# Adam Jones - Postdoctoral Researcher in Mathematics, University of Cambridge

Email: aj718@cam.ac.uk

About me

I am a postdoctoral fellow at the University of Cambridge, researching non-commutative algebra and representation theory. I am funded and supported by the Heilbronn Institute for Mathematical Research.

I completed my DPhil at the University of Oxford in 2020, after which I was a Heilbronn fellow at the University of Manchester until October 2023, when I began my current position. I believe mathematics is an unparalleled discipline, and I hope to continue practicing it in the future for as long as I can.

Research Interests

My research is primarily in non-commutative algebra, often with application to representation theory. I usually work in a p-adic or non-archimedean setting, and one of the main goals of my research is to advance the representation theory of p-adic groups.

I have worked closely with the completed group algebra, or Iwasawa algebra, of a compact p-adic Lie group, which was the main focus of my DPhil research. In several cases, I have completely classified the prime ideal structure of these algebras. I have also explored other classes of rings, arising from related constructions, including skew-power series rings and completed enveloping algebras. More recently, I have begun to explore the smooth, modular representation theory of locally compact p-adic Lie groups, and I have utilised Bruhat-Tits theory to develop a better understanding of this topic.

My other interests include Galois theory, algebraic number theory and tropical algebra.

Teaching

I have given small group tutorials and supervisions, in Oxford, Manchester and Cambridge, for a variety of undergraduate subjects including Linear Algebra, Commutative algebra, Representation theory and Lie algebras. I have also tutored Galois theory, given lectures in non-commutative algebra, and supervised an undergraduate project on profinite groups. In Lent term 2025 in Cambridge I will be lecturing a Part III course in Coxeter Groups.

Events organised

2023-present - Algebra and Representation Theory Seminar, University of Cambridge (joint with Simon Wadsley).

2022 - Focused research event: Smooth representations of GL(n,Q_p) in natural characteristic, University of Oxford.

Conferences attended

2024 - Representations of p-adic Groups and the Langlands Correspondence, King's College London

2023 - Algebra and Number theory in Conversation, University of Manchester.

2023 - Number theory meets p-adic representations, Universitat Muenster,

2022 - Recent Advances and New Directions in the Interplay of Noncommutative Algebra and Geometry, University of Washington.

2021 - The unity of mathematics: A conference in honour of Sir Michale Atiyah, University of Cambridge.

2020 - Tropical Geometry, Berkovich Spaces, Arithmetic D-modules and p-adic Local systems, Imperial College London (virtual).

2019 - Geometric Methods in p-adic Representation Theory, Trinity College Dublin.

2019 - Representation Theory and D-modules, Université de Rennes 1.

2017 - D-modules, Geometric representation theory and Arithmetic applications, University of Oxford.

Publications and Preprints

Bounded skew power series rings for inner skew derivations (preprint), joint with William Woods, arXiv.2408.10545 (2024).

Skew power series rings over a prime base ring, Journal of Pure and Applied Algebra, joint with William Woods, (2024) arXiv

Weak dimension of power series rings over valuation rings, Journal of Pure and Applied Algebra, (2024) arXiv

Filtered skew-derivations on simple artinian rings (preprint), joint with William Woods, arxiv.2301.02639 (2023).

Primitive ideals in rational, nilpotent Iwasawa algebras, Advances in Mathematics, (2022). arXiv

Affinoid Dixmier modules and the deformed Dixmier-Moeglin equivalence, Algebras and Representation theory, (2021). arXiv

A control theorem for primitive ideals in Iwasawa algebras, Journal of Algebra, (2020). arXiv

Completed group algebras of abelian-by-procyclic groups, Advances in Mathematics, (2019). arXiv

Prime ideals in Iwasawa algebras of solvable groups, DPhil thesis, University of Oxford (2020). pdf

Computation of Galois groups using p-adic approximations, MSci Dissertation, Queen Mary University of London (2016). pdf

Links

Here are some links to sites of other mathematicians whom I have worked with or alongside in the past, and whose research I continue to follow:

Konstantin Ardakov (my DPhil supervisor)

Leonard Soicher (my masters project supervisor)