Past

Let N(A) be the number of integer solutions of x^2 + y^2

We present algorithms for dense LU and QR factorizations that minimize the cost of communication. One of today's challenging technology trends is the increased communication cost. This trend predicts that arithmetic will continue to improve exponentially faster than bandwidth, and bandwidth exponentially faster than latency. The new algorithms for dense QR and LU factorizations greatly reduce the amount of time spent communicating, relative to conventional algorithms.

This is joint work with James Demmel, Mark Hoemmen, Julien Langou, and Hua Xiang.

In this talk I will introduce hyperbolic 3-manifolds, state some major conjectures about them, and discuss some group-theoretic properties of their fundamental groups.

The Rado Graph R is a graph on a countably infinite number of vertices that can be characterized by the following property: for every pair of finite, disjoint subsets of vertices X and Y, there exists a vertex that is adjacent to every vertex in X and none in Y. It is often called the Random Graph for the following reason: for any 0

Let $K \subset {\mathbb R}^d$ be a convex set. Choose $n$ random points in $K$, and denote by $P_n$ their convex hull. We call $P_n$ a random polytope. Investigations concerning the expected value of functionals of $P_n$, like volume, surface area, and number of vertices, started in 1864 with a problem raised by Sylvester and now are a classical part of stochastic and convex geometry. The last years have seen several new developments about distributional aspects of functionals of random polytopes. In this talk we concentrate on these recent results such as central limit theorems and tail inequalities, as the number of random points tends to infinity.

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.