HColl_2012

Mean curvature flow is the negative gradient flow for volume, so any hyper-surface flows through hyper-surfaces in the direction of steepest descent for volume and eventually becomes extinct in finite time. Before it becomes extinct, topological changes can occur as it goes through singularities. Thus in some sense the topology is encoded in the singularities. In this lecture I will discuss new and old results about singularities of mean curvature flow focusing on some very recent results about generic singularities.

Joachim Schwermer (Wien): On arithmetically defined hyperbolic manifolds and their Betti numbers

An orientable hyperbolic n-manifold is isometric to the quotient of hyper-bolic n-space H by a discrete torsion free subgroup of the group oforientation-preserving isometries of H. Among these manifolds, the ones originating from arithmetically defined groups form a family of special interest. Due to the underlying connections with number theory and the theory of automorphic forms, there is a fruitful interaction between geometric and arithmetic questions, methods and results. We intend to give an account of recent investigations in this area, in particular, of those pertaining to hyperbolic 3-manifolds and bounds for their Betti numbers.

Sug Woo Shin (MIT): Families of automorphic forms

I Will survey some conjectures and statistical results on families of automorphic forms, for instance concerning the equidistribution of local invariants and the growth of fields of rationality. Recent results with Nicolas Templier will be discussed.

Steve Smale (Hong Kong/Berkeley) : Amino Acid Chains and Peptide Binding Predictions

We will give some mathematical foundations for immunology via a geometry of kernels and metric spaces, finite spaces, and apply that to spaces associated to genes.

Herbert Spohn: Interface motion and the KPZ universality class

The Kardar-Parisi-Zhang (KPZ) equation is a stochastic PDE which models the motion of an interface bordering a stable against a metastable phase. In my talk I will survey the spectacular progress in our mathematical understanding of the KPZ equation in one space dimension over the recent years.

Katrin Wendland (Freiburg): A geometric approach to topological quantum field theories

From the viewpoint of geometry one is naturally led to the study of singularities, that is, of apparently discontinuous phenomena in geometry. Classical examples of such singularities include the so-called catastrophes. The mathematical structures arising in catastrophe theory, on the other hand, are also the fundamental mathematical ingredients of topological quantum field theories. This yields the geometric approach to topological quantum field theories which will be the central theme of this talk.