Geometry, Topology and Combinatorics of large random simplicial complexes

Lead Research Organisation: Queen Mary University of London
Department Name: Sch of Mathematical Sciences

Abstract

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 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.

Publications

10 25 50
publication icon
Even-Zohar C (2022) Ample simplicial complexes in European Journal of Mathematics

publication icon
Farber M (2019) Random simplicial complexes, duality and the critical dimension in Journal of Topology and Analysis

publication icon
Farber M (2020) Random simplicial complexes in the medial regime in Topology and its Applications

Studentship Projects

Project Reference Relationship Related To Start End Student Name
EP/N50953X/1 30/09/2016 29/09/2021
1936239 Studentship EP/N50953X/1 30/09/2017 30/03/2021 Lewis Mead
 
Description A generalised of a network (think the tube map) is to simplicial complexes, an object that encodes not just pairwise connections but also if 3, 4, 5, etc. things are talking to each other at once. Random graphs have been intensely studied since the 50s, the concept of random simplicial complexes first appeared in the mid 2000s. My research has developed a new model for random simplicial complexes, looked at its connections to previous models and answered some questions of what one expects a "generic" random object to look like.
Exploitation Route Potential to be used in some network science for large networks where one cares about higher dimensional interactions (e.g. neurons in the brain).
Sectors Digital/Communication/Information Technologies (including Software)