Past

*****     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.

Convergence of the Bayes posterior measure is considered in canonical statistical settings (like density estimation or nonparametric regression) where observations sit on a geometrical object such as a compact manifold, or more generally on a compact metric space verifying some conditions.

A natural geometric prior based on randomly rescaled solutions of the heat equation is considered. Upper and lower bound posterior contraction rates are derived.

A-infinity algebras arise whenever one has a multiplication which is "associative up to homotopy". There is an important theory of minimal models which involves studying differential graded algebras via A-infinity structures on their homology algebras. However, this only works well over a ground field. Recently Sagave introduced the more general notion of a derived A-infinity algebra in order to extend the theory of minimal models to a general commutative ground ring.

Operads provide a very nice way of saying what A-infinity algebras are - they are described by a kind of free resolution of a strictly associative structure. I will explain the analogous result for derived A_infinity algebras - these are obtained in the same manner from a strictly associative structure with an extra differential.

This is joint work with Muriel Livernet and Constanze Roitzheim.

Jump processes, and Lévy processes in particular, are notoriously difficult to simulate. The task becomes even harder if the process is stopped when it crosses a certain boundary, which happens in applications to barrier option pricing or structural credit risk models. In this talk, I will present novel adaptive discretization

schemes for the simulation of stopped Lévy processes, which are several orders of magnitude faster than the traditional approaches based on uniform discretization, and provide an explicit control of the bias. The schemes are based on sharp asymptotic estimates for the exit probability and work by recursively adding discretization dates in the parts of the trajectory which are close to the boundary, until a specified error tolerance is met.

This is a joint work with Jose Figueroa-Lopez (Purdue).