# 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.

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.

## People |
## ORCID iD |

Michael Farber (Primary Supervisor) | |

Lewis Mead (Student) |

### Publications

Even-Zohar C
(2022)

*Ample simplicial complexes*in European Journal of Mathematics
Farber M
(2019)

*Random simplicial complexes, duality and the critical dimension*in Journal of Topology and Analysis
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) |