MSI Weekly Bulletin - Week starting Monday 14 May, 2007

A computer search for relations of the form M1*arctan(1/X1) + M2*arctan(1/X2) + ... + Mn*arctan(1/Xn) = k*Pi/4 is described. We search for the "best" arctan n-term relation, the one were the smallest Xi is a big as possible. For example, the following 5-term relation is found: Pi/4 = + 88 arctan(1/192) + 39 arctan(1/239) + 100 arctan(1/515) - 32 arctan(1/1068) - 56 arctan(1/173932) For all (inverse) arguments X the prime factors of X^2 +1 are in the set {2,5,13,73,101}. A fast algorithm is given to determine all X in a prescribed range so that X^2 + 1 factors into a given set of primes. The resulting table of values X <= 10^14 that factor into primes p <= 761 is used to rediscover all previously known "good" relations and find new relations with up to 27 terms. The search was possible only with highly optimized methods that will be explained. The best 2-term, 3-term and 4-term relations were known before 1900 (the relations were given by Machin, Gauss, and Stormer). All records for 5-term ... 27-term relations (with the exception of the 15-term relation which is by Hwang Chien-lih) were obtained in my 1993 and 2006 computations. A warning about the addictiveness of the topic will be given.

In my colloquium talk I discussed various sufficient conditions in terms of classical function space norms, for a Banach space valued function to be stochastically integrable with respect to a Brownian motion. In this seminar talk I will present necessary and sufficient conditions in terms of a 'gaussian' function norm, and show how these conditions generalise from functions to stochastic processes in the case of UMD Banach spaces by the use of decoupling inequalities.

I'll explain how the results from last week enable us to make sense of the algebraic geometry of Witt vectors and Lambda-rings in a very general context. A word of warning: I will be making heavy use of the language of abstract algebraic geometry, usually without much explanation, and this might not be to the taste of some!

I will give a short introduction to the p-adic numbers and their relation to 'local' number theory.

One of most fundamental and basic equations in probability and mathematical physics is the heat equation. In my talk I will discuss a simple non-technical proof of Gaussian bounds for the heat equation based on Phragm\'en-Lindel\"of theorem - a version of maximum principle in complex analysis. I will describe also some new results, which can be obtained using this approach. In more technical terms I prove that in presence of $L^2$ Gaussian estimates, so-called Davies-Gaffney estimates, on-diagonal upper bounds imply precise off-diagonal Gaussian upper bounds for the kernels of analytic families of operators. The talk is based on my joint work with Thierry Coulhon http://front.math.ucdavis.edu/math.AP/0609429.