Applied Algebraic Topology of Large Random Spaces

The School of Mathematical Sciences of Queen Mary University of London invite applications for a PhD project commencing either in September 2016 (funded students) or at any point in the academic year (self-funded students).

This project belongs to the field of stochastic topology which studies properties of large random spaces and predicts their geometric, topological and combinatorial properties. The relationship between stochastic and classical topology is similar to the relationship between statistical and classical mechanics. The predictions of stochastic topology become increasingly accurate when the random space becomes “large" (in certain sense), i.e. when the methods of classical topology become inadequate. The well-developed theory of random graphs, initiated by P. Erdӧs and A. Réniy about 50 years ago, is a prelude to stochastic topology which deals with random simplicial complexes of any dimension ≥ 1. Random simplicial complexes of dimension ≥ 2 have wealth of interesting topological invariants, some of them are non-commutative (such as the fundamental group), and their study is much more challenging, compared to the theory of random graphs. The tools of stochastic topology may be used for modelling large complex systems in various practical applications. The methods and results of stochastic topology might also be useful in pure mathematics for non-constructive existence proofs in topology.

First models of random simplicial complexes and smooth compact manifolds appeared around 2006 and are the object of intensive current research. We may mention random surfaces, random 3-manifolds, configuration spaces of random mechanisms, and several different models of random simplicial complexes.

The proposed PhD project will be focused on topological properties of random simplicial complexes in the new multi-parameter model. We are interested in homological properties of random simplicial complexes and in phase transitions which happen at some critical values of the probability parameters.

The project involves tools from various areas of mathematics such as algebraic topology, combinatorial group theory, spectral analysis and elements of probability theory. Knowledge of these areas is not a prerequisite, the necessary material can be learned in the process of PhD work.

This project will be supervised by Professor Michael Farber. For full details, see the project abstract: http://www.maths.qmul.ac.uk/sites/default/files/phd%20projects%202015/G&A/farber-3.pdf

The application procedure is described on the School website. For further enquiries please contact Prof Michael Farber ([Email Address Removed]).

This project is eligible for several sources of full funding for the 2016/17 academic year, including support for 3.5 years’ study, additional funds for conference and research visits and funding for relevant IT needs. Applicants interested in the full funding will have to participate in a highly competitive selection process. The best candidates will be eligible to receive a prestigious Ian Macdonald Postgraduate Award of £1000, for which you will be considered alongside your application. The application deadline for full funding is January 31st 2016.

There is also 50% funding scheme available for students who are able to find the matching 50 % of the cost of their studies. Competition for these half-funded slots will be less intensive, and eligible students should mention their willingness to be considered for them in their application. The application deadline for 50 % funding is January 31st 2016.

This project can be also undertaken as a self-funded project, either through your own funds or through a body external to Queen Mary University of London. Self-funded applications are accepted year-round.