News:
- 08/11/2024: I have accepted a short-term position as an assistant professor at Heriot-Watt University. This coincides with a pause of my postdoctoral position, which will last for 7 months.
- 22/01/2023: I have now accepted a postdoctoral research position at Heriot-Watt University, funded by an EPSRC Doctoral Prize. In this role I will be building on my previous research in a couple of new directions, as well as developing interdepartmental relations, and involving myself with outreach to schools in the local community. If you are reading this as a member of EPS, IPAQS, or another HWU department, and you are interested in, or have ideas on, building strong interdepartmental networks with MACS, please get in touch! And vice versa. If you are a member of MACS interested in developing maths communication and outreach events, I would be glad to hear from you.
- 10/01/2023: The pre-print "A Q-Operator for Open Spin Chains II: Boundary Factorisation" has been submitted to the arXiv. In this paper, we derive a boundary factorisation relation for the XXZ chain through a natural extension of the existing work on R-matrix factorisation (found, for example, in [1,2,3]). In this paper, we connect with the universal K-matrix formalism of [4,5], allowing us to additionally reframe the short-exact-sequence approach to boundary fusion of [6] in terms of level-0 representations of the Borel subalgebras of affine Uq(sl2). The boundary factorisation relation is expected to lead to transfer matrix factorisation for the open chain, and the approach is expected to be extensible to alternative choices of boundary conditions (corresponding to alternative K-matrices/choice of co-ideal subalgebra). More details will be available on publication of my PhD thesis, currently in submission.
Research
I am currently working as a research associate in the school of MACS at Heriot-Watt University, researching topics in Quantum Integrable Systems, a particular subfield of Mathematical Physics. This has followed 4-month role as an assistant professor within the department.
With my PhD supervisors Robert Weston and Bart Vlaar, I am continuing a series of papers on the derivation of Q-operators associated to quantum integrable models with open boundaries from the quantum groups perspective. This project extends well-known results from the representation theory of quantum groups, used to derive operatorial TQ relations for quantum systems with periodic boundary conditions, in order to find operator relations for the open boundary case, and find an analogous derivation of the TQ relations in the open boundary case. This process started with their 2020 paper, which provided a route to operatorial TQ-Relations for the XXZ Heisenberg spin chain with diagonal boundary conditions, via fusion relations derived from short-exact sequences. The most recent of these papers is now available on the arXiv, in which we derive a boundary factorisation relation for the open XXZ chain, which we anticipate will soon lead to transfer matrix factorisation. This paper also connects to recent work on the universal K-matrix (see here and here), which will be key in generalising these results for different boundary conditions, and different quantum groups.
The details of the construction and properties of a second Q-operator for the XXZ chain, and parts of the route to transfer matrix factorisation, as well as the connection to the XXX chain through carefully taking the isotropic limit of relevant operator relations, are discussed in my doctoral thesis, soon to be made available on the arXiv. I am also in the process of preparing some shorter papers, following up the publication of this work.
Publications
A Q-Operator for Open Spin Chains II: Boundary Factorisation - https://arxiv.org/abs/2301.03997
Teaching
In my 7-month role as assistant professor at Heriot-Watt, I am currently lecturing the course F17SC - Discrete Mathematics, which will begin in January 2025. The course covers a range of introductory material for both Mathematicians and adjacent fields (Computer Science, Mathematical Physics etc.), including set theory and functions, probability, linear recurrence, graph theory, and linear algebra. The Canvas course pages can be found here.
If you are a student on my course with any questions, please get in touch via email, or drop by my office, CMG.02. If there is enough interest, I am more than happy to run office hours - if you are a group of students with overlapping questions, or who would be interested in attending, please get in touch via email.
In my 4-month role as assistant professor at Heriot-Watt, I lectured for the course F18XD - Mathematics for Engineers and Scientists 4, which began in January 2024. I taught weeks 1-5, covering vector geometry, vector calculus, and the Fourier and Laplace transforms. The teaching in weeks 7-12 was provided by the course leader Oana Pocovnicu - the Canvas course pages can be found here.
In this role, I also provided lecture cover and workshops for students taking the course F17CC Introduction to University Maths, run by Pamela Docherty, as well as marking for F17CA Calculus A, and F10MM/F11MM Optimisation.
Conference Attendance
NBMPS 72, Durham University, 16/11/2024
RAQIS `24, LAPTh Annecy, 02/09/2024 - 06/09/2024
Integrable Techniques in Theoretical Physics (SWI 2024), Physikzentrum Bad Honnef, 14/07/2024-19/07/2024
IPFQS 24, University of Tours, 02/07/2024 - 04/07/2024
ICFT 2024, City University London, 27/06/2024 - 28/06/2024
At The Crossroads..., IPhT Saclay, 24/06/2024 - 26/06/2024
Algebraic Structures in the Yang-Baxter Equation, Heriot-Watt University, 03/06/2024 - 07/06/2024 - [Speaker, presentation available here]
Painlevé Equations and Related Topics, Heriot-Watt University, 22/03/2024
Student Workshop on Integrability 2021, BU Wuppertal (Online), 26/04/2021 - 30/04/2021
Hypergeometry, Integrability and Lie Theory, Lorentz Center (Online), 07/12/2020 - 11/12/2020
ARTIN 56, University of Edinburgh, 28/09/2019 - 29/09/2019
NBMPS 57, Durham University, 16/09/2019
Student Workshop on Integrability 2019, Université Catholique de Louvain, 07/04/2019 - 10/04/2019 - [Speaker, abstract available here]
ICFT 2019, Oxford University, 29/03/2019 - 01/03/2019
MACS Christmas Conference 2018, Heriot-Watt University, 30/11/2018 - [Chair/Organiser]
Integrability in Atomic and Condensed Matter Physics, École De Physique Des Houches, 30/07/2018 - 24/08/2018
ICFT 22, Cardiff University, 01/06/2018 - 02/06/2018
EMS The Burn, Montrose, 21/05/2018
NBMPS 51, Durham University, 21/09/2017
Alongside these conferences, I am also a regular attendee at the seminars provided by the Edinburgh Mathematical Physics Group.
Tutoring and Marking
During my PhD at Heriot-Watt, I completed training to become an Approved Tutor and Approved Marker. In these roles, I assisted in problems classes and marking for the following courses:
F17XA Mathematics for Engineers and Scientists 1 (2020)
F19PL1 Abstract Algebra (2020)
F17XP1 Logic & Proof (2019)
F19PL1 Abstract Algebra (2019)
F17CB2 Calculus B (2019)
F17XB2 Mathematics for Engineers and Scientists 2 (2019)
F18XC1 Mathematics for Engineers and Scientists 3 (2018)
F17LP1 Logic & Proof (2018)
I have had further experience tutoring with the Maths Gym at Heriot-Watt since its inception in 2019. The Maths Gym was developed as a space for unstructured tutoring in Mathematics at Heriot-Watt, offering both drop-in sessions and 1-to-1 tutorials by appointment, catering to the needs of Mathematics and Maths-adjacent students within the university. For more information about the Maths Gym, please visit the dedicated website.
During the Covid-19 Pandemic, in response to a shift to a Responsive Blended Learning model for Mathematics teaching at Heriot-Watt, I have developed a short set of summary/revision notes on Vector Calculus for students taking the course F10AC/F11AE Applied Mathematics C/E under Professor Bernd Schroers. These notes were adapted from an excellent set of course notes kindly provided by Professor Richard Szabo, and include some short example solutions. They are available here, and a slightly less condensed version is available on request.
In 2018 and 2019, I was a PhD assistant for the Royal Institution Mathematics Masterclasses, a Mathematics outreach programme organised jointly between Heriot-Watt, Edinburgh, and Napier Universities, in order to provide opportunities for pupils from local schools to engage with concepts outside of the scope of their usual curriculum, in order to stretch the abilities of high achieving students, and to encourage an interest in Maths that can continue to grow throughout their education.
In both 2020 and 2021, I have enjoyed tutoring and marking for the SMSTC Advanced Course on Classical and Quantum Integrable Systems.
Other Responsibilities
Maxwell PhD Seminar
During 2020-2021, alongside my co-host Andrew Beckett at the University of Edinburgh, I was involved with the organisation of the Maxwell PhD Seminar, hosted online for PhD students at our two institutions, as well as interested friends and colleagues. This event developed out of the regular MACS PhD Seminar, historically held in CMT.03, before being held online during the Covid era. The ethos was intended to allow any interested PhD students to give a ~40 minute presentation, in an encouraging and friendly environment. The presentation could be on any Mathematical topic of the speaker's interest, and not necessarily be about their own research. This gave an opportunity for students to develop theirr speaking and presenting skills, especially in a time where there were limitations on physical conferences. These Seminars took place on Fridays at 13:15 BST.
I am still looking for a successor in this organisational role within the department, if you are a PhD student at Heriot-Watt interested in rekindling this seminar, I would be thrilled to hear from you.
ICFT 2020 Organiser
Alongside Robert Weston, and Bart Vlaar, I am a local organiser for the 2020 meeting of the conference on Integrable Models, Conformal Field Theories, and Related Topics, to be held at the ICMS (5th Floor, Bayes Centre, Edinburgh), on the 3rd-4th April 2020. More information about the conference can be found on the conference website. Covid Update - Sadly, due to the current global situation, it is advised that travel to conferences should be prohibited. As such, the conference has been postponed indefinitely.
MACS PGR Representative
In the academic year 2018/19, I served as the PGR representative for both Mathematics and Maths and Computer Sciences. Within this role, I was responsible for organising the MACS Christmas Conference, and the annual PhD Poster Session, alongside the representatives from Computer Sciences and AMS. I was also a point of call for any students with feedback or other issues that they wanted raised at a higher level within the department. As part of the feedback process, I attended regular meetings with the PGR co-ordinators for MACS, where the feedback I received from students was put to the committee. In my time as PGR representative, I made a move to expand the role and increase the number of social events within the MACS calendar, in order to promote a more cohesive student community across the school, and between students within the broader Edinburgh community. This expansion is being continued by my successor in the role, Bruno Barton-Singer.
If you are looking to give anonymous feedback, you can find the form here.
SMSTC Representative
In the academic year 2017/18, I was appointed the overall student representative for the Scottish Mathematical Sciences Training Centre. In this role, I collected feedback from the postgraduate student representatives of the institutions supported by SMSTC training, and presented this feedback to the organising committee at its annual meeting.
Athena Swan Committee
Until the pandemic hit, I was attending the Athena Swan meetings co-ordinated by Professor Damian Clancy. You can find more information about the MACS Equality Charter here. We should all be working towards a world in which all aspects of life are fair and diverse.
Education
I completed my PhD at Heriot-Watt University in 2023, under the supervision of Robert Weston and Bart Vlaar. The title of my thesis was "Q-Operators for Open Quantum Spin Chains", and dealt with the derivation of operatorial relations for Q-operators, including TQ-relations, in the context of the six-vertex model, corresponding to the Heisenberg XXZ and XXX spin chains with diagonal boundary conditions. I passed my viva in August 2023, under the careful examination of Des Johnston (internal) and Christian Korff (external).
I completed my Masters degree at the University of Bristol in 2016, under the supervision of Professor Mark Dennis, on a project titled "Singularities and Interfaces in Anisotropic Crystal Optics". It dealt with polarisation singularities and their relationship to refractive indices in anisotropic dielectrics, and how to account for transitions between materials with different dielectric tensors.
Contact Details
Alexander Cooper
CM G.02
School of Mathematical and Computer Sciences
Heriot-Watt University
Edinburgh
EH14 4AS
a.cooper@hw.ac.uk