The Birch and Swinnerton-Dyer conjecture, one of the millenium problems, is a bridge between algebraic invariants of an elliptic curve and its (complex analytic) L-function. In the case of low ranks, we prove this conjecture up to the finitely many bad primes and the prime 2, by proving the Iwasawa main conjecture in full generality. The ideas in the proof and formulation also lead us to new and mysterious phenomena. This talk assumes no specialized background in number theory.

Hindman’s theorem states that if one colors every natural number either red or blue, then there will be an infinite set X of natural numbers such that all finite sums of distinct elements from X have the same color. The original proof of Hindman’s theorem was a combinatorial mess and the slickest proof is via ultrafilters. In this talk, I will introduce the notion of an ultrafilter on a set, which is simply a division of the subsets of the set into two categories, “small" and “large", satisfying some natural axioms. We will then give the proof of Hindman’s theorem using ultrafilters that are idempotent with respect to a natural addition operation on the set of ultrafilters on the set of natural numbers. Finally, we will introduce an open conjecture of Erdos related to Hindman’s theorem, its reformulation in terms of ultrafilters, and some recent progress made on the problem by myself and my collaborators.

Cohomology is a basic and powerful tool that arises in many fields of geometry and topology.  I will motivate this technique and demonstrate its use in some simple examples.

In 2011 J.Streets and G.Tian introduced a family of metric flows over a complex Hermitian manifold. We consider one particular member of this family and prove that if the initial metric has Griffiths positive Chern curvature, then this property is preserved along the flow.  On a manifold with Griffiths non-negative Chern curvature this flow has nice regularization properties, in particular, for any t>0 the zero set of Chern curvature becomes invariant under certain torsion-twisted parallel transport. If time permits, we discuss applications of the results to some uniformization problems.

The sloshing problem is a Steklov type eigenvalue problem describing small oscillations of an ideal fluid. We will give an overview of some latest advances in the study of Steklov and sloshing spectral asymptotics, highlighting the effects arising from corners, which appear naturally in the context of sloshing. In particular, we will outline an approach towards proving the conjectures posed by Fox and Kuttler back in 1983 on the asymptotics of sloshing frequencies in two dimensions. The talk is based on a joint work in progress with M. Levitin, L. Parnovski and D. Sher.