# Dylan Johnston

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 May 2024.

## Projects

- During my undergraduate degree I completed an 8 week summer project titled "
**On the representation theory of GL**" which can be found hereLink opens in a new window, supervised by Jack ShottonLink opens in a new window._{3}(F_{p})

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**" which can be found hereLink opens in a new window. I was supervised by Jens FunkeLink opens in a new window.^{l}(V) - 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 SL
_{2}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 MartÃn 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]

## 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 SL_{2}(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 SL_{2}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**titled "Contramodules for algebraic groups". Slides for this talk can be found here.Link opens in a new window^{th }Ukraine Algebra Conference - 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 SL_{2}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".

- Supervisor of a group of 1
**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.

- Supervisor of 2 groups of 1
**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.

- Supervisor of 2 groups of 1

## 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