49 KAM Mathematical Colloquium
Prof. Keith Ball
University College London
ENTROPY GROWTH FOR SUMS OF INDEPENDENT RANDOM VARIABLES
December 16, 2003 12:30 Lecture hall S5
Abstract
Entropy plays a central role in the analysis of many random and physical systems: the entropy measures the disorder of such a system. For ``natural'' processes we expect entropy to grow as the process evolves. The most natural random process is the sequence of normalised sums
\[\frac{1}{\sqrt{n}} \sum_1^n X_i\]
of independent identically distributed square-integrable random
variables, which converges in distribution to a Gaussian (normal). It was proved
in the late 1940's by Shannon that the entropy for $n=2$ is at least as large as that
for $n=1$ but there seemed to be no way to determine whether the entropy continues to increase.
In this lecture I will briefly introduce the concept of entropy and
explain a new method for analysing entropy production which resolves the problem left by
Shannon and has also been used to obtain the first quantitative results on entropy
production for a wide class of random variables.
No knowledge beyond elementary probability and vector calculus will be
assumed.
THERE ARE INFINITELY MANY IRRATIONAL VALUES OF ZETA AT THE ODD INTEGERS
December 16, 2003 12:30 Lecture hall S5
Abstract
The zeta function is defined, for $s>1$ by
\zeta(s) = \sum_{n=1}^\infty \frac{1}{n^s}.
It is universally believed that its values at positive integers ($s>1$)
are irrational (and even transcendental). This is known for even integers
because the values are rational multiples of appropriate powers of $\pi$. For the
odd integers, only $\zeta(3)$ is known to be irrational: this was shown by Ap\'ery in 1978.
This talk will describe an elementary construction found by the author and
T. Rivoal which suffices to show that infinitely many of the odd-number values
are irrational. Nothing will be assumed beyond a first course in complex analysis.
