Past

This talk is regarding PDE systems from geophysical applications with multiple time scales, in which linear skew-self-adjoint operators of size 1/epsilon gives rise to highly oscillatory solutions. Analysis is performed in justifying the limiting dynamics as epsilon goes to zero; furthermore, the analysis yields estimates on the difference between the multiscale solution and the limiting solution. We will introduce a simple yet effective time-averaging technique which is especially useful in general domains where Fourier analysis is not applicable.

Random wavelet series were introduced in the mid 90s as simple and flexible models that allow to take into account observed statistics of wavelet coefficients in signal and image processing. One of their most interesting properties is that they supply random processes whose pointwise regularity jumps form point to point in a very erratic way, thus supplying examples of multifractal processes.

Interest in such models has been renewed recently under the spur of new applications coming from widely different fields; e.g.

-in functional analysis, they allow to derive the regularity properties of ``generic'' functions in a given function space (in the sense of

prevalence)

-they offer toy examples on which one can check the accuracy of numerical algorithms that allow to derive the multifractal parameters associated with signals and images.

We will give an overview of these properties, and we will focus on recent extensions whose sample paths are not locally bounded, and offer models for signals which share this property.

 In this talk I will introduce my joint work with Kreck on a classification of
certain 5-manifolds with fundamental group Z. This result can be interpreted as a
generalization of the classical Browder-Levine's fibering theorem to dimension 5.

The talk will survey some results about smooth and topological 4-manifolds obtained via surgery, and discuss some contrasting information provided by gauge theory about smooth finite group actions on 4-manifolds.

Free probability theory has been introduced by Voiculescu in the 80's for the study of the von Neumann algebras of the free groups. It consists in an algebraic setting of non commutative probability, where one encodes "non commutative random variables" in abstract (non commutative) algebras endowed with linear forms (which satisfies properties in order to play the role of the expectation). In this context, Voiculescu introduce the notion of freeness which is the analogue of the classical independence.

A decade later, he realized that a family of independent random matrices invariant in law by conjugation by unitary matrices are asymptotically free. This phenomenon is called asymptotic freeness. It had a deep impact in operator algebra and probability and has been generalized in many directions. A simple particular case of Voiculescu's theorem gives an estimate, for N large, of the spectrum of an N by N Hermitian matrix H_N = A_N + 1/\sqrt N X_N, where A_N is a given deterministic Hermitian matrix and X_N has independent gaussian standard sub-diagonal entries.

Nevertheless, it turns out that asymptotic freeness does not hold in certain situations, e.g. when the entries of X_N as above have heavy-tails. To infer the spectrum of a larger class of matrices, we go further into Voiculescu's approach and introduce the distributions of traffics and their free product. This notion of distribution is richer than Voiculescu's notion of distribution of non commutative random variables and it generalizes the notion of law of a random graph. The notion of freeness for traffics is an intriguing mixing between the classical independence and Voiculescu's notion of freeness. We prove an asymptotic freeness theorem in that context for independent random matrices invariant in law by conjugation by permutation matrices.

The purpose of this talk is to give an introductory presentation of these notions.

We discuss the superhedging problem under model uncertainty based on existence

and duality results for minimal supersolutions of backward stochastic differential equations.

The talk is based on joint works with Samuel Drapeau, Gregor Heyne and Reinhard Schmidt.

Nowadays there are a large number of satellite and airborne observations of the large ice sheet that covers Antarctica. These include maps of the surface elevation, ice thickness, surface velocity, the rate of snow accumulation, and the rate of change of surface elevation. Uncertainty in the possible rate of future sea level rise motivates using all of these observations and models of ice-sheet flow to project how the ice sheet will behave in future, but this is still a challenge. To make useful predictions, especially in the presence of potential dynamic instabilities, models will need accurate initial conditions, including flow velocity throughout the ice thickness. The ice sheet can be several kilometres thick, but most of the observations identify quantities at the upper surface of the ice sheet, not within its bulk. There is thus a question of how the subsurface flow can be inferred from surface observations. The key parameters that must be identified are the viscosity in the interior of the ice and the basal drag coefficient that relates the speed of sliding at the base of the ice sheet to the basal shear stress. Neither is characterised well by field or laboratory studies, but for incompressible flow governed by the Stokes equations they can be investigated by inverse methods analogous to those used in electric impedance tomography (which is governed by the Laplace equation). Similar methods can also be applied to recently developed 'hybrid' approximations to Stokes flow that are designed to model shallow ice sheets, fast-sliding ice streams, and floating ice shelves more efficiently. This talk will give a summary of progress towards model based projections of the size and shape of the Antarctic ice sheet that make use of the available satellite data. Some of the outstanding problems that will need to be tackled to improve the accuracy of these projections will also be discussed.

A mini-lecture series consisting of four 1 hour lectures.

It has long been a challenge to synthesize the complementary insights offered by model theory and category theory. A small fragment of that challenge is to understand ultraproducts categorically. I will show that, granted some general categorical machinery, the notions of ultrafilter and ultraproduct follow inexorably from the notion of finiteness of a set. The machine in question, known as the codensity monad, has existed in an underexploited state for nearly fifty years. To emphasize that it was not constructed specifically for this purpose, I will mention some of its other applications. This talk represents joint work with an anonymous referee. Little knowledge of category theory will be assumed.

Focused ion beam milling allows small scale single crystal cantilevers to be produced with cross-sectional dimensions on the order of microns which are then tested using a nanoindenter allowing both elastic and plastic materials properties to be measured. EBSD allows these cantilevers to be milled from any desired crystal orientation. Micro-cantilever bending experiments suggest that sufficiently smaller cantilevers are stronger, and the observation is believed to be related to the effect of the neutral axis on the evolution of the dislocation structure. A planar model of discrete dislocation plasticity was used to simulate end-loaded cantilevers to interpret the behaviour observed in the experiments. The model allowed correlation of the simulated dislocation structure to the experimental load displacement curve and GND density obtained from EBSD. The planar model is sufficient for identifying the roles of the neutral axis and source spacing in the observed size effect, and is particularly appropriate for comparisons to experiments conducted on crystals orientated for plane strain deformation. The effect of sample dimensions and dislocation source density are investigated and compared to small scale mechanical tests conducted on Titanium and Zirconium.

I will discuss conjectures relating cup products of cyclotomic units and modular symbols modulo an Eisenstein ideal. In particular, I wish to explain how these conjectures may be viewed as providing a refinement of the Iwasawa main conjecture. T. Fukaya and K. Kato have proven these conjectures under certain hypotheses, and I will mention a few key ingredients. I hope to briefly mention joint work with Fukaya and Kato on variants.

I will be taking us on a journey through low dimensional topology, starting in 2 dimensions motivating handles decompositions in a dimension that we can visualize, moving onto to a brief of note of what this means in 3 dimensions and then moving onto the wild world of 4 manifolds. I will be showing a way in which we can actually try and view a 4 manifold before moving onto a way of manipulating these diagrams to give diffeomorphic 4 manifolds. Hopefully, I will have time to go into some ways in which Kirby calculus has been used to show that certain potential exotic 4 spheres are not exotic and some results on stable diffeomorphims of 4 manifolds.

The behavior of lipid vesicles is due to a complex interplay between the mechanics of the vesicle membrane, the surrounding fluids, and any external fields which may be present. In this presentation two aspects of vesicle behavior are explored: vesicles in inertial flows and vesicles exposed to electric fields.

The first half of the talk will present work done investigating the behavior of two-dimensional vesicles in inertial flows. A level set representation of the interface is coupled to a Navier-Stokes projection solver. The standard projection method is modified to take into account not only the volume incompressibility of the fluids but also the surface incompressibility of the vesicle membrane. This surface incompressibility is enforced by using the closest point level set method to calculate the surface tension needed to enforce the surface incompressibility. Results indicate that as inertial effects increase vesicle change from a tumbling behavior to a stable tank-treading configuration. The numerical results bear a striking similarity to rigid particles in inertial flows. Using rigid particles as a guide scaling laws are determined for vesicle behavior in inertial flows.

The second half of the talk will move to immersed interface solvers for three-dimensional vesicles exposed to electric fields. The jump conditions for pressure and fluid velocity will be developed for the three-dimensional Stokes flow or constant density Navier-Stokes equations assuming a piecewise continuous viscosity and an inextensible interface. An immersed interface solver is implemented to determine the fluid and membrane state. Analytic test cases have been developed to examine the convergence of the fluids solver.

Time permitting an immersed interface solver developed to calculate the electric field for a vesicle exposed to an electric field will be discussed. Unlike other multiphase systems, vesicle membranes have a time-varying potential which must be taken into account. This potential is implicitly determined along with the overall electric potential field.

I will look at a toy model for an index in a large market. The aim is to

consider the pricing of volatility swaps on the index. This is very much

work in progress.

We consider the free boundary problem for the plasma-vacuum interface in ideal compressible magnetohydrodynamics (MHD). In the plasma region the flow is governed by the usual compressible MHD equations, while in the vacuum region we consider the pre-Maxwell dynamics for the magnetic field. At the free-interface, driven by the plasma velocity, the total pressure is continuous and the magnetic field on both sides is tangent to the boundary. The plasma density does not go to zero continuously at the interface, but has a jump, meaning that it is bounded away from zero in the plasma region and it is identically zero in the vacuum region. The plasma-vacuum system is not isolated from the outside world, because of a given surface current on the fixed boundary that forces oscillations.

Under a suitable stability condition satisfied at each point of the initial interface, stating that the magnetic fields on either side of the interface are not collinear, we show the existence and uniqueness of the solution to the nonlinear plasma-vacuum interface problem in suitable anisotropic Sobolev spaces.

The proof follows from the well-posedness of the homogeneous linearized problem and a basic a priori energy estimate, the analysis of the elliptic system for the vacuum magnetic field, a suitable tame estimate in Sobolev spaces for the full linearized equations, and a Nash-Moser iteration.

This is a joint work with Y. Trakhinin (Novosibirsk).

A generalized Baumslag-Solitar group is a group G acting co-compactly on a tree X, with all vertex- and edge stabilizers isomorphic to the free abelian group of rank n. We will discuss the $L^p$-metric and $L^p$-equivariant compression of G, and also the quasi-isometric embeddability of G in a finite product of binary trees. Complete results are obtained when either $n=1$, or the quotient graph $G\X$ is either a tree or homotopic to a circle. This is joint work with Yves Cornulier.

Group actions play an important role in both topological problems and coarse geometric conjectures. I will introduce the idea of a partial action of a group on a metric space and explain, in the case of certain classes of coarsely disconnected spaces, how partial actions can be used to give a geometric proof of a result of Willett and Yu concerning the coarse Baum-Connes conjecture.

Historically, decay rates have been used to provide quantitative and qualitative information on the solutions to hyperbolic conservation laws. Quantitative results include the establishment of convergence rates for approximating procedures and numerical schemes. Qualitative results include the establishment of results on uniqueness and regularity as well as the ability to visualize the waves and their evolution in time.

In this talk, I will present two decay estimates on the positive waves for systems of hyperbolic and genuinely nonlinear balance laws satisfying a dissipative mechanism. The result is obtained by employing the continuity of Glimm-type functionals and the method of generalized characteristics. Using this result on the spreading of rarefaction waves, the rate of convergence for vanishing viscosity approximations to hyperbolic balance laws will also be established. The proof relies on error estimates that measure the interaction of waves using suitable Lyapunov functionals. If time allows, a further application of the recent developments in the theory of balance laws to differential geometry will be addressed.

In this talk, I will discuss homotopy limits: The basics, and why you should care about them if you are a topologist, an algebraic geometer, or an algebraist (have I missed anyone?).