# Seminars & Events for 2010-2011

##### Metaphors in systolic geometry

The systolic inequality says that any Riemannian metric on an $n$-dimensional torus with volume 1 contains a non-contractible closed curve with length at most $C(n)$ - a constant depending only on $n$. One remarkable feature of the inequality is it holds for such a wide class of metrics.

##### A survey of results in universality of Wigner matrices, Part I

In the 1950's, Wigner proved the famous semicircle laws for Wigner matrices and started the study of universality results in random matrices. In these two talks, this will serve as our starting point as we surveyed the historical developments in this field.

##### Hypergraph Turan Problem

The Turan function $ex(n,F)$ of a k-graph $F$ is the maximum number of edges in an $F$-free k-graph on $n$ vertices. This problem goes back to the fundamental paper of Turan from 1941 that solved it for complete graphs (k=2). Unfortunately, very few non-trivial instances of the problem have been solved when we consider hypergraphs (k>2).

##### Transverse homology

Knot contact homology is a combinatorial Floer-theoretic knot invariant derived from Symplectic Field Theory. I'll discuss the geometry behind this invariant and a new filtered version, transverse homology, which turns out to be a fairly effective invariant of transverse knots

##### Periods of special cycles and derivatives of L-series

In this talk, I will state some conjectures and examples concerning the central derivatives of automorphic L-series in terms of heights of special cycles on Shimura varieties.

##### Sharp gradient estimates for a class of elliptic equations

I will present some results related with existence and sharp regularity for solutions to a class of singular elliptic equations with gradient dependence for which solutions may exhibit a free boundary. Once we have obtained sharp regularity, further analysis of the free boundary may be carried out with nondegeneracy and refined gradient estimates.

##### A New Formalism for Electromagnetic Scattering in Complex Geometry

We will describe some recent, elementary results in the theory of electromagnetic scattering in R3. There are two classical approaches that we will review - one based on the vector and scalar potential and applicable in arbitrary geometry, and one based on two scalar potentials, due to Lorenz, Debye and Mie, valid only in the exterior (or interior) of a sphere.

##### Around the Tate conjecture with integral coefficients

Due to the analogy with the Hodge conjecture, it has been known for a long time that the Tate conjecture for algebraic cycles on varieties over finite fields does not hold if one considers the cycle map into \'etale cohomology with Z_\ell-coefficients. Still, some cases may be expected to hold and they have interesting consequences.

##### Critical velocites in rotating Bose gases

Some of the remarkable phenomena that emerge when a trapped, ultracold Bose gas is set in rapid rotational motion will be reviewed. In anharmonic traps, where the rotational velocity can in principle be arbitrarily large, one can distinguish three critical velocities at which the flow pattern changes radically.

##### Acoustical spacetime geometry and shock formation

In 2007 I published a monograph which treated the relativistic Euler equations in three space dimensions for a perfect fluid with an arbitrary equation of state. In this monograph I considered initial data which outside a sphere coincide with the data corresponding to a constant state.

##### Canonical Kahler metrics and the K-stability of projective varieties

The "standard conjectures" in Kahler geometry state that the existence of a canonical metric in a given Hodge class is equivalent to the stability of the associated projective model(s). There are several competing definitions of stability ( mainly due to Tian and Donaldson ) and various partial results linking these definitions to the sought after metric.

##### A survey of results in universality of Wigner matrices, Part II

In the 1950's, Wigner proved the famous semicircle laws for Wigner matrices and started the study of universality results in random matrices. In these two talks, this will serve as our starting point as we surveyed the historical developments in this field.

##### Hypergraph list coloring and Euclidean Ramsey Theory

A hypergraph is simple if it has no two edges sharing more than a single vertex. It is $s$-list colorable if for any assignment of a list of $s$ colors to each of its vertices, there is a vertex coloring assigning to each vertex a color from its list, so that no edge is monochromatic. I will discuss a recent result, obtained jointly with A.

##### Somewhat simple curves on surfaces, and the mysteries of covering spaces

I will count some curves on 2-dimensional manifolds, and will discuss some related issues in geometric (and otherwise) group theory.

##### Endoscopic transfer of depth-zero supercuspidal L-packets

In a recent paper, DeBacker and Reeder have constructed a piece of the local Langlands correspondence for pure inner forms of unramified $p$-adic groups and have shown that the corresponding L-packets are stable.

##### Laplace eigenvalues via asymptotic separation of variables

We study the behavior of eigenvalues under geometric perturbations using a method that might be called asymptotic separation of variables. In this method, we use quasi-mode approximations to compare the eigenvalues of a warped product and another metric that is asymptotically close to a warped product. As one application, we shoe that the generic Euclidean triangle has simple Laplace spectrum.

##### Rigidity of critical metrics in dimension four

The general quadratic curvature functional is considered in dimension four. It is possible to "gauge" the Euler-Lagrange equations, in a self-adjoint fashion, to become elliptic. Fredholm theory may then be used to describe local properties of the moduli space of critical metrics.

##### Wavelet Frames and Applications

This talk focuses on the tight wavelet frames derived from multiresolution analysis and their applications in imaging sciences. One of the major driven forces in the area of applied and computational harmonic analysis over the last two decades is to develop and understand redundant systems that have sparse approximations of different classes of functions.

##### Normal form-type arguments in the study of dispersive PDEs

Bourgain used normal form reduction and the I-method to prove global well-posedness of one-dimensional periodic quintic NLS in low regularity. In this talk, we discuss the basic notion of normal form reduction for Hamiltonian PDEs and apply it to one-dimensional periodic NLS with general power nonlinearity.

##### Special Gamma, Zeta, Multizeta values and Anderson t-Motives

We will describe how the "special value theory" in function field arithmetic is an interesting mixture of very strong theorems determining all algebraic relations in some cases, emerging partial conjectural pictures in some cases, and quite wild phenomena often.