Past

The complement of a knot or link is a 3-manifold which admits a geometric

structure. However, given a diagram of a knot or link, it seems to be a

difficult problem to determine geometric information about the link

complement. The volume is one piece of geometric information. For large

classes of knots and links with complement admitting a hyperbolic

structure, we show the volume of the link complement is bounded by the

number of twist regions of a diagram. We prove this result for a large

collection of knots and links using a theorem that estimates the change in

volume under Dehn filling. This is joint work with Effie Kalfagianni and

David Futer

Born in France around 1780, the optimal transport problem has known a scientific explosion in the past two decades, in relation with dynamical systems and partial differential equations. Recently it has found unexpected applications in Riemannian geometry, in particular the encoding of Ricci curvature bounds

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.