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.

Optimization problems with time-periodic parabolic PDE constraints can arise in important chemical engineering applications, e.g., in periodic adsorption processes. I will present a novel direct numerical method for this problem class. The main numerical challenges are the high nonlinearity and high dimensionality of the discretized problem. The method is based on Direct Multiple Shooting and inexact Sequential Quadratic Programming with globalization of convergence based on natural level functions. I will highlight the use of a generalized Richardson iteration with a novel two-grid Newton-Picard preconditioner for the solution of the quadratic subproblems. At the end of the talk I will explain the principle of Simulated Moving Bed processes and conclude with numerical results for optimization of such a process.

I will begin by defining the space of algebraic metrics in a particular Kahler class and recalling the Tian-Ruan-Zelditch result saying that they are dense in the space of all Kahler metrics in this class.  I will then discuss the relationship between some special algebraic metrics called 'balanced metrics' and distinguished Kahler metrics (Extremal metrics, cscK, Kahler-Ricci solitons...). Finally I will talk about some numerical algorithms due to Simon Donaldson for finding explicit examples of these balanced metrics (possibly with some pictures).

I am going to introduce Thompson's groups F, T and V. They can be seen in two ways: as functions on [0,1] or as isomorphisms acting on trees.

Modern differential geometry is the art of the abstract that can be pictured. Continuum mechanics is the abstract description of concrete material phenomena. Their encounter, therefore, is as inevitable and as beautiful as the proverbial chance meeting of an umbrella and a sewing machine on a dissecting table. In this rather non-technical and lighthearted talk, some of the surprising connections between the two disciplines will be explored with a view at stimulating the interest of applied mathematicians.

On a manifold there is the graded algebra of polyvector fields with its Lie-Schouten bracket, and the module of de Rham differentials with exterior differentiation. This package is called a "calculus". The moduli

space of sheaves (or derived category objects) on a Calabi-Yau threefold has a kind of "virtual calculus" on it, at least conjecturally. In particular, this moduli space has virtual de Rham cohomology groups, which categorify Donaldson-Thomas invariants, at least conjecturally. We describe some attempts at constructing such a virtual calculus. This is work in progress.

How many triangles must a graph of density d contain? This old question due to Erdos was recently answered by Razborov, after many decades of progress by numerous authors.

We will consider the analogous question for tripartite graphs. Given a tripartite graph with prescribed edges densities between each

pair of classes how many triangles must it contain?

On a manifold there is the graded algebra of polyvector fields with its Lie-Schouten bracket, and the module of de Rham differentials with exteriour differentiation. This package is called a "calculus". The moduli space of sheaves (or derived category objects) on a Calabi-Yau threefold has a kind of "virtual calculus" on it, at least conjecturally. In particular, this moduli space has virtual de Rham cohomology groups, which categorify Donaldson-Thomas invariants, at least conjecturally. We describe some attempts at constructing such a virtual calculus. This is work in progress.

We investigate the behaviour of free-surface waves on time-varying potential flow in the limit as the Froude number becomes small. These waves are exponentially small in the Froude number, and are therefore inaccessible to ordinary asymptotic methods. As such, we demonstrate how exponential asymptotic techniques may be applied to the complexified free surface in order to extract information about the wave behaviour on the free surface, using a Lagrangian form of the potential flow equations. We consider the specific case of time-varying flow over a step, and demonstrate that the results are consistent with the steady state case.

Let $u \in W^{1,p}(\Omega,\R^N)$, $\Omega$ a bounded domain in

$\R^n$, be a minimizer of a convex variational integral or a weak solution to

an elliptic system in divergence form. In the vectorial case, various

counterexamples to full regularity have been constructed in dimensions $n

\geq 3$, and it is well known that only a partial regularity result can be

expected, in the sense that the solution (or its gradient) is locally

continuous outside of a negligible set. In this talk, we shall investigate

the role of the space dimension $n$ on regularity: In arbitrary dimensions,

the best known result is partial regularity of the gradient $Du$ (and hence

for $u$) outside of a set of Lebesgue measure zero. Restricting ourselves to

the partial regularity of $u$ and to dimensions $n \leq p+2$, we explain why

the Hausdorff dimension of the singular set cannot exceed $n-p$. Finally, we

address the possible existence of singularities in two dimensions.

We prove that no Riemannian manifold quasiisometric to

complex hyperbolic plane can have a better curvature pinching. The proof

uses cup-products in $L^p$-cohomology.

We study a simple heat-bath type dynamic for a simple model of
polymer interacting with an interface. The polymer is a nearest neighbor path in
Z, and the interaction is modelised by energy penalties/bonuses given when the
path touches 0. This dynamic has been studied by D. Wilson for the case without
interaction, then by Caputo et al. for the more general case. When the interface
is repulsive, the dynamic slows down due to the appearance of a bottleneck in the
state space, moreover, the systems exhibits a metastable behavior, and, after time
rescaling, behaves like a two-state Markov chain.

We consider a directed random polymer interacting with an interface
that carries random charges some of which attract while others repel
the polymer. Such a polymer can be in a localized or delocalized
phase, i.e., it stays near the interface or wanders away respectively.
 The phase it chooses depends on the temperature and the average bias
of the disorder. At a given temperature, there is a critical bias
separating the two phases. A question of particular interest, and
which has been studied extensively in the Physics and Mathematics
literature, is whether the quenched critical bias differs from the
annealed critical bias. When it does, we say that the disorder is
relevant.

Using a large deviations result proved recently by Birkner, Greven,
and den Hollander, we derive a variational formula for the quenched

critical bias. This leads to a necessary and sufficient condition for
disorder relevance that implies easily some known results as well as
new ones.

The talk is based on joint work with Frank den  Hollander.