Counting linear extensions of a partial order (linear orders compatible with the partial order) is a classical and computationally difficult problem in enumeration and computer science. A family of partial orders that are prevalent in enumerative and algebraic combinatorics come from Young diagrams of partitions and skew partitions. Their linear extensions are called standard Young tableaux. The celebrated hook-length formula of Frame, Robinson and Thrall from 1954 gives a product formula for the number of standard Young tableaux of partition shape. No such product formula exists for skew partitions. 

In 2014, Naruse announced a formula for skew shapes as a positive sum of products of hook-lengths using ”excited diagrams” of  Ikeda-Naruse, Kreiman, Knutson-Miller-Yong in the context of equivariant cohomology. We prove Naruse’s formula algebraically and combinatorially in several different ways. Also, we show how excited diagrams give asymptotic results and product formulas for the enumeration of certain families of skew tableaux. Lastly, we give analogues of Naruse's formula in the context of equivariant K-theory.

This is joint work with Igor Pak and Greta Panova.
 

Topology began with the study of complex functions, and the interaction of topology with algebraic geometry and number theory continues to be fertile ground.  Starting with basic examples, I will explain how the function field/number field dictionary and the remarkable framework of the Weil conjectures (as established by Weil, Grothendieck, Dworkin, Deligne and many others) allows one to use counting problems to predict and discover topological properties of manifolds, and vice versa. This is joint work with Benson Farb and Melanie Matchett Wood.

I will speak about two topics. The first one is Poincare inequality 3/2 on the Hamming cube which significantly improves the Beckner's result. The second topic will be devoted to extremal problems on BMO space where I will illustrate by dynamical algorithm how to solve all these problems together.

The study of division algebras has, from its inception, been closely tied to geometry of various kinds. This relationship has become richer with developments in both algebra and algebraic geometry. In this talk I will discuss some of the history of the theory of division algebras and some of its interactions with geometry as well as introduce some modern perspectives.

The question of global regularity remains open for many fundamental models of fluid dynamics.  In two dimensions, solutions to the incompressible Euler equations have been known to be globally regular since the 1930s, although their derivatives can grow double-exponentially with time.  On the other hand, this question has not yet been resolved for the more singular surface quasi-geostrophic (SQG) equation, which is used in atmospheric models.  The latter state of affairs is also true for the modified SQG equations, a family of PDE which interpolate between these two models.

I will present two results about the patch dynamics version of these equations on the half-plane.  The first is global-in-time regularity for the Euler patch model, even if the patches initially touch the boundary of the half-plane.  The second is local-in-time regularity for those modified SQG patch equations that are only slightly more singular than Euler, but also existence of their solutions which blow up in finite time. The latter appears to be the first rigorous proof of finite time blow-up in this type of fluid dynamics models.