Past

Klein's famous lecture proposes that to study geometry we study homogeneous spaces ie study transformation groups acting on a space. E. Cartan found a generalization now known as "Cartan geometries", these are a curved generalization of homogeneous spaces, eg Riemannian manifolds are Cartan geometries modeled on {Euclidean group}/{orthogonal group}.

Topics for my talk will be

Cartan geometries / Cartan connections

Parabolic geometries - a special class of Cartan geometries

Examples - depending on how much time but I will probably explain conformal

geometry as a parabolic geometry

Digital trees is a general structure to manipulate sequences of characters. We propose a novel approach to the structure of digital trees.

It shades some new light on the profile of digital trees, and provides a unified explanation of the relationships between different kinds of digital trees. The idea relies on the distinction of nodes based on their type, i.e., the set of their children. Only two types happen to matter when studying the number of nodes lying at a specified level: the nodes with a full set of children which constitutes the core, and the nodes with a single child producing spaghetti-like trees hanging down the core. We will explain the distinction and its applications on a number of examples related to data structures such as the TST of Bentley and Sedgewick.

This is joint work with Luc Devroye.

The theory of Special Lagrangian (SL) submanifolds is the natural point of intersection between various classical (Lagrangian and volume-minimizing submanifolds) and contemporary (Mirror Symmetry and invariants of Calabi-Yau manifolds) topics. The key problem is how to characterize the compactified moduli space of SLs. Equivalently, to understand which SL singularities admits desingularizations. Our aim is to present some explicit examples, topological results and simple observations which shed some light on the nature and complexity of this problem, and which we expect will be a useful foundation for future progress in the field. This is joint work with M. Haskins (Imperial College), cfr. arXiv:math/0609352.

A central task in conservation biology is measuring, predicting, and preserving biological diversity as species face extinction. Dating back to 1992, phylogenetic diversity is a prominent notion for measuring the biodiversity of a collection of species. This talk gives a flavour of some the combinatorial and algorithmic problems and recent solutions associated with computing this measure. This is joint work with Magnus Bordewich (Durham University, UK) and Andreas Spillner (University of East Anglia, UK).

We show how renormalisation methods similar to the ones used by physicists to make sense of Feynman integrals can be implemented to make sense of sums on infinite cones. On the basis of joint work with D. Manchon, we also discuss multiple zeta functions which can be seen as sums on a specific class of infinite cones.

I will present a general formalism for understanding coloured operads of different flavours, such as cyclic operads, modular operads and topological field theories. The talk is based on arXiv:math/0701767.

The discrete structure of the ground states of a spin system is often neglected by averaging on a mesoscopic scale and thus capturing the main features of the model while simplifying its analysis. In many cases this procedure is not rigorous and not even well understood. In this talk we show that the coarse graining procedure for the XY (N-dimensional, possibly anysotropic) spin type model can be made rigorous by using Gamma-convergence. In the two-dimesional case we show how it is possible to address the same problem for a model with long-range interactions. Finally we discuss several possible developments and present some open problems.

Abstract: Following Donaldson's approach we compute the Calabi-Yau metric on quintics, a four-generation quotient, Schoen threefolds and quotients thereof. Using the explicit Calabi-Yau metric, we then compute eigenvalues and eigenmodes of the Laplace operator.

By classical results of Tarski and Artin-Schreier, the elementary theory of the field of real numbers can be axiomatized in purely Galois-theoretic terms by describing the absolute Galois group of the field. Using work of Ax-Kochen/Ershov and a p-adic analogue of the Artin-Schreier theory the same can be proved for the field $\mathbb{Q}_p$ of p-adic numbers and for very few other fields.

Replacing, however, the absolute Galois group of a field K by that of the rational function field $K(t)$ over $K$, one obtains a Galois-theoretic axiomatiozation of almost arbitrary perfect fields. This gives rise to a new approach to longstanding decidability questions for fields like

$F_p((t))$ or $C(t)$.

The Landscape problem in String Theory is the fact that there are apparently a great many possible vacua; each leading to a very different four dimensional world. I will give a survey of the space of possibilities and then argue that we may, after all, live in a naturally defined tip of the distribution.

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.]

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.

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

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.

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.