Autumn Term 2007/08
Abstracts will be added as they become available.
Friday 19 October 2007 Boris Zilber (Oxford) Zariski geometries and quantum deformations of classical structures
I am going to start with an idea of "logically perfect" mathematical structures. This lead to the notion of a Zariski geometry which initially was hoped to become a basis for an abstract axiomatization of Algebraic Geometry. As a matter of fact the class of Zariski geometries turned out to be wider and include interesting noncommutative geometries. Moreover, the analysis of a typical class of Zariski geometries depending on parameter leads to the construction of a "classical limit", which is a real differentiable manifold, a gauge field of nonconstant curvature.
Friday 9 November 2007 Ian Roulstone (Surrey) Kaehler geometry and the Navier-Stokes equations
From slowly-evolving large-scale fluid flows, such as we observe in the atmosphere and oceans, to rapidly changing and turbulent flows, fluid mechanics is believed to be described accurately by the classical Navier-Stokes-based equations of motion. Detailed computations of the three-dimensional incompressible Navier--Stokes equations vividly illustrate the importance of vortex tubes and sheets. On larger scales (such as in the atmosphere and oceans) it can be shown that the solutions of the fluid equations stay close over finite, but useful, time intervals to the solutions of much simpler dynamical systems. These approximate models (often called 'balanced models') seek to describe flows in which there is a dominant balance between the Coriolis, buoyancy and pressure-gradient forces on fluid particles, which can be described very succinctly using vortex dynamics.
Recent research suggests that ideas from Kaehler geometry may be important in understanding the principles that govern the vortex dynamics of both the incompressible Navier--Stokes equations and the equations that govern those regimes most important to weather forecasting. In this lecture I shall show how i) the constraint of incompressibility, and ii) the constraints that govern the large-scale balanced flows, lead to pdes that can be studied from a geometric perspective. The implications of these results, and some speculative remarks concerning stability of flows, will be discussed.
Friday 16 November 2007 Pierre Vogel (Jussieu) The universal Lie algebra
The properties of simple Lie algebras or simple Lie superalgebras canbe globalized in a categorical way in order to produce an object called the universal Lie algebra. This object is a R-linear tensor category where R is a commutative ring. It is useful for understanding many properties of Lie algebras and representations of Lie algebras. It is also useful in order to produce finite type invariants for knots, links and 3-dimensional manifolds. The coefficient ring R is highly unknown but a complete description is conjectured.
Friday 30 November 2007 Philip Welch (Bristol) The Definable Continuum Problem
Cantor's Continuum Problem is well known: it was at the head of Hilbert's list of problems. Although a solution remains still out of reach, there have been many advances in talking about 'definable parts' of the continuum. Set theorists have known for a while that large sets, or measures of sufficient additivity on large sets, affect the view from analyst's perspective of the real line, but they are now in a position to give an explanation as to *why* definable sets in the projective hierarchy beyond Borel are, eg Lebesgue measurable. The connection is through determinacy assumptions on the existence of winning strategies for definable infinite two person games.
There is a body of quasi-empirical evidence that suggests that the continuum might in the end have cardinality not the first infinite cardinal number after that of the natural numbers (as Cantor surmised) but the second such.
The talk is a review of definable continuum questions with, if time permits, a review of the current state of this later evidence.
Friday 7 December 2007 Simon Donaldson (Imperial) Extremal metrics on complex surfaces
Extremal metrics are Kahler metrics satisfying a paticular elliptic equation. The basic concept is due to Calabi. The existence problem is, in general, rather challenging. In the first part of the talk we discuss the general picture and explain how, in the case of complex surfaces, the main difficulties are related to "collapsing" phenomena in Riemannian geometry. In the second part of the talk we specialise to the case of toric surfaces and outline some existence results. We also present numerical solutions and, if there is time, discuss recent work of Chen, LeBrun and Weber concerning a new Einstein 4-manifold.