Geometry, Topology and Combinatorics of large random simplicial complexes

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This project belongs to the field of stochastic topology which studies propertiesof large random spaces and predicts their geometric, topological and combinatorialproperties. The relationship between stochastic and classical topologyis similar to the relationship between statistical and classical mechanics.The predictions of stochastic topology become increasingly accurate when therandom space becomes "large" (in certain sense), i.e. when the methods ofclassical topology become inadequate. The tools of stochastic topology may be used formodelling large complex systems in various practical applications. The methodsand results of stochastic topology might also be useful in pure mathematics fornon-constructive existence proofs in topology.First models of random simplicial complexes and smooth compact manifoldsappeared around 2006 and are the object of intensive current research. We maymention random surfaces, random 3-manifolds, configuration spaces of randommechanisms, and several different models of random simplicial complexes.The proposed PhD project will be focused on topological properties of randomsimplicial complexes in the new multi-parameter model. We are interestedin homological properties of random simplicial complexes and in phase transitionswhich happen at some critical values of the probability parameters.The project involves tools from various areas of mathematics such as algebraictopology, combinatorial group theory, spectral analysis and elements ofprobability theory.