I am a third 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 March 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.

If anyone has any thoughts on how to distinguish two homogeneous spaces, both of the form G/SL2 (with different copies of SL2 corresponding to two different nilpotent orbits in Lie(G)), please get in touch as I would be very eager to make Proposition 5.24 a complete result.

Preprints

• Contramodules for algebraic groups: induction and projective covers.
  [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]

Talks

• In December 2019, I gave a short talk at the Durham Mathematics Colloquium discussing my LMS summer project, 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.

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.

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.