We show that the profinite completion (a universal algebraic
construction) and the MacNeille completion (an order-theoretic
construction) of a modal algebra $A$ coincide, precisely when the congruences of finite index of $A$ correspond to principal order filters. Examples of such modal algebras are the free K4-algebra and the free PDL-algebra on finitely many generators.
I will associate, to every pair of smooth transversal
Lagrangian spheres in a symplectic manifold having vanishing first Chern
class, its Floer cohomology groups. Hamiltonian isotopic spheres give
rise to isomorphic groups. In order to define these Floer cohomology
groups, I will make a key use of symplectic field theory.
Radial basis function (RBF) methods have been successfully used to approximate functions in multidimensional complex domains and are increasingly being used in the numerical solution of partial differential equations. These methods are often called meshfree numerical schemes since, in some cases, they are implemented without an underlying grid or mesh.
The focus of this talk is on the class of RBFs that allow exponential convergence for smooth problems. We will explore the dependence of accuracy and stability on node locations of RBF interpolants. Because Gaussian RBFs with equally spaced centers are related to polynomials through a change of variable, a number of precise conclusions about convergence rates based on the smoothness of the target function will be presented. Collocation methods for PDEs will also be considered.
Category theory is used to study structures in various branches of
mathematics, and higher-dimensional category theory is being developed to
study higher-dimensional versions of those structures. Examples include
higher homotopy theory, higher stacks and gerbes, extended TQFTs,
concurrency, type theory, and higher-dimensional representation theory. In
this talk we will present two general methods for "categorifying" things,
that is, for adding extra dimensions: enrichment and internalisation. We
will show how these have been applied to the definition and study of
2-vector spaces, with 2-representation theory in mind. This talk will be
introductory; in particular it should not be necessary to be familiar with
any category theory other than the basic idea of categories and functors.
Phylogenetics is the reconstruction and analysis of 'evolutionary'
trees and graphs in biology (and related areas of classification, such as linguistics). Discrete mathematics plays an important role in the underlying theory. We will describe some of the ways in which concepts from combinatorics (e.g. poset theory, greedoids, cyclic permutations, Menger's theorem, closure operators, chordal graphs) play a central role. As well as providing an overview, we also describe some recent and new results, and outline some open problems.
We provide a realization of Cherednik's double affine Hecke
algebras (for GL_n) as a convolution algebra of functions on moduli spaces
of coherent sheaves on an elliptic curve. As an application we give a
geometric construction of Macdonald polynomials as (traces of) certain
natural perverse sheaves on these moduli spaces. We will discuss the
possible extensions to higher (or lower !) genus curves and the relation
to the Hitchin nilpotent variety. This is (partly) based on joint work
with I. Burban and E. Vasserot.
In this talk I will look at a definition of the energy-momentum for the dynamical horizon of a black hole. The talk will begin by examining the role of a special class of observers at null infinity determined by Bramson's concept of frame alignment. It is shown how this is given in terms of asymptotically constant spinor fields and how this framework may be used together with the Nester-Witten two form to give a definition of the Bondi mass at null infinity.
After reviewing Ashtekar's concept of an isolated horizon we will look at the propagation of spinor fields and show how to introduce spinor fields for the horizon which play the role of the asymptotically constant spinor fields at null infinity, giving a concept of alignment of frames on the horizon. It turns out that the equations satisfied by these spinor fields give precisely the Dougan-Mason holomorphic condition on the cross sections of the horizon, together with a simple propagation equation along the generators. When combined with the Nester-Witten 2-form these equations give a quasi-local definition of the mass and momentum of the black hole, as well as a formula for the flux across the horizon. These ideas are then generalised to the case of a dynamical horizon and the results compared to those obtained by Ashtekar as well as to the known answers for a number of exact solutions.
In the Examination Schools