Representation-theoretic approaches to several problems in probability theory
Lead Research Organisation:
Queen Mary University of London
Department Name: Sch of Mathematical Sciences
Abstract
The project aims to solve some problems in probability using techniques from algebra, in particular the branch called representation theory. The current problem is called the Manhattan problem, because it is based on a grid that resembles the one-way streets in Manhattan. This problem is probabilistic, but we can formulate it in terms of an algebraic object, which we name the Manhattan algebra. The project studies this Manhattan algebra, and others similar to it. In particular, we conjecture that the representation theory of the Manhattan algebra will be useful in solving the Manhattan problem.
People |
ORCID iD |
| Alexander Sodin (Primary Supervisor) | |
| Kieran Ryan (Student) |
Studentship Projects
| Project Reference | Relationship | Related To | Start | End | Student Name |
|---|---|---|---|---|---|
| EP/N50953X/1 | 01/10/2016 | 30/09/2021 | |||
| 1936327 | Studentship | EP/N50953X/1 | 01/10/2017 | 30/09/2021 | Kieran Ryan |
| Description | Notable progress has been made on two problems in probability; both are very much works in progress. In both, we study probabilistic models of the movement of particles, which have applications in physics. Partial results have been obtained in both, using an area of algebra known as representation theory. We are optimistic about comprehensive results being obtained by the completion of this award. The first problem is known as the Manhattan model, and models particles moving on a 2D lattice, which is directed (similarly to how the roads in Manhattan are directed). Particles change direction when they encounter obstacles, which themselves are placed randomly. We wish to know whether a particle remains contained in some finite area. The partial result assumes, in a precise way, that there are very few obstacles. The second problem is joint work with Davide Macera, a PhD student at Roma Tre, who is visiting Queen Mary University. The problem is known as the Interchange model (with reversals). This models several particles moving on a lattice (or any graph, which is just a set of points connected by lines), which move by exchanging places with their neighbours in a random way (hence the name Interchange). This problem is well-studied, although less studied in the case when "reversals" are included, which is a variant on the way that particles exchange places. In both cases, the movement of particles can be described with certain diagrams, which can be studied algebraically. Together these diagrams form the "Brauer algebra". The representation theory of this algebra is the key tool we use in the results above. |
| Exploitation Route | If answered fully, both problems addressed have applications in physics. |
| Sectors | Other |