Chen and Ruan constructed symplectic orbifold Gromov-Witten invariants more than 20 years ago. In ongoing work with Alex Ritter, we show that moduli spaces of pseudo-holomorphic curves mapping to a symplectic orbifold admit global Kuranishi charts...

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School of Mathematics

What are the slopes of periodic billiard paths in a regular polygon?

We will connect this question and others to:

- cusps of thin groups,

- curves on Hilbert modular varieties,

- heights from Jacobians with real multiplication...

In this lecture, we will review recent works regarding spectral statistics of the normalized adjacency matrices of random $d$-regular graphs on $N$ vertices.

Denote their eigenvalues by $\lambda_1=d/\sqrt{d-1}\geq \la_2\geq\la_3\cdots\geq \la_N$...

Ratner's landmark equidstribution results for unipotent flows have had dramatic applications in many mathematical areas. Recently there has been considerable progress in the long sought for goal of getting effective equidistribution results for...

(Joint with Samuel Grushevsky, Gabriele Mondello, Riccardo Salvati Manni) We determine the maximal dimension of a compact subvariety of the moduli space of principally polarized abelian varieties Ag for any value of g. For g<16 the dimension is g−1, while for g≥16, it is determined by the larged dimensional compact Shimura subvariety, which we determine. Our methods rely on deforming the boundary using special varieties, and functional transcendence theory.

To study the asymptotic behavior of orbits of a dynamical system, one can look at orbit closures or invariant measures. When the underlying system has a homogeneous structure, usually coming from a Lie group, with appropriate assumptions a wide...

Starting with the "Leibniz" formula for π

π/4=1−1/3+1/5−1/7+…

the special values of Dirichlet L-functions have long been a source of fascination and frustration. From Euler's solution in 1734 of the Basel problem to Apery's proof in 1978 that zeta(3...

Cohen, Lenstra, and Martinet have given highly influential conjectures on the distribution of class groups of number fields, the finite abelian groups that control the factorization in number fields. Malle, using tabulation of class groups of number...

The Higher order Fourier uniformity conjecture asserts that on most short intervals, the Mobius function is asymptotically uniform in the sense of Gowers; in particular, its normalized Fourier coefficients decay to zero. This conjecture is known to...

What is the densest lattice sphere packing in the d-dimensional Euclidean space? In this talk we will investigate this question as dimension d goes to infinity and we will focus on the lower bounds for the best packing density, or in other words on...