Nearby cycles of parahoric shtukas, and a fundamental lemma for base change

Using the Langlands-Kottwitz paradigm, we compute the trace of Frobenius composed with Hecke operators on the cohomology of nearby cycles, at places of parahoric reduction, of perverse sheaves on certain moduli stacks of shtukas. Following an argument of Ngo, we then use this to give a geometric proof of a base change fundamental lemma for parahoric Hecke algebras for GL_n over local function fields. This generalizes a theorem of Ngo, who proved the base change fundamental lemma for spherical Hecke algebras for GL_n over local function fields, and extends to positive characteristic (for GL_n) a fundamental lemma originally introduced and proved by Haines for p-adic local fields.Comment: Final version, to appear in Selecta Mathematic

Guarantees of Riemannian Optimization for Low Rank Matrix Completion

We study the Riemannian optimization methods on the embedded manifold of low rank matrices for the problem of matrix completion, which is about recovering a low rank matrix from its partial entries. Assume $m$ entries of an $n\times n$ rank $r$ matrix are sampled independently and uniformly with replacement. We first prove that with high probability the Riemannian gradient descent and conjugate gradient descent algorithms initialized by one step hard thresholding are guaranteed to converge linearly to the measured matrix provided \begin{align*} m\geq C_\kappa n^{1.5}r\log^{1.5}(n), \end{align*} where $C_\kappa$ is a numerical constant depending on the condition number of the underlying matrix. The sampling complexity has been further improved to \begin{align*} m\geq C_\kappa nr^2\log^{2}(n) \end{align*} via the resampled Riemannian gradient descent initialization. The analysis of the new initialization procedure relies on an asymmetric restricted isometry property of the sampling operator and the curvature of the low rank matrix manifold. Numerical simulation shows that the algorithms are able to recover a low rank matrix from nearly the minimum number of measurements

Mirror symmetry and the Breuil-M\'ezard Conjecture

The Breuil-M\'{e}zard Conjecture predicts the existence of hypothetical "Breuil-Mezard cycles" in the moduli space of mod $p$ Galois representations of $\mathrm{Gal}(\overline{\mathbb{Q}}_q/\mathbb{Q}_q)$ that should govern congruences between mod $p$ automorphic forms. For generic parameters, we propose a construction of Breuil-M\'{e}zard cycles in arbitrary rank, and verify that they satisfy the Breuil-M\'{e}zard Conjecture for all sufficiently generic tame types and small Hodge-Tate weights. Our method is purely local and group-theoretic, and completely distinct from previous approaches to the Breuil-M\'ezard Conjecture. In particular, we leverage new connections between the Breuil-M\'ezard Conjecture and phenomena occurring in homological mirror symmetry and geometric representation theory.Comment: minor correction