Department of Mathematics

2-252B

sdlh@mit.edu

I am an NSF postdoc at MIT. My CV is here.

I received my PhD from Harvard University in 2023, under the supervision of Mark Kisin.

I am currently co-organizing the MIT Number Theory Seminar.

For the 2024–2025 academic year, I will be visiting the Max Planck Institute for Mathematics.

Cuspidal automorphic representations τ of PGL_{2} correspond to global long root *A*-parameters for G_{2}. Using an exceptional theta lift between PU_{3} and G_{2}, we construct the associated global *A*-packet and prove the Arthur multiplicity formula for these representations when τ is dihedral and satisfies some technical hypotheses. We also prove that this subspace of the discrete automorphic spectrum forms a full near equivalence class. Our construction yields new examples of quaternionic modular forms on G_{2}.

We prove that V. Lafforgue's global Langlands correspondence is compatible with Fargues–Scholze's semisimplified local Langlands correspondence. As a consequence, we canonically lift Fargues–Scholze's construction to a non-semisimplified local Langlands correspondence for positive characteristic local fields. We also deduce that Fargues–Scholze's construction agrees with that of Genestier–Lafforgue, answering a question of Fargues–Scholze, Hansen, Harris, and Kaletha. The proof relies on a uniformization morphism for moduli spaces of shtukas.

Using a mixed-characteristic incarnation of fusion, we prove an analog of Nekovář–Scholl's plectic conjecture for local Shimura varieties. We apply this to obtain results on the plectic conjecture for (global) Shimura varieties after restricting to a decomposition group. Along the way, we prove a *p*-adic uniformization theorem for the basic locus of abelian type Shimura varieties at hyperspecial level, which is of independent interest.

We prove the plectic conjecture of Nekovář–Scholl over global function fields *Q*. For example, when the cocharacter is defined over *Q* and the structure group is a Weil restriction from a geometric degree *d* separable extension *F/Q*, consider the complex computing ℓ-adic intersection cohomology with compact support of the associated moduli space of shtukas over *Q*_{I}. We endow this with the structure of a complex of (Weil(*F*)^{d} S_{d})^{I}-modules, which extends its structure as a complex of Weil(*Q*)^{I}-modules constructed by Arinkin–Gaitsgory–Kazhdan–Raskin–Rozenblyum–Varshavsky. We show that the action of (Weil(*F*)^{d} S_{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)

- From Fall 2022 to Spring 2023, I co-organized the Harvard Number Theory Seminar.