I am a second year PhD student under the supervision of Adam Thomas and Dmitriy Rumynin.

My interests include Representation Theory, Category Theory and Algebraic Topology.

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

Note: Everything below is correct (modulo typos) as of 14^th February 2023.

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 here, supervised by Jack Shotton.

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 here. 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 here. I was supervised by Professor Jens Funke.
• 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 here.

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.

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 here.
• 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 Theorem.
• 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 here.
• 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.

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.