Past

"We introduce some type of generalized Poisson formula which is equivalent 
to Langlands' automorphic transfer from an arbitrary reductive group over a 
global field to a general linear group."

The martingale optimal transportation problem is motivated by

model-independent bounds for the pricing and hedging exotic options in

financial mathematics.

In the simplest one-period model, the dual formulation of the robust

superhedging cost differs from the standard optimal transport problem by

the presence of a martingale constraint on the set of coupling measures.

The one-dimensional Brenier theorem has a natural extension. However, in

the present martingale version, the optimal coupling measure is

concentrated on a pair of graphs which can be obtained in explicit form.

These explicit extremal probability measures are also characterized as

the unique left and right monotone martingale transference plans, and

induce an optimal solution of the kantorovitch dual, which coincides

with our original robust hedging problem.

By iterating the above construction over n steps, we define a Markov

process whose distribution is optimal for the n-periods martingale

transport problem corresponding to a convenient class of cost functions.

Similarly, the optimal solution of the corresponding robust hedging

problem is deduced in explicit form. Finally, by sending the time step

to zero, this leads to a continuous-time version of the one-dimensional

Brenier theorem in the present martingale context, thus providing a new

remarkable example of Peacock, i.e. Processus Croissant pour l'Ordre

Convexe. Here again, the corresponding robust hedging strategy is

obtained in explicit form.

*****     PLEASE NOTE THAT THIS WILL TAKE PLACE ON FRIDAY 7TH JUNE     *****

Collective behaviours of coupled linear or nonlinear resonators have been of interest to engineers as well as mathematician for a long time. In this presentation, using the example of coupled resonant nano-sensors (which leads to a Linear pencil with a Jacobian matrix), I will show how previously feared and often avoided coupling between nano-devices along with their weak nonlinear behaviour can be used with inverse eigenvalue analysis to design multiple-input-single-output nano-sensors. We are using these matrices in designing micro/Nano electromechanical systems, particularly resonant sensors capable for measuring very small mass for use as environmental as well as biomedical monitors. With improvement in fabrication technology, we can design and build several such sensors on one substrate. However, this leads to challenges in interfacing them as well as introduces undesired parasitic coupling. More importantly, increased nonlinearity is being observed as these sensors reduce in size. However, this also presents an opportunity to experimentally study chains or matrices of coupled linear and/or nonlinear structures to develop new sensing modalities as well as to experimentally verify theoretically or numerically predicted results. The challenge for us is now to identify sensing modalities with chain of linear or nonlinear resonators coupled either linearly or nonlinearly. We are currently exploring chains of Duffing resonators, van der Pol oscillators as well as FPU type lattices.

Some real resource allocation problems are so large and complex that optimization would computationally infeasible, even with complete information about all the relevant values. For example, the proposal in the US to use television broadcasters' bids to determine which stations go off air to make room for wireless broadband is characterized by hundreds of thousands of integer constraints. We use game theory and auction theory to characterize a class of simple, strategy-proof auctions for such problems and show their equivalence to a class of "clock auctions," which make the optimal bidding strategy obvious to all bidders. We adapt the results of optimal auction theory to reduce expected procurement costs and prove that the procurement cost of each clock auction is the same as that of the full information equilibrium of its related paid-as-bid (sealed-bid) auction.

An externally definable set of a first order structure $M$ is a set of the form $X\cap M^n$ for a set $X$ that is parametrically definable in some elementary extension of $M$. By a theorem of Shelah, these sets form again a first order structure if $M$ is NIP. If $M$ is a real closed field, externally definable sets can be described as some sort of limit sets (to be explained in the talk), in the best case as Hausdorff limits of definable families. It is conjectured that the Shelah structure on a real closed field is generated by expanding the field with convex subsets of the line. This is known to be true in the archimedean case by van den Dries (generalised by Marker and Steinhorn). I will report on recent progress around this question, mainly its confirmation on real closed fields that are close to being maximally valued with archimedean residue field. The main tool is an algebraic characterisation of definable types in real closed valued fields. I also intend to give counterexamples to a localized version of the conjecture. This is joint work with Francoise Delon.

I will show how to associate a quaternion algebra to a hyperbolic 3-manifold. I will then go through some examples and applications of this theory

The coastal ocean contains a diversity of physical and biological

processes, often occurring at vastly different scales. In this talk,

we will outline some of these processes and their mathematical

description. We will then discuss how finite element methods are used

in coastal ocean modeling and recent research into

improvements to these algorithms. We will also highlight some of the

successes of these methods in simulating complex events, such as

hurricane storm surges. Finally, we will outline several interesting

challenges which are ripe for future research.

Consider in the phase space of classical mechanics a Radon measure that is a probability density carried by the graph of a Lipschitz continuous (or even less regular) vector field. We study the structure of the push-forward of such a measure by a Hamiltonian flow. In particular, we provide an estimate on the number of folds in the support of the transported measure that is the image of the initial graph by the flow. We also study in detail the type of singularities in the projection of the transported measure in configuration space (averaging out the momentum variable). We study the conditions under which this projected measure can have atoms, and give an example in which the projected measure is singular with respect to the Lebesgue measure and diffuse. We discuss applications of our results to the classical limit of the Schrödinger equation. Finally we present various examples and counterexamples showing that our results are sharp.

In this talk we study numerical approximations of continuous solutions to a nonlocal $p$-Laplacian type diffusion equation, \[ u_t (t, x) = \int_\Omega J(x − y)|u(t, y) − u(t, x)|^{p-2} (u(t, y) − u(t, x)) dy. \]

First, we find that a semidiscretization in space of this problem gives rise to an ODE system whose solutions converge uniformly to the continuous one, as the mesh size goes to zero. Moreover, the semidiscrete approximation shares some properties with the continuous problem: it preserves the total mass and the solution converges to the mean value of the initial condition, as $t$ goes to infinity.

Next, we discretize also the time variable and present a totally discrete method which also enjoys the above mentioned properties.

In addition, we investigate the limit as $p$ goes to infinity in these approximations and obtain a discrete model for the evolution of a sandpile.

Finally, we present some numerical experiments that illustrate our results.

This is a joint work with J. D. Rossi.

 We consider two decision problems for linear recurrence sequences (LRS) 
over the integers, namely the Positivity Problem (are all terms of a given 
LRS positive?) and the Ultimate Positivity Problem (are all but finitely 
many terms of a given LRS positive?). We show decidability of both 
problems for LRS of order 5 or less, and for simple LRS (i.e. whose 
characteristic polynomial has no repeated roots) of order 9 or less. Our 
results rely on on tools from Diophantine approximation, including Baker's 
Theorem on linear forms in logarithms of algebraic numbers. By way of 
hardness, we show that extending the decidability of either problem to LRS 
of order 6 would entail major breakthroughs on Diophantine approximation 
of transcendental numbers.

This is joint with work with Joel Ouaknine and Matt Daws.

The Einstein equation from general relativity is a

quasilinear hyperbolic, geometric PDE (when viewed in an appropriate

coordinate system) for a manifold. A particularly interesting set of

known, exact solutions describe black holes. The wave and Maxwell

equations on these manifolds are models for perturbations of the known

solutions and have attracted a significant amount of attention in the

last decade. Key estimates are conservation of energy and Morawetz (or

integrated local energy) estimates. These can be proved using both

Fourier analytic methods and more geometric methods. The main focus of

the talk will be on decay estimates for solutions of the Maxwell

equation outside a slowly rotating Kerr black hole.

I will talk about random walks on groups and define the Poisson boundary of such. Studying it gives criteria for amenability or growth. I will outline how this can be used and describe recent related results and still open questions.

Saturated fusion systems are both a convenient language in which to formulate p-local finite simple group theory and interesting structures in their own right. In this talk, we will start by explaining what is meant by a 'tree of fusion systems' and give conditions on such an object for there to exist a saturated completion. We then describe how this theory can be used to understand a class of fusion systems first considered by Bob Oliver, which are determined by modular representations of finite groups. If time permits, we will discuss joint work with David Craven towards a complete classification of such fusion systems. The talk is aimed at a general mathematical audience with some background in algebra.

Let g be the complex semisimple Lie algebra associated to a complex semisimple algebraic group G, b a Borel subalgebra of g, h the Cartan sublagebra contained in b and N the unipotent subgroup corresponding to the nilradical n of b. Extremal projection operators are projection operators onto the subspaces of n-invariants in certain g-modules the action of n on which is locally nilpotent. Zhelobenko also introduced a family of operators which are analogues to extremal projection operators. These operators are called now Zhelobenko operators.
I shall show that the explicit formula for the extremal projection operator for g obtained by Asherova, Smirnov and Tolstoy and similar formulas for Zhelobenko operators are related to the existence of a birational equivalence (N, h) -> b given by the restriction of the adjoint action. Simple geometric proofs of  formulas for the ``classical'' counterparts of the extremal projection operator and of Zhelobenko operators are also obtained.