Past

I will discuss ongoing work to provide a proof for the following

conjecture: if the development of a time symmetric, conformally flat

initial data set admits a smooth null infinity, then the initial data

is Schwarzschildean in a neighbourhood of infinity. The strategy

to construct a proof consists in a detailed analysis of a

certain type of expansions that can be obtained using H. Friedrich's

"cylinder at infinity" formalism. I will also discuss a toy model for

the analysis of the Maxwell field near the

spatial infinity of the Schwarzschild spacetime

APOLOGIES - this seminar is cancelled.

Professor Terry Lyons will talk instead on signed probability measures and some old results of Krylov.

It is an interesting exercise to compute the iterated integrals of Brownian Motion and to calculate the expectations (of polynomial functions of these integrals).

Recent work on constructing discrete measures on path space, which give the same value as Wiener measure to certain of these expectations, has led to promising new numerical algorithms for solving 2nd order parabolic PDEs in moderate dimensions. Old work of Krylov associated finitely additive signed measures to certain constant coefficient PDEs of higher order. Recent work with Levin allows us to identify the relevant expectations of iterated integrals in this case, leaving many interesting open questions and possible numerical algorithms for solving high dimensional elliptic PDEs.

Log-Sobolev inequalities with weighted square field are derived from a class of super Poincaré inequalities. As applications, stronger versions of Talagrand's transportation-cost inequality are provided on Riemannian manifolds. Typical examples are constructed to illustrate these results.

Abstract: Twistor-string theory is reformulated as a `half-twisted heterotic' theory with target $CP^3$. This in effect gives a Dolbeault formulation of a theory of holomorphic curves in twistor space and gives a clearer picture of the mathematical structures underlying the theory and how they arise from the original Witten and Berkovits models. It is also explained how space-time physics arises from the model. It intended that the lecture be, to a certain extent, pedagogical.

I will describe how atom interferometry can be used to set limits on beyond the Standard Model forces.

A new primal-dual augmented Lagrangian merit function is proposed that may be minimized with respect to both the primal and dual variables. A benefit of this approach is that each subproblem may be regularized by imposing explicit bounds on the dual variables. Two primal-dual variants of classical primal methods are given: a primal-dual bound constrained Lagrangian (pdBCL) method and a primal-dual l1 linearly constrained Lagrangian (pdl1-LCL) method.

This talk will be about the systematic simplification of differential equations.

After giving a geometric reformulation of the concept of a differential equation using prolongations, I will show how we can prolong group actions relatively easily at the level of Lie algebras. I will then discuss group-invariant solutions.

The key example will be the heat equation.

Abstract: Self-dual connections on ALF spaces can be encoded in terms of bow diagrams, which are natural generalizations of quivers. This provides a convenient description of the moduli spaces of these self-dual connections. We make some comments about the associated twistor data. Via the Nahm transform we construct two explicit examples: a single instanton and a single monopole on a Taub-NUT space.

The Lovasz Local Lemma is known to have an extension for cases where the dependency graph requirement is relaxed to negative dependency graph (Erdos-Spencer 1991). The difficulty is to find relevant negative dependency graphs that are not dependency graphs. We provide two generic constructions for negative dependency graphs, in the space of random matchings of complete and complete bipartite graphs. As application, we prove existence results for hypergraph packing and Turan type extremal problems. We strengthen the classic probabilistic proof of Erdos for the existence of graphs with large girth and chromatic number by prescribing the degree sequence, which has to satisfy some mild conditions. A more surprising application is that tight asymptotic lower bounds can be obtained for asymptotic enumeration problems using the Lovasz Local Lemma. This is joint work with Lincoln Lu.

Local Bifurcation Theory (II): Principle of exchange of stability, Lyapunov-Schmidt reduction, Theorem of Ize.

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).