# Seminars & Events for 2010-2011

##### Random maximal isotropic subspaces and Selmer groups

We show that the $p$-Selmer group of an elliptic curve is naturally the intersection of two maximal isotropic subspaces in an infinite-dimensional locally compact quadratic space over $F_p$.

##### Partial regularity of a minimizer of the relaxed energy for biharmonic maps

In 1999, Chang, Wang and Yang established the fundamental result on the partial regularity stationary biharmonic maps into spheres. Since then, the study of biharmonic maps has attracted much attention. In this talk, we will discuss some new result on the relaxed energy for biharmonic maps from an $m$-dimensional domain into spheres for an integer $m\geq 5$.

##### Tournament heroes

The chromatic number of a tournament $T4 is the smallest number of transitive tournaments that partition $V(T)$. Let us say that a tournament $S$ is a hero if for every tournament $T$ not containing $S$, the chromatic number of $T$ is at most a constant $c(S)$.

##### Rational points on algebraic varieties

I will discuss several geometric techniques and constructions that emerged in the study of rational points on higher-dimensional algebraic varieties over global fields.

##### Bifurcations of solutions of the 2-dimensional Navier-Stokes system

I will explain recent joint work with Sinai on the bifurcations of solutions to the 2-dimensional Navier-Stokes system.

##### An integral lift of the Gamma-genus

The Hirzebruch genus of a complex-oriented manifold $M$ associated (by Kontsevich) to Euler's Gamma-function has an analytic interpretation as the index of a family of deformations of a Dirac operator, parametrized by the homogeneous space $Sp_U$; in more homotopy-theoretic terms, it is the homomorphism $\mu \rightarrow \mu_{MSp} KO$ of ring spectra.

##### Pseudo-Anosov maps with small dilatation

Fix an orientable surface $S$. It is known that the set of dilatations of all pseudo-Anosov maps acting on $S$ is a family of real numbers that is bounded below by 1, and has a minimum value $\lambda_{min,S}>1$ which is realized geometrically.

##### Geometrical variational problems in economics

The monopolist's problem of deciding what types of products to manufacture and how much to charge for each of them, knowing only statistical information about the preferences of an anonymous field of potential buyers, is one of the basic problems analyzed in economic theory.

##### Product Formulas for Measures and Applications to Analysis

Joint Analysis Seminar and PACM Colloquium

We will discuss elementary product formalisms for positive measures. These appeared in analysis for purposes of examining "harmonic measures" related to elliptic equations (work of R. Fefferman, J. Pipher, C. Kenig).

##### Compact aspherical manifolds whose fundamental groups have center

Classical work of Borel had shown that an action of the circle on a manifold with contractible universal cover yields non-trivial center in the manifold's fundamental group. In the early 70's, Conner and Raymond made further deep investigations which led them to conjecture a converse to Borel's result.

##### Stirring Tails of Evolution

One of the most fundamental issues in biology is the nature of evolutionary transitions from single cell organisms to multicellular ones. Not surprisingly for microscopic life in a fluid environment, many of the processes involved are related to transport and locomotion, for efficient exchange of chemical species with the environment is one of the most basic features of life.

##### Endoscopic transfer of the Bernstein center

The Langlands-Shelstad theory of endoscopy plays a central role in the study of Shimura varieties and the Arthur-Selberg trace formula. The fundamental lemma and a deep consequence, endoscopic transfer, have now been established in works of Ngo, Waldspurger, and Hales.

##### The group number function: the number of groups of a given order

The group number, gnu(n) of n, is defined to be the number of groups of order n. It is now known for all n < 211. I shall discuss the peculiar properties of this function and of the related function moa(n), defined to be the least number N for which gnu(N)=n.

##### Moment-angle complexes from simplicial posets

The construction of moment-angle complexes may be extended from simplicial complexes to simplicial posets. As a result, a certain $T^m$-space $Z_S$ is associated to an arbitrary simplicial poset S on m vertices. Face rings $Z[S]$ of simplicial posets generalise those of simplicial complexes, but have much more complicated algebraic structure.

##### Geometric structures on moment-angle manifolds

Moment-angle complexes are spaces acted on by a torus and parametrised by finite simplicial complexes. They are central objects in toric topology, and currently are gaining much interest in the homotopy theory. Due the their combinatorial origins, moment-angle complexes also find applications in combinatorial geometry and commutative algebra.

##### Minimal fillings and boundary rigidity - a survey

A Riemannian manifold with boundary is said to be boundary rigid if its metric is uniquely determined by the boundary distance function, that is the restriction of the distance function to the boundary. Loosely speaking, this means that the Riemannian metric can be recovered from measuring distances between boundary points only.

##### From (basic) image denoising to surface evolution

It is relatively easy to make a connection between the implicit time-discrete approaches for the mean curvature flow and the "Rudin-Osher-Fatemi" total variation based approach for image denoising.

##### Global dynamics for the ion equation in the Euler-Poisson system

We prove that small perturbations of a constant background in the Euler-Poisson equation for the ions lead to global smooth solutions. This is a manifestation of the stabilization effect of the electric field as the corresponding result does not hold for the closely related compressible Euler equation.

##### The Geometry of Music

In my talk, I explain how to translate basic concepts of music theory into the language of contemporary topology and geometry.

##### Eigenfunctions and nodal sets

Nodal sets are the zero sets of eigenfunctions of the Laplacian on a Riemannian manifold (M, g). If $\Delta \phi = \lambda^{2 \phi}$, then $\phi$ is somewhat analogous to a polynomial of degree $\lambda$ and its nodal set is somewhat analogous to a real algebraic variety of this degree. The analogy is closest if (M, g) is real analytic.