Let $A(s) = \sum_n a_n n^{-s}$ be a Dirichlet series with meromorphic continuation. Say we are given information on the poles of $A(s)$ with $|\Im s| \leq T$ for some large constant $T$. What is the best way to use such finite spectral data to give explicit estimates for sums $\sum_{n\leq x} a_n$? The problem of giving explicit bounds on the Mertens function $M(x) = \sum_{n\leq x} \mu(n)$ illustrates how open this basic question was. Bounding $M(x)$ might seem equivalent to estimating $\psi(x) = \sum_{n\leq x} \Lambda(n)$ or the number of primes $\leq x$. However, we have long had fairly good explicit bounds on prime counts, while bounding $M(x)$ remained a notoriously stubborn problem. We prove a sharp, general result on sums $\sum_{n\leq x} a_n n^{-\sigma}$ for $a_n$ bounded, giving a optimal way to use information on the poles of $A(s)$ with $|\Im s|\leq T$ and no data on the poles above. Our bounds on $M(x)$ are stronger than previous ones by many orders of magnitude. We also give a sharp result on such sums for a_n non-negative and not necessarily bounded, and apply it to obtain optimal bounds on psi(x)-x given finite verifications of RH. Our proofs mixe a Fourier-analytic approach in the style of Wiener--Ikehara with contour-shifting, using optimal approximants of Beurling--Selberg type as in (Graham--Vaaler, 1981) and (Carneiro--Littmann, 2013). While our approach does not depend on existing explicit work in number theory, our method has an important step in common with work on another problem by (Ramana–Ramare, 2020).

A locally symmetric variety is a non-compact complex algebraic variety obtained as the quotient of a Hermitian symmetric domain by the action of an arithmetic group. I will start by reviewing the theory of toroidal compactifications of these varieties, originally due to Ash-Mumford-Rapoport-Tai. Building on this construction, we define the tropicalization of a locally symmetric variety to be a combinatorial (polyhedral) object encoding the boundary strata of a toroidal compactification of the variety. I will discuss applications of this theory to the cohomology of moduli spaces and arithmetic groups, with an emphasis on the case of moduli of abelian varieties and general linear groups. Based on joint work with Assaf, Brandt, Bruce, and Chan.

We study the problem of approximating the partition function of the transverse-field Ising model (TFIM), a widely studied quantum many-body model with important applications in quantum simulation and quantum annealing. Despite its fundamental importance, the algorithmic landscape for computing the TFIM partition function has remained poorly understood beyond restricted parameter regimes. We provide a precise characterization of the temperature regimes in which efficient approximation is possible, establishing a sharp computational phase transition. Let $J$ denote the symmetric interaction matrix and $\Delta(J) = \lambda_{\max}(J)-\lambda_{\min}(J)$ denote its spectral width. We show that, for all inverse temperatures $\beta \in [0,1/\Delta(J)]$, there exists an efficient classical randomized algorithm that approximates the partition function $\text{tr}(e^{-\beta H})$ to within an arbitrarily small multiplicative factor. To obtain this result, we apply the standard Trotter decomposition to map the quantum model to a classical spin system, and then leverage new techniques in Markov chain analysis to derive an efficient algorithm that samples from and computes the partition function of the resulting distribution.  This temperature threshold is tight: for $\beta > 1/\Delta(J)$, we show that approximating the partition function, even within an exponential factor, is NP-hard and thus is unlikely to admit an efficient classical or quantum algorithm. Joint work with Alistair Sinclair.

We study a continuum Lagrangian alignment system for interacting agents with weak initial data. We first prove global well-posedness of the Lagrangian dynamics and derive quantitative flocking estimates. We then pass from the Lagrangian description to an Eulerian one, and obtain an Euler-Reynolds-alignment system involving a nonnegative Reynolds stress and, in the nonlinear velocity-coupling case, an additional defect force caused by microscopic velocity fluctuations. Under a heavy-tailed interaction assumption, we show that these defect terms vanish asymptotically, leading to mono-kinetic closure at large times. In the linear velocity-coupling case, we further prove the global existence of weak solutions to the Euler-alignment system, including a sharp critical-threshold result in one dimension and a global existence result in higher dimensions under a large-coupling condition. We also establish mean-field convergence results for the underlying particle system, including uniform-in-time convergence in the linear case.