School of Mathematics

Since Hironaka's famous resolution of singularities in characteristics zero in 1964, it took about 40 years of intensive work of many mathematicians to simplify the method, describe it using conceptual tools and establish its functoriality. However...

High-dimensional expanders (HDX) are a generalization of expander graphs which have seen various applications in coding theory, PCPs, pseudorandomness, derandomization, approximate sampling, and beyond. One technique for proving a complex is an HDX...

The Poisson summation conjecture of Braverman-Kazhdan, L. Lafforgue, Ngo, and Sakellaridis is an ambitious proposal to prove analytic properties of quite general Langlands L-functions using vast generalizations of the Poisson summation formula. In...

Many geometric spaces carry natural collections of special submanifolds that encode their internal symmetries. Examples include abelian varieties and their sub-abelian varieties, locally symmetric spaces with their totally geodesic subspaces, period...

Ben Bakker explained to us how to construct moduli spaces of polarized Hodge structures, and then produced period maps associated to families of smooth projective varieties. However, in practice one often encounters a family of smooth varieties...

In this talk I will describe new results establishing the existence of a strong spectral gap for  geometrically finite, n dimensional, hyperbolic manifold with critical exponent greater than (n−1)/2. This settles a conjecture of Mohammadi and Oh...

Last week we defined the Linial--Meshulam model for random 2 dimensional simplicial complexes and discussed two notions of connectivity for it: Vanishing of its 1st cohomology with F2 coefficients, and vanishing of its fundamental group. This time...

In this talk, I will explore the fascinating landscape of assumptions in quantum cryptography—especially, how little we need to assume to build secure quantum protocols. We will cover key cryptographic primitives including quantum encryption...

We compute the subleading asymptotics of the ECH and elementary ECH capacities of toric domains and show that they recover the perimeter in the liminf, without any genericity required.  As an application, we give the first examples of the failure of...