Past

Extend the affine Weyl group action in Lecture I to double affine Hecke algebra action, and (hopefully) more examples.

Gravitational instantons are complete hyperkaehler 4-manifolds whose Riemann curvature tensor is square integrable. They can be viewed as Einstein geometry analogs of finite energy Yang-Mills instantons on Euclidean space. Classical examples include Kronheimer's ALE metrics on crepant resolutions of rational surface singularities and the ALF Riemannian Taub-NUT metric, but a classification has remained largely elusive. I will present a large, new connected family of gravitational instantons, based on removing fibers from rational elliptic surfaces, which contains ALG and ALH spaces as well as some unexpected geometries.

Introduce the parabolic Hitchin fibration, construct the affine Weyl group action on its fiberwise cohomology, and study one example.

It will be shown how the critical mass classical Keller-Segel system and

the critical displacement convex fast-diffusion equation in two

dimensions are related. On one hand, the critical fast diffusion

entropy functional helps to show global existence around equilibrium

states of the critical mass Keller-Segel system. On the other hand, the

critical fast diffusion flow allows to show functional inequalities such

as the Logarithmic HLS inequality in simple terms who is essential in the

behavior of the subcritical mass Keller-Segel system. HLS inequalities can

also be recovered in several dimensions using this procedure. It is

crucial the relation to the GNS inequalities obtained by DelPino and

Dolbeault. This talk corresponds to two works in preparation together

with E. Carlen and A. Blanchet, and with E. Carlen and M. Loss.

Probability measures in infinite dimensional spaces especially that induced by stochastic processes are the main objects of the talk. We discuss the role played by measures on analysis on path spaces, Sobolev inequalities, weak formulations and local versions of such inequalities related to Brownian bridge measures.

We relate the partition function associated with a certain Brownian directed polymer model to a diffusion process which is closely related to a quantum integrable system known as the quantum Toda lattice. This result is based on a `tropical' variant of a combinatorial bijection known as the Robinson-Schensted-Knuth (RSK) correspondence and is completely analogous to the relationship between the length of the longest increasing subsequence in a random permutation and the Plancherel measure on the dual of the symmetric group.

Abstract: describe several results on the convergence of approximation schemes for possibly degenerate, linear or nonlinear parabolic equations which apply in particular to equations arising in option pricing or portfolio management. We address both the questions of the convergence and the rate of convergence.

This is the session for industrial sponsors of the MSc in MM and SC to present the project ideas for 2010-11 academic year. Potential supervisors should attend to clarify details of the projects and meet the industrialists.

We consider certain q-series depending on parameters (A,B,C), where A is

a positive definite r times r matrix, B is a r-vector and C is a scalar,

and ask when these q-series are modular forms. Werner Nahm (DIAS) has

formulated a partial answer to this question: he conjectured a criterion

for which A's can occur, in terms of torsion in the Bloch group. For the

case r=1, the conjecture has been show to hold by Don Zagier (MPIM and

CdF). For r=2, Masha Vlasenko (MPIM) has recently found a

counterexample. In this talk we'll discuss various aspects of Nahm's conjecture.

A kinetic theory for swarming systems of interacting individuals will be described with and without noise. Starting from the the particle model \cite{DCBC}, one can construct solutions to a kinetic equation for the single particle probability distribution function using distances between measures \cite{dobru}. Analogously, we will discuss the mean-field limit for these problems with noise.

We will also present and analys the asymptotic behavior of solutions of the continuous kinetic version of flocking by Cucker and Smale The large-time behavior of the distribution in phase space is subsequently studied by means of particle approximations and a stability property in distances between measures. It will be shown that the solutions concentrate exponentially fast their velocity to their mean while in space they will converge towards a translational flocking solution.