# Seminars & Events for 2009-2010

##### Surface Comparison With Mass Transportation

A method for transportation of metric between simply-connected surfaces with boundary is presented. The method is based on classical uniformization and optimal mass transportation. One application of the method is a novel definition of distance function between simply-connected surfaces with boundaries.

##### Solving High-Dimensional Stochastic Optimization Problems using Approximate Dynamic Programming

There are many stochastic resource allocation problems arising in transportation, energy and health that involve high-dimensional state and action variables in the presence of dierent forms of uncertainty. These might involve discrete or continuous resources, and generally involve vectors of random variables that preclude exact computation of expectations.

##### Quasi-local horizons

##### The spectral edge of random band matrices

We consider random periodic $N\times N$ band matrices of band width $W$. If the band is wide $(W>>N^{5/6})$, the spectral statistics at the edge behave similarly to those of GUE matrices; in particular, the largest eigenvalue converges in distribution to the Tracy—Widom law. Otherwise, a different limit appears.

##### Rigidity properties of Fano varieties

From the point of view of the Minimal Model Program, Fano varieties constitute the building blocks of uniruled varieties. Important information on the biregular and birational geometry of a Fano variety is encoded, via Mori theory, in certain combinatorial data corresponding to the Neron–Severi space of the variety.

##### Concentration Compactness for critical Wave Maps

This talk will discuss a recent result on global regularity and asymptotic behavior of large critical wave maps with hyperbolic target, obtained jointly with W. Schlag. The proof relies on an adaptation of the recently developed method of Kenig-Merle to the case of systems of wave equations, as well as the harmonic analytic methods devised by Klainerman-Machedon, Tataru and Tao.

##### Looking over the painter’s shoulder

Just microns below their paint surface lies a wealth of information on Old Master Paintings. Hidden layers can include the underdrawing, the underpainting or compositional alterations by the artist. All too often artists simply re-used their canvases and painted a new composition on top. Thus, a look *through* the paint layer provides a look *over* the painter’s shoulder.

##### A splitter theorem for induced subgraphs

A homogeneous set in a graph $G$ is a subset $X$ of $V(G)$, such that no vertex of $V(G)/X$ has both a neighbor and a non-neighbor in $X$. Let us say that a graph is prime if it has no homogeneous set $X$ with $1<|X|<|V(G)|$.

##### Generic singularities of mean curvature flow

Mean curvature flow (or MCF) is a nonlinear heat equation for hyper-surfaces, where the surface evolves by moving in the direction where volume locally decreases the fastest. The simplest non-static examples are round concentric spheres, where the radius shrinks until it becomes zero at "extinction" (a singularity of the flow). Singularities are unavoidable as the flow contracts any closed su

##### Bigger is Better

Originally, Hardy and Littlewood developed their "circle method" to study Waring's problem on the representation of numbers as the sums of $k^th-powers$. In the circle method, one decomposes the circle into "major" and "minor" arcs. Some rough estimates on the minor arcs give a power saving, and the work is then to study the major arcs. The guiding principle is "Bigger is better", i.e.

##### Unbiased Random Perturbations of Navier-Stokes Equation

A random perturbation of a deterministic Navier-Stokes equation is considered in the form of an Stochastic PDE with Wick product in the nonlinear term. The equation is solved in the space of generalized stochastic processes using the Cameron-Martin version of the Wiener chaos expansion. The generalized solution is obtained as an inverse of solutions to corresponding quantized equations.

##### Codes, arithmetic and topology

##### Hilbert modular surfaces through K3 surfaces

We describe how to use Shioda-Inose structures on K3 surfaces to write down explicit equations for Hilbert modular surfaces, which parametrize principally polarized abelian surfaces with real multiplication by the ring of integers in $Q(\sqrt{D})$. In joint work with Elkies, we have computed several of these (for fundamental discriminants less than 100), including some of general type.

##### Knots with small rational genus

If $K$ is a rationally null-homologous knot in a 3-manifold $M$ then there is a compact orientable surface $S$ in the exterior of $K$ whose boundary represents $p[K]$ in $H_1(N(K))$ for some $p > 0$. We define $\Vert K \Vert$, the *rational genus* of $K$, to be the infimum of $-\chi^-(S)/2p$ over all $S$ and $p$. If $M$ is a homology sphere then this is essentially the genus of $K$.

##### Conformal Structure of Minimal Surfaces with Finite Topology

The recent construction of a genus-one helicoid verified the existence of a second example of a complete, embedded minimal surface with finite topology and infinite total curvature in $\mathbb{R}3$. We determine the conformal structure and asymptotic Weierstrass data of all surfaces with these properties.

##### Imaging Techniques and the Rejuvenation of Artwork

Advances in digital imaging within the visible spectrum enable the accurate color rendering of artwork. It is possible to generate a colorimetric image with high spatial resolution and high image quality (appropriate sharpness and low noise). When the number of sensor channels exceeds three, it is also possible to generate spectral images.

##### Universality of Random Matrices and Dyson Brownian Motion

The universality for eigenvalue spacing distributions is a central question in the random matrix theory. In this talk, we introduce a new general approach based on comparing the Dyson Brownian motion with a new related dynamics, the local relaxation flow.

##### Rational curves on hypersurfaces

This talk is on the geometry of spaces of rational curves on Fano hypersurfaces. I will talk about some of the known results on the dimension, irreducibility, and the Kodaira dimension of these spaces. I will also discuss the problem of bounding the dimension of the cones of non-free rational curves on general hypersurfaces.

##### Random Matrices: Universality of Local Eigenvalues Statistics

One of the main goals of the theory of random matrices is to establish the limiting distributions of the eigenvalues. In the 1950s, Wigner proved his famous semi-cirle law (subsequently extended by Anord, Pastur and others), which established the global distribution of the eigenvalues of random Hermitian matrices.

##### Proof of the Bollobas-Catlin-Eldridge conjecture

We say that two graphs $G$ and $H$ pack if $G$ and $H$ can be embedded into the same vertex set such that the images of the edge sets do not overlap. Bollobas and Eldridge, and independently Catlin conjectured that if the graphs $G$ and $H$ on $n$ vertices with maximum degree $M(G)$ and $M(H)$, respectively, satisfy $(M(G) + 1)(M(H) + 1) ≤ n + 1$ then $G$ and $H$ pack.