# Seminars & Events for 2009-2010

##### Central Limit Theorem for Random Poisson polygons in an Arbitrary convex set in $R^2$

Miniconference on Dynamical Systems at Princeton

##### Regular or stochastic dynamics in higher degree families of unimodal maps

Miniconference on Dynamical Systems at Princeton

##### Some recent convergence results for nonconventional ergodic averages

The phenomenon of multiple recurrence in ergodic theory occurs when several of the images of one fixed positive-measure set under some probability-preserving transformations, indexed by the points of some finite configuration in the acting group, all overlap in a set of positive measure.

##### On the structure of singularities of solutions to the Einstein equations with toroidal symmetry

I will present recent results concerning the study of the global Cauchy problem in relativity, without restrictions on the size of the data but under certain symmetry and topological assumptions. More specifically, I will focus on the issue of the structure of singularities for solutions with toroidal symmetry, in relation to the so-called strong cosmic censorship conjecture.

##### White noise for KdV, mKdV, and cubic NLS on the circle

We discuss two methods for establishing the invariance of the white noise for the periodic KdV. First, we briefly go over the basic theory of Gaussian Hilbert spaces and abstract Wiener spaces and show that the Fourier-Lebesgue space $\mathcal{F}L^{s, p}$ captures the regualrity of the white noise for $sp < -1$.

##### Mathematical Theory of Cryo-Electron Microscopy

The importance of determining three dimensional macromolecular structures for large biological molecules was recognized by the Nobel Prize in Chemistry awarded this year to V. Ramakrishnan, T. Steitz and A. Yonath for studies of the structure and function of the ribosome.

##### The spectral dichotomy for one-frequency Schrödinger operators

In the theory of one-frequency Schrödinger operators, the best understood potentials have been those that can be somehow considered either small or large. Roughly, small potentials tend to inherit the behavior of the Laplacian and present absolutely continuous spectral measures (leading to good transport properties), while for large potentials it is Anderson localization that prevails.

##### Quasigeodesic pseudo-Anosov flows

A quasigeodesic is a curve which is uniformly efficient in measuring distance in relative homotopy classes or equivalently efficient up to a bounded multiplicative distortion in measuring distance when lifted to the universal cover. A flow is quasigeodesic if all flow lines are quasigeodesics.

##### Generalized modular functions

Generalized modular functions are holomorphic functions on the complex upper half-plane, meromorphic at the cusps that satisfy the usual defintion of a modular function, however with the important exception that the character need not be unitary. The theory is partly motivated from conformal field theory in physics. In my talk I will report on recent joint work with G.

##### On m-Quasi Einstein metrics

We say an $n$-dimensional Riemannian manifold is an $m$-Quasi Einstein metric if it is the base of an $(n+m)$-dimensional warped product Einstein manifold. We view the $m$-Quasi Einstein equation as a generalization of the Einstein equation (since an Einstein manifold is the base of a trivial product Einstein manifold).

##### Generalized principal eigenvalue of elliptic operators in unbounded domains and applications

In this talk, I will discuss several notions that are extensions of the principal eigenvalue of a linear elliptic operator to the framework of unbounded domains along with some of their properties. I will describe applications to semi-linear elliptic equations and to propagation in reaction-diffusion equations of the KPP type.

##### Dynamics of renormalization operators

It is a remarkable characteristic of some classes of low-dimensional dynamical systems that their long time behavior at a short spatial scale is described by an induced dynamical system in the same class. The renormalization operator that relates the original and the induced transformations can then be iterated.

##### Disconnection and Random Interlacements

The general theme of the talk pertains to the question of understanding how paths of random walks can create large separating interfaces.

##### The Kervaire invariant problem

The of existence of framed manifolds of Kervaire invariant one is one of the oldest unresolved problems in algebraic topology. Important questions about smooth structures on spheres and on the homotopy groups of spheres depend on its solution. In this talk I will describe joint work with Mike Hill and Doug Ravenel which solves this problem in all dimensions except 126.

##### Grothendieck's problem for 3-manifold groups

##### Convergence of renormalizations

##### Equivariant Computations and the Gap Theorem

I'll show how elementary computations with equivariant chain complexes and homology can be used to prove the vanishing of certain homotopy groups. Together with the detection theorem, this shows that the group in which the Kervaire classes would be detected is the zero group.

##### Graph/Group Random Elements and Applications to Group-Based Cryptanalysis

We introduce the notion of the mean-set (expectation) of a graph-(group-) valued random element $\xi$, generalize the strong law of large numbers to graphs and groups, and consider analogues of Chebyshev and Chernoff type bounds. In addition, we discuss several results about configurations of mean-sets in graphs and reflect on an algorithm for computing sample mean-sets.

##### The proof of the Periodicity Theorem

The Periodicity Theorem is one of the key steps in the proof of the Kervaire Invariant Theorem. Its proof involves methods from equivariant stable homotopy theory including computations with $RO(G)$-graded homotopy groups.

##### Some Remarks on quadratic Twists of L-Functions

Suppose $E$ is a rational elliptic curve and $p$ is a given prime. It is of interest to know that there exists a square free integer $D$ such that the $D$-th quadratic twist $E_D$ of $E$ such that the p-Selmer group of $E_D$ is trivial. The existence of such a $D$ seems not yet known in general.