L6

Little is known about C_exp, the complex field with the exponential function. Model theoretically it is difficult due to the definability of the integers (so its theory is not stable), and a lack of clear algebraic structure; for instance, it is not known whether or not pi+e is irrational. In order to study C_exp, Boris Zilber constructed a class of pseudo-exponential fields which satisfy all the properties we desire of C_exp. This class is categorical for every uncountable cardinal, and other more general classes have been defined. I shall define the three main classes of exponential fields that I study, one of which being Zilber's class, and show that they exhibit "stable-like" behaviour modulo the integers by defining a notion of independence for each class. I shall also explicitly apply one of these independence relations to show that in the class of exponential fields ECF, types that are orthogonal to the kernel are exactly the generically stable types.
 

The aim of this talk is to provide a general setting in which a number of important dualities in mathematics can be framed uniformly.  The setting comes about as a natural generalisation of the Galois connection between ideals of polynomials with coefficients in a field K and affine varieties in K^n.  The general picture that comes into sight is that the topological representations of Stone, Priestley, Baker-Beynon, Gel’fand, or Pontryagin are to their respective classes of structures just as affine varieties are to K-algebras.

A G\"odel sentence is often described as a sentence saying about itself that it is not provable, and a Henkin sentence as a sentence stating its own provability. We discuss what it could mean for a sentence to ascribe to itself a property such as provability or unprovability. The starting point will be the answer Kreisel gave to Henkin's problem. We describe how the properties of the supposedly self-referential sentences depend on the chosen coding, the formulae expressing the properties and the way a fixed point for the formula is obtained. Some further examples of self-referential sentences are considered, such as sentences that \anf{say of themselves} that they are $\Sigma^0_n$-true (or $\Pi^0_n$-true), and their formal properties are investigated.

The fundamental results about definable groups in o-minimal structures all suggested a deep connection between these groups and Lie groups. Pillay's conjecture explicitly formulates this connection in analogy to Hilbert's fifth problem for locally compact topological groups, namely, a definably compact group is, after taking a suitable the quotient by a "small" (type definable of bounded index) subgroup, a Lie group of the same dimension. In this talk we will report on the proof of this conjecture in the remaining open case, i.e. in arbitrary o-minimal structures. Most of the talk will be devoted to one of the required tools, the formalism of the six Grothendieck operations of o-minimals sheaves, which might be useful on it own. 

Functions of bounded variation, abbreviated as BV functions, are defined in the Euclidean setting as very weakly differentiable functions that form a more general class than Sobolev functions. They have applications e.g. as solutions to minimization problems due to the good lower semicontinuity and compactness properties of the class. During the past decade, a theory of BV functions has been developed in general metric measure spaces, which are only assumed to be sets endowed with a metric and a measure. Usually a so-called doubling property of the measure and a Poincaré inequality are also assumed. The motivation for studying analysis in such a general setting is to gain an understanding of the essential features and assumptions used in various specific settings, such as Riemannian manifolds, Carnot-Carathéodory spaces, graphs, etc. In order to generalize BV functions to metric spaces, an equivalent definition of the class not involving partial derivatives is needed, and several other characterizations have been proved, while others remain key open problems of the theory.

Panu is visting Oxford until March 2015 and can be found in S2.48

When solving Einstein's equations with negative cosmological constant, the natural setting is that of an initial-boundary value problem. Data is specified on the timelike conformal boundary as well as on some initial spacelike (or null) hypersurface. At the PDE level, one finds that the boundary data is typically prescribed on a surface at which the equations become singular and standard energy estimates break down. I will discuss how to handle this singularity by introducing a renormalisation procedure. I will also talk about the consequences of different choices of boundary conditions for solutions of Einstein’s equations with negative cosmological constant.

In this talk I will present several results connected with the idea of magnetic relaxation for MHD, including some new commutator estimates (and a counterexample to the estimate in the critical case). (Joint work with various subsets of  D. McCormick, J. Robinson, C. Fefferman and J-Y. Chemin.)