University of Oxford

I'll start by defining the zeta function and stating the Weil conjectures (which have actually been theorems for some time now). I'll then go on by saying things like "Weil cohomology", "standard conjectures" and "Betti numbers of the Grassmannian". Hopefully by the end we'll all have learned something, including me.

As with conventional Higgs bundles, calculating Betti numbers of twisted Higgs bundle moduli spaces through Morse theory requires us to

study holomorphic chains. For the case when the base is P^1, we present a recursive method for constructing all the possible stable chains of a given type and degree by representing a family of chains by a quiver. We present the Betti numbers when the twists are O(1) and O(2), the latter of which coincides with the co-Higgs bundles on P^1. We offer some open questions. In doing so, we mention how these numbers have appeared elsewhere recently, namely in calculations of Mozgovoy related to conjectures coming from the physics literature (Chuang-Diaconescu-Pan).

Mikhail Borovoi's theorem states that any simply connected compact semisimple Lie group can be understood (as a group) as an amalgam of its rank 1 and rank 2 subgroups. Here we present a recent extension of this, which allows us to understand the same objects as a colimit of their rank 1 and rank 2 subgroups under a final group topology in the category of Lie groups. Loosely speaking, we obtain not only the group structure uniquely by understanding all rank 1 and rank 2 subgroups, but also the topology.

The talk will race through the elements of Lie theory, buildings and category theory needed for this proof, to leave the audience with the underlying structure of the proof. Little prior knowledge will be assumed, but many details will be left out.

We will give a quick and dirty introduction to Gromov-Witten theory and discuss some integrality properties of GW invariants. We will start by briefly recalling some basic properties of the Deligne Mumford moduli space of curves. We will then try to define GW invariants using both algebraic and symplectic geometry (both definitions will be rather sloppy, but hopefully the basic idea will become visible), talk a bit about the axiomatic definition due to to Kontsevich and Manin, and discuss some applications like quantum cohomology. Finally, we will talk a bit about integrality and the Gopakumar-Vafa conjecture. Just as a word of warning: this talk is intended as an introduction to the

subject and should give an overview, so we will perhaps be a bit sloppy here and there...

: Backward error analysis is a technique that has been extremely successful in understanding the behaviour of numerical methods for ordinary differential equations.  It is possible to fit an ODE (the so called modified equation) to a numerical method to very high accuracy. Backward error analysis has been of particular importance in the numerical study of Hamiltonian problems, since it allows to approximate symplectic numerical methods by a perturbed Hamiltonian system, giving an approximate statistical mechanics for symplectic methods. 

Such a systematic theory in the case of numerical methods for stochastic differential equations (SDEs) is currently lacking. In this talk we will describe a general framework for deriving modified equations for SDEs with respect to weak convergence. We will start by quickly recapping of how to derive modified equations in the case of ODEs and describe how these ideas can be generalized in the case of SDEs. Results will be presented for first order methods such as the Euler-Maruyama and the Milstein method. In the case of linear SDEs, using the Gaussianity of the underlying solutions, we will derive a SDE that the numerical method solves exactly in the weak sense. Applications of modified equations in the numerical study of Langevin equations and in the calculation of effective diffusivities will also be discussed, as well as the use of modified equations  as a tool for constructing higher order methods for stiff stochastic differential equations.

This is joint work with A. Abdulle (EPFL). D. Cohen (Basel), G. Vilmart (EPFL).

In this talk, I will discuss various aspects of approximation by radial basis functions on spheres. After a short introduction to the subject of scattered data approximation on spheres and optimal recovery, I will particularly talk about error analysis, a hybrid approximation scheme involving polynomials and radial basis functions and, if time permits, solving nonlinear parabolic equations on spheres.

Please note that this is a short notice change from the originally advertised talk by Dr Shahrokh Shahpar (Rolls-Royce plc.)

The new talk "Multilevel Monte Carlo method" is given by Mike Giles, Oxford-Man Institute of Quantitative Finance, Mathematical Institute, University of Oxford.

Joint work with Rob Scheichl, Aretha Teckentrup (Bath) and Andrew Cliffe (Nottingham)

Words are building blocks of sentences, yet the meaning of a sentence goes well beyond meanings of its words. Formalizing the process of meaning assignment is proven a challenge for computational and mathematical linguistics; with the two most successful approaches each missing on a key aspect: the 'algebraic' one misses on the meanings of words, the vector space one on the grammar.

I will present a theoretical setting where we can have both! This is based on recent advances in ordered structures by Lambek, referred to as pregroups and the categorical/diagrammatic approach used to model vector spaces by Abramsky and Coecke. Surprisingly. both of these structures form a compact category! If time permits, I will also work through a concrete example, where for the first time in the field we are able to compute and compare meanings of sentences compositionally. This is collaborative work with E. Greffenstete, C. Clark, B. Coecke, S. Pulman.

Solving partial differential equations (PDEs) on curved surfaces is

important in many areas of science. The Closest Point Method is a new

technique for computing numerical solutions to PDEs on curves,

surfaces, and more general domains. For example, it can be used to

solve a pattern-formation PDE on the surface of a rabbit.

A benefit of the Closest Point Method is its simplicity: it is easy to

understand and straightforward to implement on a wide variety of PDEs

and surfaces. In this presentation, I will introduce the Closest

Point Method and highlight some of the research in this area. Example

computations (including the in-surface heat equation,

reaction-diffusion on surfaces, level set equations, high-order

interface motion, and Laplace--Beltrami eigenmodes) on a variety of

surfaces will demonstrate the effectiveness of the method.