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Alexander Belton supervises 1 postgraduate research students. If these students have produced research profiles, these are listed below:

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Dr Alexander Belton CMath, CSci, FIMA, SFHEA

Head of Department, Senior Lecturer

Alexander Belton

Fylde College

LA1 4YF

Lancaster

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.

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Research Interests

The majority of my research takes place within the field of quantum probability, a branch of mathematics which involves functional analysis, probability theory and quantum mechanics. At present I am investigating Feynman-Kac perturbations of quantum dynamical semigroups, the general theory of quantum random walks and the theory of non-commutative stopping times.

I also have a strong interest in the classical theory of stochastic processes and its applications to financial mathematics.

Recently I have started a new research theme, in the area of matrix positivity.

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

  • Department representative 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)
  • Department representative and Northern Regional Committee member for the London Mathematical Society (March 2012 - August 2017)

PhD Supervisions Completed

  • Michał Gnacik (2010-14)

Business and Enterprise

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

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