Video Lectures

We will present two results in complex geometry: (1) A Kähler compactification of ℂn with a smooth divisor complement must be ℙn, which confirms a conjecture of Brenton and Morrow under the Kähler assumption; (2) Any complete asymptotically conical...

The construction by Mikhalkin of a non-planar tropical cubic curve in R^3 of genus 1 marked a significant breakthrough in the study of combinatorial tropical varieties. It was the first known example of a non-realizable tropical variety, with the...

The conjecture in combinatorics that has received perhaps the most attention over the last 50 years is McMullen's g-conjecture. It provides a complete characterisation of the number of i-dimensional faces in a triangulation of an (d−1)-dimensional...

The reverse Khovanskii-Teissier inequality is a three term inequality for nef divisors which first appeared in the context of Kähler geometry. It provides an upper bound on a product of two divisors in terms of products with the third, hence its...

Consider a contact three-manifold, and within it a symplectic surface with boundary on Reeb orbits. We show that assuming a certain inequality on the rotation numbers of the boundary Reeb orbits, there must exist Reeb orbits which intersect the...

Every word in a free group induces a probability distribution on every finite group by substituting the letters of w by uniformly random elements of the group. The connection between such distributions on the symmetric group and the poset of...

Pretannakian categories, that is k-linear abelian categories with finite dimensional Hom groups, given with a commutative and associative tensor product, with a unit object (such that End(1)=k) and duals, can be viewed as generalizations of linear...

What can be said about a system of polynomial equations over a finite field if its number of solutions deviates from that of a random system? Answer: Roughly, one of the polynomials (or a linear combination thereof) can be written in terms of “few”...

I will describe a new lower bound on the number of intersection points of a Lagrangian pair, in the exact setting, using Steenrod squares on Lagrangian Floer cohomology which are defined via a Floer homotopy type.

In contact geometry, a systolic inequality aims to give a uniform upper bound on the shortest period of a periodic Reeb orbit for contact forms with fixed volume on a given manifold. This generalizes a well-studied notion in Riemannian geometry. It...