Past

We shall review recent progress in the understanding of

isoperimetric inequalities for product probability measures (a very tight

description of the concentration of measure phenomeonon). Several extensions

of the classical result for the Gaussian measure were recently derived by

functional analytic methods.

The Hopfield model took his name and its popularity within the theory

of formal neural networks. It was introduced in 1982 to describe and

implement associative memories. In fact, the mathematical model was

already defined, and studied in a simple form by Pastur and Figotin in

an attempt to describe spin-glasses, which are magnetic materials with

singular behaviour at low temperature. This model indeed shows a very

complex structure if considered in a slightly different regime than

the one they studied. In the present talk we will focus on the

fluctuations of the free energy in the high-temperature phase. No

prior knowledge of Statistical mechanics is required to follow the

talk.

The first seminar will be given with the new students in

mind. It will begin with a brief overview of quantifier elimination and its

relation to the back-and-forth property.I shall then discuss complexity issues

with particular reference to algebraically closed fields.In particular,how much

does the height and degree of polynomials in a formula increase when a

quantifier is eliminated? The precise answer here gave rise to the definition

of a `generic' transcendental entire function,which will also be

discussed.

A cell is a wonderously complex object. In this talk I will

give an overview of some of the mathematical frameworks that are needed

in order to make progress to understanding the complex dynamics of a

cell. The talk will consist of a directed random walk through discrete

Markov processes, stochastic differential equations, anomalous diffusion

and fractional differential equations.

Joe Doob, who died recently aged 94, was the last survivor of the

founding fathers of probability. Doob was best known for his work on

martingales, and for his classic book, Stochastic Processes (1953).

The talk will combine an appreciation of Doob's work and legacy with

reminiscences of Doob the man. (I was fortunate to be a colleague of

Doob from 1975-6, and to get to know him well during that year.)

Following Doob's passing, the mantle of greatest living probabilist

descends on the shoulders of Kiyosi Ito (b. 1915), alas now a sick

man.