University of Oxford

Factorization is a property of global objects that can be built up from local data. In the first half, we introduce the concept of factorization spaces, focusing on two examples relevant for the Geometric Langlands programme: the affine Grassmannian and jet spaces.

In the second half, factorization algebras will be defined including a discussion of how factorization spaces and commutative algebras give rise to examples. Finally, chiral homology is defined as a way to give global invariants of such objects.

Ever wondered how the log function in your code is computed? This talk, which was prepared for the 400th anniversary of Napier's development of logarithms, discusses the computation of reciprocals, exponentials and logs, and also my own work on some special functions which are important in Monte Carlo simulation.

Abstract: We propose a term structure power price model that, in contrast to widely accepted no-arbitrage based approaches, accounts for the non-storable nature of power. It belongs to a class of equilibrium game theoretic models with players divided into producers and consumers. Consumers' goal is to maximize a mean-variance utility function subject to satisfying inelastic demand of their own clients (e.g households, businesses etc.) to whom they sell the power on. Producers, who own a portfolio of power plants each defined by a running fuel (e.g. gas, coal, oil...) and physical characteristics (e.g. efficiency, capacity, ramp up/down times, startup costs...), would, similarly, like to maximize a mean-variance utility function consisting of power, fuel, and emission prices subject to production constraints. Our goal is to determine the term structure of the power price at which production matches consumption. In this talk we outline that such a price exists and develop conditions under which it is also unique. Under condition of existence, we propose a tractable quadratic programming formulation for finding the equilibrium term structure of the power price. Numerical results show performance of the algorithm when modeling the whole system of UK power plants.

Partial differential equations defined on surfaces appear in various applications, for example image processing and reconstruction of non-planar images. In this talk, I will present a penalty method for evolution equations, based on an implicit representation of the surface. I will derive a simple equation in the surrounding space, formulated with an extension operator, and then show some analysis and applications of the method.

Several problems lead to the question of how well can a fine grid function be approximated by a coarse grid function, such as preconditioning in finite element methods or data compression and image processing. Particular challenges in answering this question arise when the functions may be only piecewise-continuous, or when the coarse space is not nested in the fine space. In this talk, we solve the problem by using a stable approximation from a space of globally smooth functions as an intermediate step, thereby allowing the use of known approximation results to establish the approximability by a coarse space. We outline the proof, which relies on techniques from the theory of discontinuous Galerkin methods and on the theory of Helmholtz decompositions. Finally, we present an application of our to nonoverlapping domain decomposition preconditioners for hp-version DGFEM.

Cox processes arise as a natural extension of inhomogeneous Poisson Processes, when the intensity function itself is taken to be stochastic. In multiple applications one is often concerned with characterizing the posterior distribution over the intensity process (given some observed data). Markov Chain Monte Carlo methods have historically been successful at such tasks. However, direct methods are doubly intractable, especially when the intensity process takes values in a space of continuous functions.

In this talk I'll be presenting a method to overcome this intractability that is based on the idea of "thinning" and that does not resort to approximations.

We state a conjecture about the size of the intersection between a bounded-rank progression and a sphere, and we prove the first interesting case, a result of Chang. Hopefully the full conjecture will be obvious to somebody present.

The Balog-Szemeredi-Gowers theorem states that, given any finite subset of an abelian group with large additive energy, we can find its large subset with small doubling constant. We can ask how this constant depends on the initial additive energy. In the talk, I will give an upper bound, mention the best existing lower bound and, if time permits, present an approach that gives a hope to improve the lower bound and make it asymptotically equal to the upper bound from the beginning of the talk.