Video Lectures

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...

In this talk, I will present joint work with Charles Bordenave and Mostafa Sabri establishing quantum ergodicity and quantum weak mixing for sequences of finite Schreier graphs converging, in the Benjamini–Schramm sense, to an infinite Cayley graph...

The regularity lemma says that every discrete object can be partitioned into a small number of random-like subobjects. But how small is small? And can we make small smaller if we assume that our given object is simple? And what does it mean for a...

Essentially optimal estimates have been obtained for mean values of Vinogradov’s exponential sum as a consequence of the decoupling method (by Bourgain, Demeter and Guth), and the efficient congruencing method (by the speaker). Such work makes...

The very first result ever proved about tournaments is due to Rédei, who nearly 100 years ago proved that every tournament contains a Hamiltonian directed path. Since then, questions and results about directed paths in tournaments have become a...

In this talk I will introduce the idea of Floer homotopy theory and show how it can be used to give lower bounds on degenerate Lagrangian intersections, in the case of plumbings of cotangent bundles along a submanifold. The strength of the invariant...

In this talk, we begin by recalling Arnold’s geometric formulation of hydrodynamics and then extend this framework to a broader class of Hamiltonian systems, incorporating various PDEs arising in mathematical physics. This motivates the study of...

We consider the symplectic area functional, constrained to loops of vanishing Hamiltonian mean value: It has the same critical points as the Rabinowitz action functional, and can be used to define a similar Floer homology. In contrast to RFH, it...

Consider a collection of forms of odd degree with rational coefficients. Birch proved in 1957 that if the number of variables is sufficiently large, then the forms must have a nontrivial rational zero. The bounds resulting from Birch's proof...