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Professor Alexander Belton CMath, CSci, FIMA, SFHEA

Visiting Professor

Alexander Belton

Fylde College



Tel: +44 1524 592371

PhD supervision

Applications to join the department's thriving PhD programme are welcome from students with interests in analysis, probability theory or mathematical physics. The area in which I work, non-commutative probability, is an exciting combination of these three subjects. Knowledge of all of them is not necessary, but an interest to discover more is.

At present I'm particularly interested in the following topics.

  1. Quantum random walks. A classical random walk consists of repeatedly flipping a coin and moving left or right accordingly. This simple model illustrates many important ideas in probability theory. Its quantum generalisation corresponds to a system interacting with a sequence of identical particles; limit theorems have been obtained, but many interesting questions remain unanswered.
  2. Non-commutative stopping times. Stopping times are random times which, at any given moment, are known to have occurred or not. The time of the first rainfall this week is a stopping time; the time of the last rainfall is not. The theory of stopping times is vital for developing classical theories of stochastic integration. The proper non-commutative generalisation is known, but is yet to be exploited fully.
  3. Exotic forms of independence. The concept which separates probability from analysis is stochastic independence. Once one moves to the non-commutative world, more than one form of independence exists. Free independence was introduced by Voiculescu in the 1980s, and has important connections to random matrix theory, quantum information theory and representation theory. Connections for other forms of independence remain to be explored.

Research Interests

My research takes place within two fields: quantum probability, a branch of mathematics involving functional analysis, probability theory and quantum mechanics, and matrix positivity, an area of linear algebra with diverse applications.

Research Grants

  • Leverhulme Trust RPG-2014-196 Quantum Random Walks and Quasi-free Quantum Stochastic Calculus (January 2014 - January 2016)

Current Teaching

  • MATH412 Topology and Fractals

Professional Role

  • Departmental representative and Northwest Branch committee member for the Institute of Mathematics and its Applications
  • Deputy Head of Department (January - August 2017)
  • Head of Pure Mathematics Section (January - August 2017)
  • Head of Undergraduate Teaching (Lent Term 2015)
  • Head of Undergraduate Admissions (August 2008 - August 2012)
  • Part I Director of Studies (September 2013 - August 2017)
  • Departmental representative and Northern Regional Committee member for the London Mathematical Society (March 2012 - August 2017)

PhD Supervisions Completed

  • Michał Gnacik (2010-14)
  • Jason Hancox (2014-18)

Business and Enterprise

  • Member of the Faculty of Science and Technology Marketing Committee (January 2010 - November 2012)
  • Published

    Totally positive kernels, Pólya frequency functions, and their transforms

    Belton, A., Guillot, D., Khare, A. & Putinar, M., 30/09/2023, In: Journal d'Analyse Mathématique. 150, 1, p. 83-158 76 p.

    Research output: Contribution to Journal/MagazineJournal articlepeer-review

  • Published


    Belton, A. & Franz, U., 31/12/2022, In: Infinite Dimensional Analysis, Quantum Probability and Related Topics. 25, 4, 2 p., 2202001.

    Research output: Contribution to Journal/MagazineEditorialpeer-review

  • Published

    Hirschman-Widder densities

    Belton, A., Guillot, D., Khare, A. & Putinar, M., 30/09/2022, In: Applied and Computational Harmonic Analysis. 60, p. 396-425 30 p.

    Research output: Contribution to Journal/MagazineJournal articlepeer-review

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