School of Mathematics

In this talk, we construct a Zariski open dense locus in Mg

on which the Beilinson-Bloch height of the Gross-Schoen and Ceresa cycles is a Weil height, i.e. it has a lower bound and satisfies the Northcott property. This implies a generic positivity...

Applications of o-minimality to unlikely intersection problems usually begin with the observation that the relevant analytic covering maps are definable.  However, this observation is almost never literally true in that the maps are definable on a...

Which complex analytic spaces can arise as the universal cover of a complex algebraic variety? Motivated by this question, Shafarevich asked whether the universal cover of a smooth projective variety X is always holomorphically convex — that is...

Why is it so hard to prove P != NP, or even to prove super-linear circuit lower bounds? While we often blame a lack of combinatorial ingenuity, the bottleneck might be more fundamental: the logical strength of our mathematical tools.

This series of...

Given an algebraic variety over a number field F, one can attach to it its etale cohomology groups, etale fundamental group, and higher etale homotopy groups, all equipped with an action of the absolute Galois group of F. The Galois action on etale...

We will discuss a proof that the integral Hodge conjecture is false for a very general abelian variety of dimension ≥ 4. Associated to any regular matroid is a degeneration of principally polarized abelian varieties. We introduce a new combinatorial...

We revisit the question of whether it is possible to build succinct non-interactive arguments (SNARGs) for all of NP under standard cryptographic assumptions. In particular, we give a candidate non-adaptive SNARG for NP and prove its soundness under...

A result of Simpson characterizes subspaces that are (irreducible) algebraic varieties on both sides of the rank one Riemann-Hilbert correspondence. This result suggests an underlying Ax-Schanuel statement. In joint work with Jacob Tsimerman we...

We prove that in any dimension n there exists an origin-symmetric ellipsoid of volume c n^2 that contains no points of Z^n other than the origin. Here c > 0 is a universal constant. Equivalently, there exists a lattice sphere packing in R^n whose...

In this talk, I will discuss the barcode entropy—the exponential growth rate of the number of not-too-short bars—of the persistence module associated with the relative symplectic cohomology SHM(K) of a Liouville domain K embedded in a symplectic...