Beijing Institute of Mathematical Sciences and Applications

We describe the solution to two problems concerning indefinite integral ternary quadratic forms. The first about anisotropic forms was popularized by Margulis following his solution of the Oppenheim Conjecture. The second about the density of isotropic forms was raised by Serre. Joint work with A.Gamburd, A.Ghosh and J.Whang.

Einstein's field equations relate the curvature of spacetime to the distribution of matter and energy. Since Ollivier's influential work on Ricci curvature for Markov chains on metric spaces, there has been renewed interest in synthetic and coarse notions of curvature. In this talk, I shall discuss constructions of coarse curvature in both Riemannian and Lorentzian geometry. In the Riemannian setting, I shall describe coarse analogues of intrinsic Ricci curvature and of extrinsic curvature for embedded submanifolds. I shall then turn to the Lorentzian setting and discuss coarse notions of Ricci curvature associated with timelike directions.

Several problems in additive combinatorics and Diophantine approximation can be reduced to the task of obtaining one-sided estimates for trigonometric sums. Such estimates are generally significantly harder to establish than their two-sided counterparts, and are deeply connected to the additive structure of the spectrum. I will discuss recently developed Fourier-analytic and combinatorial methods for obtaining them, and illustrate their applicability in resolving a longstanding question of Erdős on large sum-free subsets of sets of integers, obtaining polynomial bounds for Chowla’s cosine problem, and improving the best bound towards the Lonely Runner Conjecture.

In this talk, I will discuss recent results on the behavior of Dirichlet L-functions in families, focusing on simultaneous non-vanishing and value distribution questions. For simultaneous non-vanishing, I will consider two settings: families of twists of Dirichlet L-functions of prime moduli, and families arising from Galois orbits of Dirichlet characters. The main question I will focus on is whether one can find many members of these families whose L-functions are simultaneously nonzero at the central point. I will also discuss the simultaneous fluctuations of these L-functions, proving weighted central limit theorems that let us quantify how large or small several Dirichlet L-functions can be at the same time. This talk is based on joint works with Hung Bui and Micah Milinovich, and Hung Bui and Hieu Ngo.

This talk mainly reviews and introduces recent progress on the typical optimization of dynamical systems: the symbolic systems case. The contents include: the theory of maximizing sets in dynamical systems, for the study of ergodic optimization in systems with weak hyperbolicity, but where the Mañé cohomology lemma does not hold. The theory yields a structural theorem that isolates the part of the system responsible for any robust non-periodic optimization. The structural theorem is developed further in the setting of symbolic dynamics: given any shift space, for typical Lipschitz functons, the maximizing measure is shown to be either periodic or supported on the Markov boundary of the shift space. It follows that Contreras' Typical Periodic Optimization theorem for shifts of finite type can be extended to a wide class of shift spaces, including every sofic shift. This is joint work with Oliver Jenkinson, Leiye Xu, and Yiwei Zhang.

The Alon-Jaeger-Tarsi conjecture states that for any finite field $\mathbb{F}$ of size at least 4 and any nonsingular matrix $A$ over $\mathbb{F}$ there exists a vector $x$ such that neither $x$ nor $Ax$ has a 0 component. In this talk we discuss the proof of this result for $|\mathbb{F}|>79$ and further applications of our method about coset covers and additive bases. Joint work with János Nagy and István Tomon.

There is a close analogy between the theory of closed geodesics on a hyperbolic surface and prime number theory. This has been known since the 1950s. After surveying the two theories, I will discuss a recent perspective, when we consider the surface as being random in the moduli space of hyperbolic surfaces of large genus.

Given a bounded submanifold M in $R^n$, how many rational points with common bounded denominator are there in a small thickening of $M$? How does this counting function behave if we let the size of the denominator go to infinity? The study of the density of rational points near manifolds has seen significant progress in the last couple of years. In this talk I will explain why we might be interested in this question, focusing on applications in Diophantine approximation and the (quantitative) arithmetic of projective varieties.

I will discuss extending classical estimates for character sums over intervals to character sums over general additively structured sets. These problems turn out to be naturally connected to the sum-product phenomenon in additive combinatorics. In particular, I will describe a Burgess-type estimate for character sums over generalized arithmetic progressions (GAPs) of rank 2. A crucial ingredient in the proof is a sharp upper bound for the multiplicative energy of such GAPs. This is joint work with Ali Alsetri.