Department of Mathematics

Harvard University

sli@math.harvard.edu

I am a third-year graduate student at Harvard University advised by Mark Kisin.

For a year, I was a graduate student at the University of Chicago. Before that, I was an undergraduate at Princeton University, where my senior thesis was advised by Sophie Morel.

Let *Q* be a global field of positive characteristic. We prove the plectic conjecture of Nekovář–Scholl for moduli spaces of shtukas over *Q*. For example, when the cocharacter is defined over *Q* and the structure group is a Weil restriction from a degree *d* separable extension *F/Q* with the same constant field, we construct an action of (S_{d} Weil(*F*)^{d})^{I} on the ℓ-adic intersection cohomology with compact support of the associated moduli space of shtukas over *Q*_{I}. This extends the action of Weil(*Q*)^{I} constructed by Xue along the *I*-fold product of the map Weil(*Q*) → S_{d} Weil(*F*)^{d}. We show that the action of (S_{d} Weil(*F*)^{d})^{I} commutes with the Hecke action, and we give a moduli-theoretic description of the action of Frobenius elements in Weil(*F*)^{d×I}.

Let *F* be a local field of characteristic *p* > 0. By adapting methods of Scholze, we give a new proof of the local Langlands correspondence for GL_{n} over *F*. More specifically, we construct ℓ-adic Galois representations associated with many discrete automorphic representations over global function fields, which we use to construct a map π→rec(π) from isomorphism classes of irreducible smooth representations of GL_{n}(*F*) to isomorphism classes of *n*-dimensional semisimple continuous representations of *W*_{F}. Our map rec is characterized in terms of a local compatibility condition on traces of a certain test function *f*_{τ,h}, and we prove that rec equals the usual local Langlands correspondence (after forgetting the monodromy operator).

- Introduction to étale cohomology
- From elliptic modules to excursion operators
- Almost étale algebras
- Automorphic representations and the Ruziewicz problem

- Functional Analysis (Charlie Smart)
- Height Functions in Number Theory (Kazuya Kato)
- Geometric Satake (Victor Ginzburg)