$ be a triple of positive integers and let $\pi:=\otimes_{i=1}^3\pi_i$ where the $\pi_i$ are all cuspidal automorphic representations of $\mathrm{GL}_{r_i}(\mathbb{A}_F)$. We denote by $L(s,\pi, \otimes^3)=L(s, \pi_1\times \pi_2 \times \pi_3)$ the corresponding triple product $L$-function. It is the Langlands $L$-function defined by the tensor product representation $\otimes^3:{}^L(\mathrm{GL}_{r_1} \times \mathrm{GL}_{r_2} \times \mathrm{GL}_{r_3}) \to \mathrm{GL}_{r_1r_2r_3}(\mathbb{C})$. In this talk I will present a family of Eulerian period integrals, which are holomorphic multiples of the triple product -function in a domain that nontrivially intersects the critical strip. We expect that they satisfy a local multiplicity one statement and a local functional equation. This is joint work with Jayce Getz, Chun-Hsien Hsu and Spencer Leslie.

This talk is an overview of my thesis work, which consists of 3 projects exploring the effect of multiscale structure on a class of interacting particle systems called weakly interacting diffusions. In the absence of multiscale structure, we have a collection of N particles, with the dynamics of each being described by the solution to a stochastic differential equation (SDE) whose coefficients depend on that particle's state and the empirical measure of the full particle configuration. It is well known in this setting that as N approaches infinity, the particle system undergoes the ``propagation of chaos,'' and its corresponding sequence of empirical measures converges to the law of the solution to an associated McKean-Vlasov SDE. Meanwhile, in our multiscale setting, the coefficients of the SDEs may also depend on a process evolving on a timescale of order 1/\epsilon faster than the particles. As \epsilon approaches 0, the effect of the fast process on the particles' dynamics becomes deterministic via stochastic homogenization. We study the interplay between homogenization and the propagation of chaos via establishing large deviations and moderate deviations results for the multiscale particles' empirical measure in the combined limit as N approaches infinity and \epsilon approaches 0. Along the way, we derive rates of homogenization for slow-fast McKean-Vlasov SDEs.