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Mathematics PhD: p-adic group representations and arithmetic

Project 2692 · University of East Anglia · School of Engineering, Mathematics and Physics; Algebra, Number Theory, Logic and Representations group

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Country or region
United Kingdom
Application deadline
2026-11-30 (confirm the exact time with the source)
Funding
The listing indicates a UK-student scholarship, but the amount, stipend, duration, tuition coverage and eligibility details are not specified; confirm the terms.

The research opportunity

This mathematics PhD at the University of East Anglia explores connections between p-adic group representations, arithmetic and the local Langlands correspondence. The project is supervised by Prof Shaun Stevens. The specific research question will be selected within this area; the description does not prescribe a single theorem or proof plan.

Research context and possible work

The local Langlands correspondence broadly relates representations of the absolute Galois group of a p-adic field to representations of a matrix group over that field. It partitions irreducible group representations into L-packets. For general linear groups, each L-packet is a singleton. For more general groups, local factors—including L-functions and epsilon factors—are conjectured to characterise packets, but these factors are currently defined only for generic representations, and the conjecture that each packet contains exactly one generic representation remains open.

Most representations of the group are infinite-dimensional and difficult to analyse directly. One approach is to restrict a representation to a compact open subgroup K and study which K-representations occur. This can help identify properties such as unramified, tame or generic status, and may support more explicit descriptions when combined with arithmetic data. The project may investigate these or related questions and how results connect, through Langlands correspondence, to absolute Galois groups. The exact direction remains open.

Eligibility, funding and dates

The minimum stated qualification is a 2:1 bachelor's degree in mathematics. The listing describes a UK-student scholarship, but gives no amount, stipend, duration, tuition coverage or detailed eligibility definition. The position is full-time, planned to start on 1 February 2027, with a 30 November 2026 deadline; no deadline time or time zone is stated. Application route and required materials are not listed. Confirm the complete award, eligibility, application instructions and registration arrangements before applying.

Research and application brief

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Show report contents

Mathematics PhD: p-adic group representations and arithmetic

Quick Decision

Strong but verify fit. This is a valuable specialist mathematics studentship for someone with substantial proof-based preparation who wants to work on local Langlands, p-adic groups and arithmetic. It is broader than a single theorem: likely directions include using compact-open subgroup restrictions to understand representations and their arithmetic data, while the precise problem is to be developed with Shaun Stevens. His current research and UEA's algebra group fit the topic well. The advert says funding is for UK students, but the amount and terms are not specified. Confirm funding and the first-year question promptly; the application deadline is 30 November 2026 and the planned start is 1 February 2027.

Research Problem and Doctoral Work

The local Langlands correspondence relates representations of a p-adic field's absolute Galois group to representations of a reductive matrix group over that field. The project description highlights several live mathematical issues: extending beyond complex coefficients; understanding L-packets for groups other than general linear groups; and the fact that local factors are defined only for generic representations, while the claim that each packet has one generic member remains conjectural in the general setting.

Because most group representations are infinite-dimensional, one useful approach is to restrict them to compact-open subgroups and examine which smaller representations occur. Such information can identify properties such as unramified, tame or generic behavior, and can help produce explicit descriptions tied to arithmetic data. These are project-context examples, not a fixed list of thesis deliverables. The advert leaves the specific question open within local Langlands or a related problem. Discuss which group, representation family and tractable first result would anchor the doctorate.

School, Programme and Field Context

The degree-awarding institution is the University of East Anglia (UEA), within the School of Engineering, Mathematics and Physics. UEA's REF 2021 Mathematical Sciences profile reports 28% 4-star and 66% 3-star work overall. REF is a peer-assessed research profile, not a league-table rank. The unit-level environment statement lists 20 permanent academics, £4.2m research income and 36 PhD completions since REF2014. Its algebra group connects representation theory with number theory and category theory. This subject environment is unusually relevant to the advertised work.

UEA's general PGR guide describes a three-year PhD with primary and secondary supervision in a research group. The vacancy names Stevens as the supervisor. The available materials do not specify a taught module list, formal coursework requirements, annual project milestones or whether a second supervisor will be assigned. Ask what structured training exists for local fields, p-adic groups and proof development, and whether seminar participation or a reading course is expected.

Supervisor and Lab Assessment

Stevens is a professor of pure mathematics and group lead in algebra, number theory, logic and representations. His official profile describes explicit representation theory of p-adic groups and local Langlands, lists ten completed PhD supervisions, and invites prospective students to discuss project and funding options. Recent papers demonstrate sustained work in the exact field: a 2024 paper determines Langlands parameters for a studied class of symplectic representations; a 2025/26 paper studies sharpness in local converse theorems; and a 2026 paper studies reducibility for simple supercuspidal representations over positive-characteristic local fields.

The wider group has current activity in algebra, number theory, logic and higher representation theory. A current EPSRC project on 2-representations is led by Vanessa Miemietz; it provides complementary group context, not evidence that this studentship is funded by that project. Supervision frequency, project scoping and the exact seminar/training environment remain to be discussed.

Funding, Application, and Living Cost

The vacancy is titled a studentship and lists UK students as the funding population, with the funding amount not specified. This is evidence of a restricted studentship rather than an unfunded or self-funded position, but it leaves the stipend, award length, fee coverage and research allowance unknown. UKRI figures are a comparison only: the 2026/27 minimum full-time stipend is £21,805 and the minimum fee contribution is £5,238 for home fees. Do not attribute those numbers to this offer unless the provider confirms UKRI funding. UEA lists £21,650 for first-year overseas Mathematics research fees in 2026/27, so overseas-fee coverage is a material question.

UEA's current campus accommodation prices provide only a partial benchmark: 41-week en-suite room rates are about £7,318–£8,590, and doctoral accommodation may use a different term length. The UK visa maintenance amount is a visa threshold, not a full doctoral budget, and could change before a February 2027 start. The short interval from the 30 November deadline to the 1 February start makes written funding and visa timing confirmation important.

Applicant Fit and Preparation

Advertised minimum: a 2:1 bachelor's degree in mathematics. Suggested preparation, not additional formal eligibility: proof-based algebra, representation theory, number theory and comfort with abstract infinite-dimensional objects; prior exposure to p-adic fields or local Langlands would help an applicant explain a research direction. The advert does not require programming or a laboratory background. As the topic is open-ended, prepare a concise research-interest note naming a tractable question and the mathematical background that supports it. A candidate whose experience is mainly applied mathematics should explain concrete proof-based work and why this theory is the intended next step.

Next Actions and Questions to Verify

Before applying: ask Stevens for the application route/materials, the exact home-fee eligibility definition, award amount and term, stipend, fee coverage and any international funding route. Send a brief summary of mathematical preparation and ask which possible research angle is most realistic for a February start.

At interview: clarify the first-year reading/research plan, how the first theorem will be selected, supervision frequency, opportunities to join local seminars, and how the question can be narrowed if one avenue stalls.

Before accepting: obtain the written funding and fee terms, confirmed start/registration dates, training plan and expected research costs. This project should not be filtered as low value merely because its field is specialized: its value depends on a candidate's mathematical interests and the verified funding package.

Evidence and Sources

Project and research environment

Supervision and publications

Training, funding and living costs

Field and career context

Sources

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