I'm a postgraduate research student at the University of Leeds in the School of Mathematics. I am supervised by Dr Oleg Chalykh and co-supervised by Professor Karin Baur. I completed my undergraduate studies at the University of Leeds in July 2020, obtaining the MMath, BSc Mathematics degree.
Research
My primary research areas of interest are algebra, geometry, topology, representation theory and integrable systems (note that this is certainly not an exhaustive list). During my time as a postgraduate student, the focus has been on representations of Cherednik algebras and quivers. I have, however, studied the interplay between these and the other topics mentioned in the list prior.
The main goal of my project is to prove a conjecture by Etingof-Gan-Oblomkov (2006), namely that the variety of finite-dimensional irreducible representations of the double affine Hecke algebra of type \(C^\vee C_n\) is isomorphic to a Calogero-Moser space. This isomorphism is known for type \(A_{n-\!1}\) by Oblomkov (2003), so we wish to develop some theory that allows us to adapt this proof for our setting. The next goal is to show that the isomorphism respects Poisson structures, and thence interpret things from the point-of-view of integerable systems.
Papers
DAHAs of Type \(C^\vee C_n\) and Character Varieties (with O. Chalykh), preprint (2024), [arXiv].
Projects
Knot Theory, Master's Thesis, University of Leeds, May 2020, [PDF].
Hyperbolic Geometry, Bachelor's Thesis, University of Leeds, May 2019, [PDF].
Presentations
DAHA, Character Varieties and van Diejen's System: Leeds Integrable Systems Seminar Series, University of Leeds, November 2024, [Slides].
Double Affine Hecke Algebras and Calogero-Moser Space: Junior London Algebra Colloquium, City, University of London, November 2023, [Slides, Notes].
Double Affine Hecke Algebras and Character Varieties: Leeds Algebra Seminar, University of Leeds, October 2023, [Slides, Notes].
Character Varieties and Symmetric Polynomials: School of Mathematics Postgraduate Research Conference, University of Leeds, June 2023, [Slides].
A Deligne-Simpson Problem: School of Mathematics Postgraduate Research Conference, University of Leeds, June 2022, [Slides].
Representations of Quivers via Double Affine Hecke Algebras: Pure Mathematics Postgraduate Research Seminar (online), University of Leeds, November 2021, [Slides].
Knots and the Fundamental Group: Pure Mathematics Postgraduate Research Seminar (online), University of Leeds, May 2021, [Slides].
Conferences
Quantisation of Integrable Systems: Old and New, University of Leeds, May 2024.
Painlevé Equations and Related Topics, The University of Edinburgh/Heriot-Watt (online), March 2024.
Dualities and Symmetries in Integrable Systems, Isle of Skye, June 2023.
School of Mathematics Postgraduate Research Conference, University of Leeds, June 2023.
Poisson and Quantum Structures in Integrable Systems, University of Leeds, May 2023.
74th British Mathematical Colloquium, University of Bath, April 2023.
The Geometry of DAHAs and Coulomb Branches, The University of Edinburgh, March 2023.
60th ARTIN Meeting, University of Nottingham (online), August 2022.
Poisson Structures and Noncommutative Integrability, University of Kent, June 2022.
School of Mathematics Postgraduate Research Conference, University of Leeds, June 2022.
ARTIN New Arrivals Meeting, The University of Edinburgh, May 2022.
58th ARTIN Meeting, The University of Manchester (online), March 2022.
Integrability and Applied Algebraic Geometry, The University of Edinburgh (fully online), March 2022.
Integrable Day, Loughborough University (online), November 2021.
British Early Career Mathematicians' Colloquium, University of Birmingham (fully online), July 2021.
Integrable Days, Loughborough University/The University of Edinburgh (fully online), November 2020.
London Mathematical Society Undergraduate Summer School, University of Leeds, July 2019.
Teaching
This is an exhaustive list of my official teaching responsibilities, both past and present. You can also find condensed versions of the lecture notes (each internally dubbed a cheatsheet) for the tutored and marked modules. These are based on notes I made as an undergraduate student, with updates to reflect the current versions of each module.
Academic Year 2023/24
MATH1025 Number Systems, Tutor, University of Leeds, [Notes].
MATH1026 Sets, Sequences and Series, Tutor and Exam Marker, University of Leeds, [Notes].
MATH1225 Introduction to Geometry, Tutor and Exam Marker, University of Leeds, [Notes].
MATH1400 Modelling with Differential Equations, Exam Marker, University of Leeds.
MATH2017 Real Analysis, Module Marker, University of Leeds, [Notes].
MATH2022 Groups and Vector Spaces, Exam Marker, University of Leeds.
MATH2027 Rings and Polynomials, Module Marker, University of Leeds, [Notes].
Academic Year 2022/23
MATH1060 Introductory Linear Algebra, Tutor, University of Leeds, [Notes].
MATH1225 Introduction to Geometry, Exam Marker, University of Leeds.
MATH1331 Linear Algebra with Applications, Tutor, University of Leeds, [Notes].
MATH1400 Modelling with Differential Equations, Tutor, University of Leeds, [Notes].
MATH2027 Rings and Polynomials, Module Marker, University of Leeds, [Notes].
MATH3044 Number Theory, Module Marker and Exam Marker, University of Leeds, [Notes].
MATH3071 Groups and Symmetry, Exam Marker, University of Leeds.
MATH3143 Combinatorics, Exam Marker, University of Leeds.
Academic Year 2021/22
MATH1005 Core Mathematics, Exam Marker, University of Leeds.
MATH1025 Number Systems, Exam Marker, University of Leeds.
MATH1026 Sets, Sequences and Series, Tutor, University of Leeds, [Notes].
MATH1331 Linear Algebra with Applications, Tutor, University of Leeds, [Notes].
MATH2230 Discrete Mathematics, Exam Marker, University of Leeds.
Academic Year 2020/21
MATH1026 Sets, Sequences and Series, Exam Marker, University of Leeds.
MATH1060 Introductory Linear Algebra, Tutor and Exam Marker, University of Leeds, [Notes].
MATH1225 Introduction to Geometry, Exam Marker, University of Leeds.
Tutoring
As much as I care about understanding and developing new mathematics in a research setting, something which is equally a passion of mine is the process of communicating mathematics with others, especailly if they are not understanding it. After all, what is really the purpose of knowledge if not to share it?
Online Tutoring
Since starting my undergraduate studies in September 2016, I have been an online tutor of mathematics. Having led one-on-one tuition with over twenty students, from GCSE to undergraduate level, I got used to communicating both basic and complicated mathematical concepts to people. I am more than happy to provide references to parents/guardians and mature students who want to see if I would be a good tutoring fit for them. The subjects I tutor are A-Level (Further) Mathematics and GCSE Mathematics, from all examination boards. To date, I have not tutored anybody studying mathematics pre-GCSE level but this is something I would like to do.
Prison Math Project
I have been involved in the Prison Math Project (homepage) since August 2021, during which time I have had correspondence with a number of inmates who each have a passion for mathematics and want guidance with learning degree-level concepts. This has led me to write what really amounts to lecture notes for them, and then discussing how they find the new material by giving examples and providing exercises. In February 2022, Brady Haran released a podcast with founder Christopher Havens on the YouTube channel Numberphile2 (podcast).
My main contributions to this venture are to act as a penpal with inmates, and to help design so-called modules for inmates to work through. Below is an exhaustive list of resources that I have made for this organisation.
Differential Geometry I, Module Creator, Prison Math Project, [PDF (in progress)].
Contact Me
Please feel free to get in touch to talk about the research I am doing or enquire about tutoring.
My university email which is used for academic correspondence: B.J.Ryan@leeds.ac.uk.
