# Video Lectures

The Markoff equation x2+y2+z2=3xyz, which arose in his spectacular thesis (1879), is ubiquitous in a tremendous variety of contexts. After reviewing some of these, we will discuss joint work with Bourgain and Sarnak establishing forms of strong...

In this talk, we prove the invariance of the Gibbs measure for the three-dimensional cubic nonlinear wave equation, which is also known as the hyperbolic Φ43-model.

In the first half of this talk, we illustrate our main objects and questions...

Quantum extremal surfaces and islands have led us to rethink the very meaning of microscopic entropy in the presence of dynamical gravity. I will present some work elucidating how the island/QES prescription emerges in a simple doubly holographic...

Training a predictor to minimize a loss function fixed in advance is the dominant paradigm in machine learning. However, loss minimization by itself might not guarantee desiderata like fairness and accuracy that one could reasonably expect from a...

Because of the presence of non-trivial automorphisms of stable maps, Gromov-Witten invariants of a general symplectic manifold are usually rational-valued. Realizing a proposal of Fukaya-Ono back in the 1990s, I will explain how to construct integer...

For any regularity exponent β<12, we construct non-conservative weak solutions to the 3D incompressible Euler equations in the class C0t(Hβ∩L1(1−2β)). By interpolation, such solutions belong to C0tBs3,∞ for s approaching 13 as β approaches 12. Hence this result provides a new proof of the flexible side of the Onsager conjecture, which is independent from that of Isett. Of equal importance is that the intermittent nature of our solutions matches that of turbulent flows, which are observed to possess an L2-based regularity index exceeding 13. The proof employs an intermittent convex integration scheme for the 3D incompressible Euler equations. We employ a scheme with higher-order Reynolds stresses, which are corrected via a combinatorial placement of intermittent pipe flows of optimal relative intermittency.

I will present two constructions of Kähler manifolds, endowed with Hamiltonian torus actions of infinite dimension. In the first example, zeroes of the moment map are related to isotropic maps from a surfaces in ℝ2n. In the second example, which is...

In his seminal work, Leray demonstrated the existence of global weak solutions, with nonincreasing energy, to the Navier-Stokes equations in three dimensions. In this talk we exhibit two distinct Leray solutions with zero initial velocity and...

We determine the average size of the 3-torsion in class groups of G-extensions of a number field when G is any transitive 2-group containing a transposition, for example, D4. It follows from the Cohen--Lenstra--Martinet heuristics that the average...

Whether the 3D incompressible Euler equations can develop a finite-time singularity from smooth initial data is an outstanding open problem. The presence of vortex stretching is the primary source of a potential finite-time singularity. However, to...