# Seminars & Events for 2012-2013

##### Adams and Moser-Trudinger inequalities: from concrete to abstract, and back

In this talk I will briefly review Moser-Trudinger inequalities, and David Adams' important contributions to the subject. I will then talk about recent work with Luigi Fontana, where we extend, unify and improve Adams' results in an abstract, measure-theoretic setting.

##### Open-closed Gromov-Witten theory

Lagrangian Floer theory studies boundary valued problem for the maps from bordered Riemann surface with boundary condition given Lagrangian submanifold. Gromov-Witten theory studies pseudo-holomorphic maps from Riemann surface. Various relations between them is found and is expected to be useful both in symplectic geometry and the study of Mirror symmetry.

##### Regularity Results for Optimal Transport Maps

Knowing whether optimal maps are smooth or not is an important step towards a qualitative understanding of them. In the 90's Caffarelli developed a regularity theory on R^n for the quadratic cost, which was then extended by Ma-Trudinger-Wang and Loeper to general cost functions which satisfy a suitable structural condition.

##### Hypersurfaces in hyperbolic space and conformal metrics on domains in sphere

In this talk I will introduce a global correspondence between properly immersed horospherically convex hyper surfaces in hyperbolic space and complete conformal metrics on subdomains in the boundary at infinity of

##### Scattering for the 3d Zakharov system

We prove global existence and scattering for small localized solutions of the Cauchy problem for the Zakharov system in 3 space dimensions. Joint work with Zaher Hani and Jalal Shatah.

##### The Erdos-Stone Theorem for finite geometries

For any class of graphs, the growth function h(n) of the class is defined to be the maximum number of edges in a graph in the class on n vertices. The Erdos-Stone Theorem remarkably states that, for any class of graphs that is closed under taking subgraphs, the asymptotic behaviour of h(n) can (almost) be precisely determined just by the minimum chromatic number of a graph not in the class.

##### Action-Dimension of Groups

**Please note change of time and location. This is a joint Topology/Algebraic Topology seminar. ** For a group G, we define a notion of dimension in terms of dimension change of the of the top homology between a free G space X and it's quotient X/G. We show that this is well defined and calculate this "Action-dimension" for certain groups, including finitely generated solvable gr

##### Action-Dimension of Groups

*This is a combined Topology/Algebraic Topology seminar.*

##### Towards weak p-adic Langlands for GL(n)

For GL(2) over Q_p, the p-adic Langlands correspondence is available in its full glory, and has had astounding applications to Fontaine-Mazur, for instance. In higher rank, not much is known. Breuil and Schneider put forward a conjecture, which is a somewhat coarse version of p-adic Langlands for GL(n).

##### On the loss of continuity for supercritical drift-diffusion equations

We consider the (linear) drift-diffusion equation $\partial_t \theta + u \cdot \nabla \theta + (-\Delta)^s \theta = 0$. Here the divergence free drift $u$ belongs to a supercritical space, and $0 < s \leq 1$. We prove that starting with smooth initial data solutions may become discontinuous in ﬁnite time. For $s < 1$ this may even be achieved with autonomous drift.

##### F-Signature and Relative Hilbert-Kunz Multiplicity Revisited

In this talk, building on an observation of Yao, we will sketch a partial answer to a question of Watanabe and Yoshida by showing that the F-signature and relative Hilbert-Kunz multiplicity (for cyclic modules) coincide. The method of proof also suggests a number of generalizations of F-signature which we will present as time allows.

##### Renaissance of the h-principle in symplectic topology

Flexible and rigid methods coexisted in Symplectic Topology since its inception. The developmnet during last 20 years was dominated by rigid results, with accidental flexible breakthroughs (such as Donaldson's theory of almost holomorphic sections). In the talk I will discuss some recently found new remarkable examples of symplectic flexibility.

##### Asymptotic Representation Theory

I will offer a glimpse of asymptotic representation theory, focusing on the typical and extremal behavior of representations of S_n as n approaches infinity.

##### Duchon-Robert Solutions for a Two-Fluid Interface

Duchon and Robert constructed a class of global vortex sheet solutions to the Euler equations, where the vorticity is concentrated on an analytic curve for all positive time. In this talk, I will first discuss the main ideas behind their construction and general properties of vortex sheet solutions.

##### Apollonian structure in the abelian sandpile

The Abelian sandpile is a diffusion process on configurations of chips on the integer lattice, in which a vertex with at least 4 chips can "topple", distributing one of its chips to each of its 4 neighbors.

##### Splittings of the polyhedral product functor

I will talk about joint work with Bahri, Cohen and Gitler. We showed that the polyhedral product functor (a generalization of the Moment Angle Complex) stably splits into pieces which are understood for restricted cases. I will talk about further splitting of the pieces that appear in the splitting and describe the homology group.

##### An introduction to the rational genus of a knot

What is the "simplest" knot in a given three-manifold $Y$?

##### A Paneitz-type operator for CR pluriharmonic functions

Paul Yang and I recently found a new fourth order CR-invariant operator on CR pluriharmonic functions in three dimensions which generalizes to the abstract setting the operator on the sphere discovered by Branson, Fontana, and Morpurgo. I will describe the similarities between this operator and the (conformal) Paneitz operator, and how this helps to better understand some problems in CR geomet

##### Algebraic structures associated to Weinstein manifolds

##### A Large Data Regime for non-linear Wave Equations

For semi-linear wave equations with null form non-linearities on $\mathbb{R}^{3+1}$, we exhibit an open set of initial data which are allowed to be large in energy spaces, yet we can still obtain global solutions in the future. This is a joint work with Jinhua Wang.