# Video Lectures

Many important consequences of the Riemann Hypothesis would remain true even if there were some zeros off the critical line, provided these exceptions to the Riemann Hypothesis are suitably rare. We can unconditionally prove some results on the...

Lévy matrices are symmetric random matrices whose entries are in the domain of attraction of an \alpha stable law. For \alpha

We present recent developments in symplectic geometry and explain how they motivated new results in the study of cluster algebras. First, we introduce a geometric problem: the study of Lagrangian surfaces in the standard symplectic 4-ball bounding...

Let G be a simply-connected complex semisimple algebraic group and let C be a smooth projective curve of any genus. Then, the moduli space of semistable G-bundles on C admits so called determinant line bundles. E. Verlinde conjectured a remarkable...

Given any non-negative function \f:ℤ→ℝ, it follows from basic ergodic ideas that either 100% of real numbers α have infinitely many rational approximations a/q with a,q coprime and |α−a/q|

I'll describe a recent resolution of this conjecture, which recasts the problem in combinatorial language, and then uses a general 'structure vs randomness' principle combined with an iterative argument to solve this combinatorial problem.

The celebrated product theorem says if A is a generating subset of a finite simple group of Lie type G, then |AAA| \gg \min \{ |A|^{1+c}, |G| \}. In this talk, I will show that a similar phenomenon appears in the continuous setting: If A is a subset...

The talk will consists of a long historical introduction to the topic of deviation of ergodic averages for locally Hamiltonian flows on compact surafces as well as some current results obtained in collaboration with Corinna Ulcigrai and Minsung...

The theory of graph quasirandomness studies sequences of graphs that "look like" samples of the Erdős--Rényi random graph. The upshot of the theory is that several ways of comparing a sequence with the random graph turn out to be equivalent. For...

Given several real numbers α1,...,αk, how well can you simultaneously approximate all of them by rationals which each have the same square number as a denominator? Schmidt gave a clever iterative argument which showed that this can be done...

Many algorithms and heuristics that work well in practice have poor performance under the worst-case analysis, due to delicate pathological instances that one may never encounter. To bridge this theory-practice gap, Spielman and Teng introduced the...