# Seminars & Events for 2009-2010

##### Optimal conditions for the extension of the mean curvature flow

In this talk, we will discuss several optimal (global) conditions for the existence of a smooth solution to the mean curvature flow. Our focus will be on quantities involving only the mean curvature. We will also discuss several applications of a local curvature estimate which is a parabolic analogue of Choi-Schoen estimate for minimal submanifolds. This is joint work with Natasa Sesum.

##### A Database Schema for the Global Dynamics of Multiparameter Nonlinear Systems

Prof. Mischaikow will discuss new computational tools based on topological methods that extracts coarse, but rigorous, combinatorial descriptions of global dynamics of multiparameter nonlinear systems. This techniques are motivated by several observations which we claim can, at least in part, be addressed:

##### Quantized Poincarè maps in chaotic scattering

I will sketch a recent approach to study the resonance spectrum of scattering Schrödinger operators, in cases where the trapped set of the corresponding classical dynamics (near some positive energy) is a fractal chaotic repeller. In that situation, we are interested in the distribution of resonances in the vicinity of the real axis, in the semiclassical limit.

##### Nodal sets for eigenfunctions of the Laplacian and lattice points on circles and spheres

##### Friedgut's theorem for the continuous cube

A celebrated theorem of Friedgut says that every boolean function on the discrete cube can be approximated by a function which depends only on a number of variables that depends on the sum of the influences of the variables of f. Dinur and Friedgut conjectured an analogue of this theorem for the continuous cube.

##### Motivic invariants of the rational numbers

We report on joint work in progress with Kyle Orsmby. Via local-to-global techniques and Adams spectral sequence computations we access motivic invariants such as K-theory, cobordism and stable stems of the rational numbers.

##### Real quadratic analogues of values of the j-function at CM points

An interesting new class of modular forms has emerged in the last several years that generalize Ramanujan's mock theta functions. The generalization is based on an observation of of Zwegers who showed that mock theta functions occur as holomorphic parts of harmonic Maass forms of weight 1/2 having singularities in cusps.

##### Complete Calabi-Yau metrics from rational elliptic surfaces

A rational elliptic surface is the blow-up of P2 in the nine base points of a pencil of cubics. The pencil then lifts as a fibration of the surface by elliptic curves. I show that the complement of any fiber F admits families of complete Calabi-Yau metrics, whose asymptotic geometry depends in a delicate way on the monodromy of the fibration around F.

##### Superconcentration

We introduce the term 'superconcentration' to describe the phenomenon when a function of a Gaussian random field exhibits a far stronger concentration than predicted by classical concentration of measure. We show that when superconcentration happens, the field becomes chaotic under small perturbations and a 'multiple valley picture' emerges. Conversely, chaos implies superconcentration.

##### Detection of Faint Edges in Noisy Images

One of the most intensively studied problems in image processing concerns how to detect edges in images. Edges are important since they mark the locations of discontinuities in depth, surface orientation, or reflectance, and their detection can facilitate a variety of applications including image segmentation and object recognition.

##### Infinite Ergodic Theory and Continued Fractions

I will compare (classical) Ergodic Theory and Infinite Ergodic Theory, i.e. when the space has infinite measure. In particular I will describe how to modify the Birkhoff Ergodic Theorem in the infinite setting. As examples, I will discuss the (classical) Euclidean continued fractions and the (less classical) continued fractions with even partial quotients.

##### Pretentiousness in the analytic theory of numbers

Following the brilliant insight of Riemann, that a good understanding of the distribution of prime numbers is equivalent to a good understanding of the location of zeros of pertinent L-functions, analytic number theory has traditionally centered on developing this point-of-view.

##### Limit theorems for sticky particle systems and positivity of integrated random walks

Consider the model of a one-dimensional gas, whose particles have random initial positions and random initial velocities. Particle attract each other due to gravitation, and stick together at collisions. As time goes, the number of particles decreases while their sizes increase until there forms a giant single particle of the total mass.

##### Pointed torsors and Galois groups

Suppose that $H$ is an algebraic group which is defined over a field $k$, and let $L$ be the algebraic closure of $k$. The canonical stalk for the etale topology on $k$ induces a simplicial set map from the classifying space $B(H-tors)$ of the groupoid of $H$-torsors (aka. principal $H$-bundles) to the space $BH(L)$. The homotopy fibres of this map are groupoids of pointed torsors.

##### Heegaard Floer Homology and Knot Surgeries

Wallace and Lickorish showed that any 3-manifold can be realized as surgery on a link in S3; however, fifty years later, we still have a rather poor understanding of which manifolds can be constructed from surgery on a knot and which knots give these surgeries. For example, the Berge Conjecture attempts to list all knots which can give rise to a lens space.

##### On Eisenstein series and the cohomology of arithmetic groups

The automorphic cohomology of a reductive $\mathbb{Q}$--group $G$, defined in terms of the automorphic spectrum of $G$, captures essential analytic aspects of the arithmetic subgroups of $G$ and their cohomology.

##### Conformally Warped Manifolds and quasi-Einstein metrics

The concept of a smooth metric measure space has recently arisen as a useful object within Riemannian geometry, for example in Perelman's formulation of Ricci flow as a gradient flow. In this setting, a key objective is to find a suitable generalization of Ricci curvature, and to understand the associated ``quasi-Einstein'' metrics.

##### Probabilistic approach to high order assignment problems

A variety of computer vision and engineering problems can be cast as high order matching problems, where one considers the affinity of two or more assignments simultaneously. The spectral matching approach of Leordeanu and Hebert (2005) was shown to provide an approximate solution of this np-hard problem.

##### On the Boltzmann limit of a homogeneous Fermi gas

##### Rational simple connectedness and Serre's "Conjecture II"

In the early 1960's Serre formulated two conjectures about Galois cohomology. The first was proved by Steinberg shortly thereafter, but the second remains open. I will discuss the proof of Serre's Conjecture II in the "geometric case": every principal homogeneous space for a bundle of simply connected, semisimple groups over a surface has a rational section.