Video Lectures

In 1986, Hooley applied (what practically amounts to) the general Langlands reciprocity (modularity) conjecture and GRH in a fresh new way, over certain families of cubic 3-folds. This eventually led to conditional near-optimal bounds for the number...

Let f=∑anxn∈ℚ[x] be a power series which is also a meromorphic function in some neighborhood of the origin. The subject of the talk will be how certain conditions on f(x) as a meromorphic function actually guarantee that f(x) is an algebraic...

In these two talks, I will describe how the classification of locally homogeneous geometric structures (closely related to flat connections) leads to interesting dynamical systems.

Many interesting dynamical systems arise from the classification of...

The 2011 PhD thesis of Farris outlined a scheme to show that the ECH of a prequantization bundle over a Riemann surface is isomorphic as a Z/2Z-graded group to the exterior algebra of the homology of its base. In addition to providing an overview of...

The unbounded denominators conjecture, first raised by Atkin and Swinnerton-Dyer, asserts that a modular form for a finite index subgroup of SL2(ℤ) whose Fourier coefficients have bounded denominators must be a modular form for some congruence...

Given a finite graph, the arboreal gas is the measure on forests (subgraphs without cycles) in which each edge is weighted by a parameter β>0. Equivalently this model is bond percolation conditioned to be a forest, the independent sets of the...

Somehow, despite the title, SL(2,Z) is the poster child for arithmetic groups not satisfying the congruence subgroup property, which is to say that it has finite index subgroups which can not be defined by congruence conditions on their coefficients...

In these two talks, I will describe how the classification of locally homogeneous geometric structures (closely related to flat connections) leads to interesting dynamical systems.

Many interesting dynamical systems arise from the classification...

I will present an elementary introduction to the regularity/uniqueness issues for the Navier-Stokes equations. Thanks to the convex integration technique, the h-principle is starting to take shape for the uniqueness problem. It might be possible...