I am a final-year PhD student under the supervision of Adam ThomasLink opens in a new window and Dmitriy RumyninLink opens in a new window.

My interests include Representation Theory, Category Theory and Homological Algebra.

You can contact me at dylan(dot)johnston(at)warwick(dot)ac(dot)uk.

Note: Everything below is correct (modulo typos) as of October 2024.

Projects

• During my undergraduate degree, I completed an 8-week summer project titled "On the representation theory of GL₃(F_p)" which can be found hereLink opens in a new window, supervised by Jack ShottonLink opens in a new window.

Note: This was funded by the London Mathematical Society (LMS) as part of their Undergraduate Research bursary, more information on the bursary can be found hereLink opens in a new window. This is a great opportunity for any undergraduates who are potentially interested in pursuing mathematics further, if you wish to chat to me about this please get in touch!

• My Master's thesis was titled "The relative Lie algebra cohomology of the Weil representation of o(p,1) x sl(2) with coefficients in Sym^l(V)" which can be found hereLink opens in a new window. I was supervised by Jens FunkeLink opens in a new window.
• My 1^st year PhD report was titled "Homotopy type of some homogeneous spaces of simple Lie groups
  of dimension 75" and this can be found hereLink opens in a new window.

Publications and preprints

• Contramodules for algebraic groups: induction and projective covers.
  [Journal of Algebra 664A (Feb 2025): pp 604-630]
  [arXiv:2304.07525Link opens in a new window, submitted April 2023]
• Topological rigidity of SL₂ quotients,
  joint with Dmitriy RumyninLink opens in a new window.
  [arXiv:2309.07238Link opens in a new window, submitted September 2023]
• Global representation ring and Knutson Index,
  joint with Diego Martín DuroLink opens in a new window and Dmitriy RumyninLink opens in a new window.
  [arXiV: 2403.18498Link opens in a new window, submitted March 2024]
• Contramodules for algebraic groups: the existence of mock projectives.
  [arXiv:2404.16618Link opens in a new window, submitted April 2024]
• Disconnected reductive groups: classification and representations,
  joint with Diego Martín DuroLink opens in a new window and Dmitriy RumyninLink opens in a new window.
  [arXiV: 2409.06375Link opens in a new window, submitted September 2024]

Talks

• In December 2019, I gave a short talk at the Durham Mathematics Colloquium discussing my LMS summer project. The slides for this talk can be found hereLink opens in a new window.
• In November 2021, I gave a talk in the Algebra reading group on "Nilpotent Orbits of Semisimple Lie algebras" by Collingwood & McGovern. The talk focused on the proof of the Jacobson-Morozov TheoremLink opens in a new window.
• In November 2022, I gave a talk in the Algebra reading group discussing some of section 4 of the paper of Helfgott titled "Growth and Generation in SL₂(Z/pZ)".
• In January 2023, I gave a talk in the Algebra reading group on Hecke algebras. The talk focused on the Kazhdan-Lusztig basis of a Hecke algebra, my preparation notes can be found hereLink opens in a new window.
• In February 2023, I gave a talk at the Warwick Algebra Seminar titled "Homotopy type of SL₂ quotients of simple complex Lie groups". This talk discussed some of the concepts and results in my 1^st year PhD report.
• In June 2023, I gave a talk at the Warwick Algebra Seminar titled "Representations of algebraic groups: A journey into the world of contramodules".
• In July 2023, I gave a short talk at the 14^th Ukraine Algebra Conference titled "Contramodules for algebraic groups". Slides for this talk can be found here.Link opens in a new window
• In October/November 2023 I gave two talks at a Quantum groups reading group. These talks discussed the representation theory of Uq(sl2) and Uq(g) respectively.
• In January 2024 I gave a short talk at the LMS South West and South Wales regional meeting and workshop held at the University of Bath.
• In February 2024 I gave a talk at the Oxford Junior Algebra and Representation Theory Seminar where I discussed my joint work on topological rigidity of SL₂ quotients.
• In October 2024 I gave a talk at the Warwick Algebra Seminar titled "Classification of disconnected reductive algebraic groups". The talk focused on the classification part of my joint work with Diego and Dmitriy, as found in our paper, "Disconnected Reductive Groups: Classification and Representations."

Teaching

• 21/22 Academic Year
  • Supervisor of a group of 1^st year Mathematics students.
    • Mark homework and discuss content from modules including Analysis, Linear algebra, Geometry and Motion, and Differential Equations.
  • Teaching assistant for "MA3K4: Introduction to Group Theory".
• 22/23 Academic Year
  • Supervisor of 2 groups of 1^st year Math/Phys students.
    • Mark homework and discuss content from modules including Analysis, Linear algebra, and Mathematical Modelling.
• 23/24 Academic Year
  • Supervisor of 2 groups of 1^st year MORSE students.
    • Mark homework and discuss content from modules including Calculus, Sets and Numbers, and Linear algebra.
• 24/25 Academic Year
  • Supervisor of 2 groups of 1^st year Math/Stats students.
    • Mark homework and discuss content from modules including Sets and Numbers, Analysis and Linear algebra.

Other

• Co-organiser (alongside DiegoLink opens in a new window) of a reading group on Quantum groups, roughly following Jantzen's book on the topic.
• Link to personal Github.Link opens in a new window
• A web app I created to visualise the surjective map from $S_4$ to $S_3$.Link opens in a new window