LMS-EPSRC Short Course 2011
Mathematical Institute, University of Oxford
3 – 8 April 2011
Organiser: Professor Alison Etheridge
Course outline and prerequisites
In the last fifty years, probability theory has emerged both as a core mathematical discipline, sitting alongside geometry, algebra and analysis, and as a fundamental way of thinking about the world. This five day residential school will provide intensive courses on some of the most exciting developments in modern probability theory. They will be accessible to first year PhD students in mathematics.
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 di fferential 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 di ffusion 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
These will be supplemented by tutorial sessions and guest lectures.
Participants might like to note that the workshop 'Random structures and dynamics', with minicourses by Alain Sznitman, Louigi Addario-Berry and Martin Barlow and a range of invited talks from across probability theory will take place in Oxford during the following week. For further information, see: www.maths.ox.ac.uk/events/conferences/random-structures-and-dynamics
Application
Applications should be made using the registration form available via the Society’s website at: www.lms.ac.uk/activities/rmc/sc/55poster.html
The closing date for applications is Friday 25 February 2011. Numbers will be limited and those interested are advised to make an early application. All applicants will be contacted approximately two weeks after this deadline; we will not be able to give information about individual applications before then.
Fees
All research students registered at a UK university will be charged a registration fee of £100. They will not be charged for subsistence costs.
UK-based postdocs will be charged a registration fee of £100, plus half the subsistence costs (£212) £312 in total.
All others (overseas students and postdocs, those working in industry) will be charged a registration fee of £250 plus the full subsistence costs (£424), £674 in total.
All participants must pay their own travel costs (for EPSRC funded students, this should be covered by their DTA). Fees are not payable until a place on the course is offered.
In the event of over-subscription preference will be given to UK-based research students.
FINAL ARRANGEMENTS
Accommodation: Check-in time at St John's College is from 12 noon on Sunday 3rd April. Please report to the Porter's Lodge upon arrival (see map). Check-out from St John's should be done by 4.00 p.m. on Friday 8th April.
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