Atlanta, GA | Posted: March 30, 2005
Keith Ball obtained his PhD from Cambridge University in 1987. He is currently a professor in the mathematics department of University College London. His primary research interests are in high-dimensional geometry, probability and information theory, and combinatorial geometry. He has just finished his tenure as a Royal Society Leverhulme Senior Research Fellow and is spending this year visiting the Theory Group at Microsoft Corporation. He is the author of a recreational mathematics book "Strange Curves, Counting Rabbits and other Mathematical Explorations" published by PUP in 2003.
The famous second law of thermodynamics states that the entropy of a closed physical system increases with time. The convergence of simple thermodynamic systems toward equilibrium parallels the convergence of sums of independent random variables to the normal distribution: the convergence in the central limit theorem. It has long been believed that there should be an analogue of the second law for the central limit process. The problem was recently solved using a variational principle inspired by high-dimensional geometry. I will begin by recalling the second law and the central limit theorem and then provide a brief introduction to information theory and the Brunn-Minkowski inequality. Finally I will outline how they come together to yield a variational characterization of entropy and how this can be used to establish the second law for the central limit process.
The values of Riemann's zeta function, zeta, at the (positive) even integers are rational multiples of integral powers of pi, so that their transcendence was established at the end of the 19th century with the transcendence of pi. The values at odd integers are harder to understand. 25 years ago Apéry proved that zeta(3) is irrational. This talk outlines the proof, found 5 years ago, that infinitely many of the odd-number values of zeta are irrational.