University of Oxford

Networks provide a convenient way to represent complex systems of interacting entities. Many networks contain "communities" of nodes that are more strongly connected to each other than to nodes in the rest of the network. Most methods for detecting communities are designed for static networks. However, in many applications, entities and/or interactions between entities evolve in time. To incorporate temporal variation into the detection of a network's community structure, two main approaches have been adopted. The first approach entails aggregating different snapshots of a network over time to form a static network and then using static techniques on the resulting network. The second approach entails using static techniques on a sequence of snapshots or aggregations over time, and then tracking the temporal evolution of communities across the sequence in some ad hoc manner. We represent a temporal network as a multilayer network (a sequence of coupled snapshots), and discuss  a method that can find communities that extend across time. 

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.