# Seminars & Events for 2016-2017

##### Localization of interacting fermions with quasi-random disorder

We consider interacting electrons in a one dimensional lattice with an incommensurate Aubry-Andre' potential in the regime when the single-particle eigenstates are localized.

##### The Cauchy-Riemann equations in complex manifolds

In this talk we will discuss the Cauchy-Riemann equations on domains in complex manifolds with positive or negative curvature. We will also report some recent new results on the $L^2$ closed range property for $\bar{\partial}$ on an annulus between two pseudoconvex domains, when the inner domain is not smooth.

##### Speed of random walks on finitely generated groups

We discuss a flexible construction of groups where the speed (rate of escape) of simple random walk can follow any sufficiently nice function between diffusive and linear. When the speed of the \mu-random walk is sub-linear, all bounded \mu-harmonic functions are constant.

##### A proof of Onsager's Conjecture

In an effort to explain how anomalous dissipation of energy occurs in hydrodynamic turbulence, Onsager conjectured that weak solutions to the incompressible Euler equations may violate the law of conservation of energy if their spatial regularity is below 1/3-Hölder. I will discuss a proof of this conjecture that shows that there are nonzero, (1/3-\epsilon)-Hölder Euler flows in 3D that have c

##### Quantum mechanics and low-dimensional topology

We'll discuss some applications of quantum mechanics (and, time permitting, quantum field theory) to Morse theory and low-dimensional topology, offering proof sketches of the Morse inequalities and new viewpoints on topological invariants.

##### Shuffling large decks of cards and the Bernoulli-Laplace urn model

In boardgames, in Casino games with multiple decks and in cryptography, one is sometimes faced with the practical problem: how can a human (as opposed to the computer) shuffle a big deck of cards. One natural procedure (used by casino’s) is to break the deck into several reasonable size piles, shuffle each throughly, assemble, do some simple deterministic thing (like a cut) and repeat. G.

##### Diameter bounds for Cayley graphs of finite simple groups of large rank

Given any non-abelian finite simple group G and any generating set S, it is conjectured by Laszlo Babai that its Cayley graph should always have diameter (log|G|)^O(1). This conjecture has been verified for all finite simple groups of Lie type with bounded rank, but little progress has been made in the cases with large rank.

##### The Grothendieck -Teichmuller lie algebra and homotopy equivalences of configuration spaces

The Grothendieck - Teichmuller Lie algebra grt is a Lie algebra, over the rational numbers Q, which is clearly very interesting and equally clearly not very well-understood. It crops up in many different areas of mathematics.

##### The Hasse-Weil zeta functions of the intersection cohomology of minimally compactified orthogonal Shimura varieties

Initiated by Langlands, the problem of computing the Hasse-Weil zeta functions of Shimura varieties in terms of automorphic L-functions has received continual study. We will discuss how recent progress in various aspects of the field has allowed the extension of the project to some Shimura varieties not treated before.

##### Strichartz estimates and local regularity for gravity-capillary water waves

We will consider the water waves problem for 2D and 3D incompressible, irrotational, inviscid fluid flows subject to both gravity and surface tension. The PDE is dispersive, quasilinear and nonlocal. The questions on local existence and uniqueness in the lowest regularity spaces are still in progress.

##### Stein fillability in higher dimensions

We recall the definition of the surgery obstruction for an almost contact manifold to be Stein fillable. This class can be used to prove various fillability and non-fillability results. This is joint work with J. Bowden and D. Crowley.

##### Blow up analysis of solutions of conformally invariant fully nonlinear elliptic equations

We establish blow-up profiles for any blowing-up sequence of solutions of general conformally invariant fully nonlinear elliptic equations on Euclidean domains. We prove that (i) the distance between blow-up points

##### A fully nonlinear Sobolev trace inequality

The $k$-Hessian operator $\sigma_k$ is the $k$-th elementary symmetric function of the eigenvalues of the Hessian. It is known that the $k$-Hessian equation $\sigma_k(D^2 u)=f$ with Dirichlet boundary condition

##### Moduli of Riemann surface and Bers conjecture

**THIS IS A SPECIAL ANALYSIS / GEOMETRY SEMINAR. **It was Keobe who first proved that closed Riemann surface can be uniformized by Schottky groups.

##### Nonuniqueness of weak solutions to the SQG equation

##### Counting Connected Graphs

Let C(n,k) be the number of labelled connected graphs with n vertices and n-1+k edges. For k=0 (trees) we have Cayley's Formula. We examine the asymptotics of C(n,k). There are several approaches involving supercritical dominant components in random graphs, local limit laws, Brownian excursions, Parking functions and other topics.

##### Towards a theory of singular symplectic varieties

Singular algebraic (sub)varieties are fundamental to the theory of smooth projective manifolds. In parallel with his introduction of pseudo-holomorphic curve techniques into symplectic topology 30 years ago, Gromov asked about the feasibility of introducing notions of singular (sub)varieties suitable for this field.

##### Old and new formulas for degeneracy loci

A very old problem asks for the degree of a variety defined by rank conditions on matrices. The story of the modern approach begins in the 1970's, when Kempf and Laksov proved that the degeneracy locus for a map of vector bundles is given by a certain determinant in their Chern classes.

##### Can one hear the shape of a random walk?

We consider a Gibbs distribution over random walk paths on the square lattice, proportional to a random weight of the path’s boundary . We show that in the zero temperature limit, the paths condensate around an asymptotic shape.

##### The free-boundary Brakke flow

A surface has geometric free-boundary if it meets some barrier hypersurface orthogonally, like a bubble on a bathtub. We extend Brakke's weak notion of mean curvature flow to have a free-boundary condition, which allows the surface to ``pop'' upon tangential contact with the barrier.