Noncommutative statistical mechanics: probability at the confluence
Lead Research Organisation:
Lancaster University
Department Name: Mathematics and Statistics
Abstract
The asymmetric simple exclusion process (ASEP) is a fundamental mathematical construction in statistical mechanics that combines the notion of stochastic motion with a physical requirement that matter cannot coexist in the same location. As such, it has wide-ranging applications, from traffic and fluid flow to biological processes. However, recent advances, particularly the emerging parallelism between ASEP-type processes and well-known families of orthogonal polynomials, hint at a hidden probabilistic phenomenon that promises to greatly expand the applicability of the ASEP.
The key is in a common computational artifact, referred to as the matrix ansatz. In this project, we explore a new perspective on the ASEP, based on the relationship between its matrix ansatz and noncommutative infinite-dimensional phenomena. In this manner, we aim to transform the interface between the 'classical' and noncommutative probability theories, arrive at a more fundamental understanding of the ASEP, and open up innovative approaches to a long-standing open question of mathematical analysis, the free group factors isomorphism problem.
The key is in a common computational artifact, referred to as the matrix ansatz. In this project, we explore a new perspective on the ASEP, based on the relationship between its matrix ansatz and noncommutative infinite-dimensional phenomena. In this manner, we aim to transform the interface between the 'classical' and noncommutative probability theories, arrive at a more fundamental understanding of the ASEP, and open up innovative approaches to a long-standing open question of mathematical analysis, the free group factors isomorphism problem.
People |
ORCID iD |
| Natasha Blitvic (Principal Investigator) |
Publications
| Description | This grant is being transferred, please see EP/V048902/2 |
| Exploitation Route | This grant is being transferred, please see EP/V048902/2 |
| Sectors | Other |
| Description | This grant is being transferred, please see EP/V048902/2 |
| Description | "Positivity problems in combinatorics" (PI: Natasha Blitvic) |
| Amount | £7,000 (GBP) |
| Organisation | Heilbronn Institute for Mathematical Research |
| Sector | Academic/University |
| Country | United Kingdom |
| Start | 05/2022 |
| End | 03/2023 |
| Description | Renewal of funding (for calendar year 2022) for collaborative travel between Lancaster and Universite de Franche-Comte (France): "Noncommutative Probability, Matrix Analysis and Quantum Groups" |
| Amount | £2,190 (GBP) |
| Funding ID | 608420427 |
| Organisation | British Council |
| Sector | Charity/Non Profit |
| Country | United Kingdom |
| Start | 01/2021 |
| End | 12/2022 |
| Description | Simons Foundation Pivot Fellowship |
| Amount | $162,601 (USD) |
| Organisation | Simons Foundation |
| Sector | Charity/Non Profit |
| Country | United States |
| Start | 08/2024 |
| End | 08/2025 |
| Description | Bringing new intra-disciplinary ideas to a wider combinatorial audience |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | The PI was an invited speaker and participant at several combinatorics conferences, where she presented and popularized the ideas pursued in this funded project. These included the 2023 Scottish Combinatorics Meeting, the collaborative workshop Dagstuhl Seminar 23121: Pattern Avoidance, Statistical Mechanics and Computational Complexity (https://www.dagstuhl.de/seminars/seminar-calendar/seminar-details/23121) and the Oberwolfach Mini-Workshop: Permutation Patterns (https://www.mfo.de/occasion/2405c/www_view). She contributed open problems and conjectures to the proceedings of both collaborative workshops, which in turn led to new collaborations. |
| Year(s) Of Engagement Activity | 2022,2023,2024 |
| Description | Creation of a new interdisciplinary research seminar |
| Form Of Engagement Activity | A formal working group, expert panel or dialogue |
| Part Of Official Scheme? | No |
| Geographic Reach | Regional |
| Primary Audience | Professional Practitioners |
| Results and Impact | The PI spearheaded the creation of a new interdiscipliniary seminar at QMUL, focused on mathematical modeling. One of the goals of the seminar is to explore possible applications of cutting-edge pure mathematics to biology, health, and environment. While the seminar is based at QMUL, the audience includes other London institutions. |
| Year(s) Of Engagement Activity | 2023 |
| URL | https://www.qmul.ac.uk/maths/news-and-events/news-/items/school-of-mathematical-sciences-at-queen-ma... |
| Description | Disseminating ideas and findings in research seminars across several mathematical sub-disciplines |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | The PI delivered a number of research seminars at leading specialist venues, in which she popularized and disseminated the key ideas and results arising from this funded research, namely the unexpected positivity properties underlying many algebraic or combinatorial objects. Venues include MSRI (USA), GGI (Italy), University of Washington (USA) probability seminar, Texas A&M University (USA) combinatorics seminar, Georgia Institute of Technology (USA) analysis seminar, ENS Lyon (France) probability seminar, University of Warwick (UK) probability seminar, University of Oxford (UK) random matrix seminar. |
| Year(s) Of Engagement Activity | 2022,2023,2024 |
| Description | Research Membership at the Mathematical Sciences Research Institute |
| Form Of Engagement Activity | A formal working group, expert panel or dialogue |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | The PI spent six weeks as an invited Research Member at the Mathematical Sciences Research Institute (Berkeley, California, USA) at the program "Universality and Integrability in Random Matrix Theory and Interacting Particle Systems". This was a unique opportunity to discuss with leading experts exploring the interface between random matrix theory and interacting particle systems, which provides the backdrop for this funded project. The PI delivered a seminar talk on an adjacent topic to an audience of experts, postdocs and postgraduate students, which sparked further discussions and has led to subsequent invitations. |
| Year(s) Of Engagement Activity | 2021 |
| URL | https://www.msri.org/programs/328 |
| Description | Research membership at the Galileo Galilei Institute for Theoretical Physics (Florence, Italy) |
| Form Of Engagement Activity | A formal working group, expert panel or dialogue |
| Part Of Official Scheme? | No |
| Geographic Reach | International |
| Primary Audience | Professional Practitioners |
| Results and Impact | The PI spent two weeks as an invited member at the Galileo Galilei Institute for Theoretical Physics (Florence, Italy) at the program "Randomness, Integrability and Universality". This was an opportunity to discuss with leading experts in interacting particle systems, which provides the backdrop for this funded project. The PI delivered a seminar talk to an audience of experts, postdocs and postgraduate students. This residency sparked further discussions, has led to subsequent invitations, and has generated new collaborations on this funded project. |
| Year(s) Of Engagement Activity | 2022 |
| URL | https://www.ggi.infn.it/showevent.pl?id=366 |
| Description | Summer schools and other engagement with early-career mathematicians |
| Form Of Engagement Activity | A talk or presentation |
| Part Of Official Scheme? | No |
| Geographic Reach | National |
| Primary Audience | Postgraduate students |
| Results and Impact | The PI delivered a three-hour lecture series at NSF-funded summer school "Probability and Algebra: New Expressions In Mathematics" at Texas A&M University. The topics covered included research funded by this grant and introduced these ideas to new generations of mathematicians. The PI delivered a plenary talk at the 2022 Early Career Researchers in Mathematics conference held at UCL in which she discussed some of the key mathematical ideas behind this project, as well as ways in which we can build a more inclusive mathematical culture. |
| Year(s) Of Engagement Activity | 2022,2023 |
| URL | https://www.math.tamu.edu/conferences/functional_analysis/PANEM.html |