Past

We discuss the convergence in distribution of rescaled random planar maps viewed as random metric spaces. More precisely, we consider a random planar map M(n), which is uniformly distributed over the set of all planar maps with n faces in a certain class. We equip the set of vertices of M(n) with the graph distance rescaled by the factor n to the power 1/4. We then discuss the convergence in distribution of the resulting random metric spaces as n tends to infinity, in the sense of the Gromov-Hausdorff distance between compact metric spaces. This problem was stated by Oded Schramm in his plenary address paper at the 2006 ICM, in the special case of triangulations.

In the case of bipartite planar maps, we first establish a compactness result showing that a limit exists along a suitable subsequence. Furthermore this limit can be written as a quotient space of the Continuum Random Tree (CRT) for an equivalence relation which has a simple definition in terms of Brownian labels attached to the vertices of the CRT. Finally we show that any possible limiting metric space is almost surely homomorphic to the 2-sphere. As a key tool, we use bijections between planar maps and various classes of labelled trees.

We study the dynamic of a very simple chain of three anharmonic oscillators with linear nearest-neighbour couplings. The first and the last oscillator furthermore interact with heat baths through friction and noise terms. If all oscillators in such a system are coupled to heat baths, it is well-known that under relatively weak coercivity assumptions, the system has a spectral gap (even compact resolvent) and returns to equilibrium exponentially fast. It turns out that while it is still possible to show the existence and uniqueness of an invariant measure for our system, it returns to equilibrium much slower than one would at first expect. In particular, it no longer has compact resolvent when the potential of the oscillators is quartic and the spectral gap is destroyed when it grows even faster.

Abstract: Toric geometry provides powerful and efficient combinatorial tools for the construction and analysis of Calabi-Yau manifolds. After recollections of the hypersurface case I present recent results on new Calabi-Yau 3-folds and their mirrors via conifold transitions, ideas for generalizations to higher codimensions and applications to string theory.

Random planar curves arise in a natural way in statistical mechanics, for example as the boundaries of clusters in critical percolation or the Ising model. There has been a great deal of mathematical activity in recent years in understanding the measure on these curves in the scaling limit, under the name of Schramm-Loewner Evolution (SLE) and its extensions. On the other hand, the scaling limit of these lattice models is also believed to be described, in a certain sense, by conformal field theory (CFT). In this talk, after an introduction to these two sets of ideas, I will give a theoretical physicist's viewpoint on possible direct connections between them.

John Cardy studied Mathematics at Cambridge. After some time at CERN, Geneva he joined the physics faculty at Santa Barbara. He moved to Oxford in 1993 where he is a Senior Research Fellow at All Souls College and a Professor of Physics. From 2002-2003 and 2004-2005 he was a member of the IAS, Princeton. Among other work on the applications of quantum field theory, in the 1980s he helped develop the methods of conformal field theory. Professor Cardy is a Fellow of the Royal Society, a recipient of the 2000 Paul Dirac Medal and Prize of the Institute of Physics, and of the 2004 Lars Onsager Prize of the American Physical Society "for his profound and original applications of conformal invariance to the bulk and boundary properties of two-dimensional statistical systems."

To understand the definable sets of a theory, it is helpful to have some invariants, i.e. maps from the definable sets to somewhere else which are invariant under definable bijections. Denef and Loeser constructed a very strong such invariant for the theory of pseudo-finite fields (of characteristic zero): to each definable set, they associate a virtual motive. In this way one gets all the known cohomological invariants of varieties (like the Euler characteristic or the Hodge polynomial) for arbitrary definable sets.

I will first explain this, and then present a generalization to other fields, namely to perfect, pseudo-algebraically closed fields with pro-cyclic Galois group. To this end, we will construct maps between the set of definable sets of different such theories. (More precisely:

between the Grothendieck rings of these theories.) Moreover, I will show how, using these maps, one can extract additional information about definable sets of pseudo-finite fields (information which the map of Denef-Loeser loses).

Model reduction (also called system reduction, order reduction) is an ubiquitous tool in the analysis and simulation of dynamical systems, control design, circuit simulation, structural dynamics, CFD, etc. In the past decades many approaches have been developed for reducing the complexity of a given model. In this introductory talk, we will survey some of the most prominent methods used for linear systems, compare their properties and highlight similarities. In particular, we will emphasize the role of recent developments in numerical linear algebra in the different approaches. Efficiently using these techniques, the range of applicability of some of the methods has considerably widened.

The performance of several approaches will be demonstrated using real-world examples from a variety of engineering disciplines.

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