Video Lectures

An important open problem is to classify rational points on all modular curves, or equivalently to classify the possible adelic Galois images of elliptic curves over Q, as envisioned in Mazur’s Program B. However, this problem is inherently infinite...

I will discuss the bulk mechanism of the stringy exclusion principle in the AdS3/CFT2 case. Consider the duality between  string theory on AdS3 × S3 × M4 and Sym^N(M4). At finite N, both the chiral-chiral spectrum and the number of the polar terms...

A G-function, in the sense of Siegel, is a power series with rational coefficients that on the one hand has nice arithmetic properties (the LCM of the denominators of the first \(n\) coefficients grows at most exponentially) and on the other hand...

The nature of mathematical thought has been a central concern in the development of artificial intelligence since the creation of the computer in the 1940s. The expansion of AI in the past decade appears to have sharply reduced the distinction...

Let A be a simple abelian variety, and X and Y be two subvarieties. We say X and Y are in the Bezout range if \dim X + \dim Y >= \dim A, and outside of the Bezout range otherwise. It is known that two varieties in the Bezout range in A always...

Let G be a perfect Lie group, Γ be a lattice in G and U be a unipotent subgroup of G. A celebrated theorem of Ratner says that for any x in G/Γ the orbit U.x is equidistributed in a periodic orbit of some subgroup U≤L≤G. Establishing a quantitative...

For the passive vector equation, the fast dynamo conjecture predicts exponential-in-time growth of the L2 norm of the solution under the Lipschitz flow generated by a vector field, at a rate independent of the resistivity. We prove this conjecture...

Rieser and I introduced a variation on the Hofer-Zehnder capacity, relative to a coisotropic submanifold. Revisiting this construction in the case in which the coisotropic submanifold is Lagrangian, we use a persistent homology approach to filtered...

The theory of higher Reidemeister torsion yields characteristic classes of (stable) fiber bundles of smooth manifolds. We use this theory to define a new family of invariants for Legendrians in 1-jet spaces which we collectively call Legendrian...