Short Course Lectures
There will be three lecture courses of 5 lectures each:
Random networks: the preferential attachment paradigm, Peter Mörters, University of Bath
The idea behind the preferential attachment paradigm is that the topology of large networks, such as the World-Wide-Web, social interaction or collaboration networks can be explained by the principle that these networks are built dynamically, and new nodes prefer to be attached to those old nodes that already have a high degree in the existing network.
In the lectures I will present a selection from the rigorous mathematical literature available on this subject. Our discussion will include the principal features of large networks such as asymptotic degree distributions, emergence of a giant component, diameter of components, and the existence of phase transitions. The proofs rely on a wide range of ideas from discrete and continuous probability.
Preparatory reading: For motivation for some of the research in this area - "Linked: The new science of networks" by A.L. Barabasi.
Aggregation and Coalescence, James Norris, University of Cambridge
Many fascinating structures are formed in nature formed by random aggregation. The challenge to describe such structures mathematically is largely open. Taking cues from the Schramm-Loewner theory, we will show how to encode and then analyse a simple planar aggregation model using conformal maps. For large clusters, it turns out that this two-dimensional model is well described in terms of a family of coalescing one-dimensional Brownian motions. The course will involve a number of topics related to this result, including: conformal invariance of planar Brownian motion, Arratia's coalescing Brownian flow, estimates for conformal maps, and scaling limits for Markov processes.
Bayesian approach to inverse problems, Andrew Stuart, University of Warwick.
Inverse problems in differential equations are ubiquitous in applications and provide formidable mathematical challenges due to their ill-posedness. One approach to regularization of inverse problems is to adopt a Bayesian framework for the problem. I will develop this Bayesian approach in a Banach space setting, leading to an interesting class of problems for probability measures on function space, defined via their Radon-Nikodym derivative with respect to a reference (prior) measure. I will develop a stability theory for these measures, showing that they are Lipschitz in the data, with respect to the Hellinger metric. I will then use this theory as the basis to quantify approximations of the measure, using finite dimensional subspaces. I will also show that a wide range of problems fit into the general framework, including inverse problems for the diffusion coefficient in an elliptic PDE, the wavespeed in a wave equation and the initial condition for the heat equation and nonlinear generalizations.
Preparatory reading: Inverse Problems: A Bayesian Perspective, Acta Numerica Volume 19, 2010
Guest Lecturers:
Tuesday 5th April, 1500 hours
The expected signature of a stochastic process. Some new PDE’s, Professor Terry Lyons, Department of Mathematics, Oxford University
How can one describe a probability measure of paths? And how should one approximate to this measure so as to capture the effect of this randomly evolving system. Markovian measures were efficiently describes by Strook and Varadhan through the Martingale problem. But there are many measures on paths that are not Markovian and a new tool, the expected signature provides a systematic ways of describing such measures in terms of their effects.
We explain how to calculate this expected signature I the case of the measure on paths corresponding to a Brownian motion started at a point x in the open set and run until it leaves the same set. A completely new (at least to the speaker) PDE is needed to characterise this expected signature.
Joint work with Ni Hao.
Thursday 7th April, 1330 hours
Scaling limits of anisotropic random growth models Dr Amanda Turner, Department of Mathematics and Statistics, Lancaster University
In 1998 Hastings and Levitov proposed a model for planar random growth such as diffusion-limited aggregation (DLA) and the Eden model, in which clusters are represented as compositions of conformal mappings. I shall introduce an anisotropic version of this model, and discuss some of the natural scaling limits that arise. I shall show that very different behaviour can be seen to that in the isotropic case.
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