Imperial College

We prove an analogue of the Tate isogeny conjecture and the
semi-simplicity conjecture for overconvergent crystalline Dieudonne modules
of abelian varieties defined over global function fields of characteristic
p, combining methods of de Jong and Faltings. As a corollary we deduce that
the monodromy groups of such overconvergent crystalline Dieudonne modules
are reductive, and after base change to the field of complex numbers they
are the same as the monodromy groups of Galois representations on the
corresponding l-adic Tate modules, for l different from p.

In this talk I will discuss recent developments in information theoretical approaches to fundamental

molecular processes that affect the cellular decision making processes. One of the challenges of applying

concepts from information theory to biological systems is that information is considered independently from

meaning. This means that a noisy signal carries quantifiably more information than a unperturbed signal.

This has, however, led us to consider and develop new approaches that allow us to quantify the level of noise

contributed by any molecular reactions in a reaction network. Surprisingly this analysis reveals an important and hitherto

often overlooked role of degradation reactions on the noisiness of biological systems. Following on from this I will outline

how such ideas can be used in order to understand some aspects of cell-fate decision making, which I will discuss with

reference to the haematopoietic system in health and disease.

We describe how one can recover the Mori--Mukai

classification of smooth 3-dimensional Fano manifolds using mirror

symmetry, and indicate how the same ideas might apply to the

classification of smooth 4-dimensional Fano manifolds. This is joint

work in progress with Corti, Galkin, Golyshev, and Kasprzyk.