Past

I will address the usage of the elliptic reconstruction technique (ERT) in a posteriori error analysis for fully discrete schemes for parabolic partial differential equations. A posteriori error estimates are effective tools in error control and adaptivity and a mathematical rigorous derivation justifies and improves their use in practical implementations.

The flexibility of the ERT allows a virtually indiscriminate use of various parabolic PDE techniques such as energy methods, duality methods and heat-kernel estimates, as opposed to direct approaches which leave less maneuver room. Thanks to ERT parabolic stability techniques can be combined with different elliptic a posteriori error analysis techniques, such as residual or recovery estimators, to derive a posteriori error bounds. The method has the merit of unifying previously known approaches, as well as providing new ones and providing us with novel error bounds (e.g., pointwise norm error bounds for the heat equation). [These results are based on joint work with Ch. Makridakis and A. Demlow.]

Another feature, which I would like to highlight, of the ERT is its simplifying power. It allows us to derive estimates where the analysis would be very complicated otherwise. As an example, I will illustrate its use in the context of non-conforming methods, with a special eye on discontinuous Galerkin methods. [These are recent results obtained jointly with E. Georgoulis.]

Local Bifurcation Theory (I): Theorem of Crandall and Rabinowitz

We will prove an extended Poincaré - Hopf theorem, identifying several invariants of a manifold $M$. These are its Euler characteristic $\chi(M)$, the sum $\sum_{x_i} ind_V(x_i)$ of indices at zeroes of a vector field $V$ on $M$, the self-intersection number $\Delta \cap \Delta$ of the diagonal $\Delta \subset M \times M$ and finally the integral $\int_M e(TM)$ of the Euler class of the tangent bundle.

We show that the diameter of G(n,p) is concentrated on one of three values provided the average degree p(n-1) goes to infity with n. This is joint work with N. Fountoulakis even though he refuses to admit it.

It has been known for some time that in more than 4 spacetime dimensions, there is a considerably larger variety of black "hole" solutions, having e.g. different horizon topology. In particular, they are no longer fully characterized by their asymptotic charges (mass, angular momenta) alone. We give a partial classification theorem for higher dimensions, for solutions with sufficiently many axial Killing fields. We show that higher dimensional black holes may be fully characterized by their asymptotic charges, together with certain "moduli" and "winding numbers" that are analogous to those of Seiffert fibrations. In particular, we find constraints on the possible horizon topologies. In 5 dimensions, they may be either a black "hole" (sphere), black "ring", or a black "lens".

Electrical impedance tomography (EIT) technique has been an active research topic since the early 1980s. In EIT, one measures the boundary voltages due to multiple injection currents to reconstruct images of the conductivity distribution. However, these boundary voltages are insensitive to a local change of the conductivity distribution and the relation between them is highly nonlinear. Medical imaging has been one of the important application areas of EIT. Indeed, biological tissues have different electrical properties that change with cell concentration, cellular structure, and molecular composition. Such changes of electrical properties are the manifestations of structural, functional, metabolic, and pathological conditions of tissues, and thus provide valuable diagnostic information. Since all the present EIT technologies are only practically applicable in feature extraction of anomalies, improving EIT calls for innovative measurement techniques that incorporate structural information. The core idea of the approach presented in this talk is to extract more information about the conductivity from data that has been enriched by coupling the electric measurements to localized elastic perturbations. More precisely, we propose to perturb the medium during the electric measurements, by focusing ultrasonic waves on regions of small diameter inside the body. Using a simple model for the mechanical effects of the ultrasound waves, we show that the difference between the measurements in the unperturbed and perturbed configurations is asymptotically equal to the pointwise value of the energy density at the center of the perturbed zone. In practice, the ultrasounds impact a spherical or ellipsoidal zone, of a few millimeters in diameter. The perturbation should thus be sensitive to conductivity variations at the millimeter scale, which is the precision required for breast cancer diagnostic. The material presented in this talk concerning the imaging by perturbation approach, is based on a joint work with Habib Ammari, Eric Bonnetier, Michael Tanter & Mathias Fink and on an ongoing collaboration with Frédéric de Gournay, Otared Kavian and Jérôme Fehrenbach. I will also discuss recent results concerning perturbation of asymptotically small volume fraction which are based on joint works with Michael Vogelius.

Roughly speaking, a quasiregular map is a possibly-branched covering map with bounded distortion. The theory of such maps was developed in the 1970s to carry over to higher dimensions the more geometric aspects of the theory of complex analytic functions of the plane. In this talk I shall outline the proof of rigidity theorems describing the quasiregular self-maps of hyperbolic manifolds. These results rely on an extension of Sela's work concerning the stability of self-maps of hyperbolic groups, and on older topological ideas concerning discrete-open and light-open maps, particularly their effect on fundamental groups. I shall explain how these two sets of ideas also lead to topological rigidity theorems. This talk is based on a paper with a similar title by Bridson, Hinkkanen and Martin (to appear in Compositio shortly). http://www2.maths.ox.ac.uk/~bridson/papers/QRhyp/

Abstract: Supersymmetric gauge theories have a spectrum of chiral operators which are preserved under at least 2 supercharges. These operators are sometimes called BPS operators in the chiral ring. The problem of counting operators in the chiral ring is reasonably simple and reveals information about the moduli space of vacua for the supersymmetric gauge theory. In this talk I will review the counting problem and present exact results on the moduli space of both mesonic and baryonic operators for a large class of gauge theories

We consider the computational complexity of optimizing various classes

of continuous functions over a simplex, hypercube or sphere. These

relatively simple optimization problems arise naturally from diverse

applications. We review known approximation results as well as negative

(inapproximability) results from the recent literature.

OxMOS Team Meetings are held regularly during term and are open to all. Two members of OxMOS will give a short update on their recent research.

In this talk we discusss some notions of small sets in Polish groups. We give some examples and applications of these notions in analysis and group theory. Moreover, we introduce a new notion of smallness which we call Haar meager sets. This notion coincides with the meager sets in locally compact groups. However, it is strictly stronger in the setting of nonlocally compact groups. We argue that this is the right topological analogue of Christian's Haar null sets. The speaker gratefully acknowledges the support of the LMS under a Scheme 2 Grant.

In this talk I'll explore the meaning of the Hankel determinant formula for the general solutions of the Painleve' equations both from the analytic and the geometric point of view. I'll start with the simple example of PII and I'll show how the generating function for the Hankel determinant satisfies two Riccati equations. These linearize into the Jimbo-Miwa-Ueno isomonodromic deformation problem. Indeed this occurs for all the Painleve' equations PII,..,PVI and it is due to the link between their solutions and the infinite Toda lattice equation. I'll then explore the geometric meaning of the Hankel determinants by looking at the (suitably defined) spectral curve of the Toda lattice equation.

Chakravarty and Ablowitz (PRL vol.76 p.857, 1996). showed that a fifth-order equation arizing from the Bianchi IX system can be described asmonodromy evolving (non-preserving) deformations. In my talk, we will show that general Halphen-type systems, which comes from generic DH-IX systems, can be represented as monodromy evolving deformations.

  In this talk we will discuss some recent developments in the study of nonlinear partial differential equations of mixed type, including the mixed parabolic-hyperbolic type and mixed elliptic-hyperbolic type. Examples include nonlinear degenerate diffusion-convection equations and transonic flow equations in fluid mechanics, as well as nonlinear equations of mixed type in a fluid mechanical formulation for isometric embedding problems in differential geometry. Further ideas, trends, and open problems in this direction will be also addressed.  

We present a continuous-time equilibrium-based model for large economic agent, where she trades with market makers at their utility indifference prices. The presentation is based on a joint project with Peter Bank.

There will be three discussion meetings based on aspects of the

programme open to all internal project members. Others interested in

attending should contact Carlos Mora-Corral.

Team meetings, held roughly every four weeks, are open to anyone who is

interested. OxMOS post docs and Dphil students will give updates on the

research.