Mathematics PhD: Function-Field Number Theory and Random-Matrix Statistics

Mathematics PhD: Function-Field Number Theory and Random-Matrix Statistics

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This Exeter EPSRC DLA project connects three problems in function-field number theory and random-matrix statistics: special-value moments of quadratic Dirichlet L-functions, shifted divisor correlations in Fq[T], and moments of moments of characteristic polynomials. Its mathematical focus closely matches Julio Andrade’s documented research, but the entry profile is narrow: the project PDF requests a first-class undergraduate degree and a Mathematics Master’s with substantial analytic number theory and random-matrix preparation. Confirm eligibility and clarify the corrupted parameter-range sentence before applying.

Research Problem and Doctoral Work

Why function fields are useful

Function fields over finite fields Fq allow some questions about L-functions to be studied using tools unavailable over the integers. The project points to the Riemann Hypothesis for curves over finite fields, proved by Weil, and Katz–Sarnak equidistribution. These results support exact averaging and symmetry comparisons between arithmetic families and random matrix ensembles. The project spans analytic number theory, arithmetic statistics and random matrix theory. Official project PDF, Exeter's Number Theory, Algebra and Geometry group.

Three advertised strands

1. Moments at s = 1 and class numbers. For quadratic characters χ_D attached to squarefree polynomials D in Fq[T], the analytic class-number formula connects L(1, χ_D) with the class number h(D) of the corresponding function field. The advertised quantity is M_k(q) = Σ_{deg D = 2g+1} L(1, χ_D)^k. The work would seek asymptotic formulae as q or deg D grows and interpret them through the distribution of function-field class numbers and related counting problems. The project also allows analogous families. Andrade and Shamesaldeen's prior class-number study provides a direct baseline, while later twisted-moment work shows continuing activity in function-field L-functions. The exact family normalization and whether the main limit is fixed-q genus growth, q-aspect growth or both should be agreed with the supervisors. Class-number moments study, Andrade and MacMillan on twisted moments.

2. Shifted divisor correlations. The project proposes sums of the form Σ_{deg f=n} d_k(f)d_k(f+a), where d_k is the k-fold divisor function and a is a fixed polynomial shift. It connects these sums to strand 1 through multiplicative structure and to analogues of Ingham/Titchmarsh divisor-correlation questions. A 2025 function-field paper proves an asymptotic for the d(f)d(f+h) case in a stated large-degree range and studies related mixed correlations. This establishes active work nearby, but does not settle the project's full d_k setting or exact range. The project material does not specify whether f is restricted to monic polynomials, how a is normalized or which variables tend to infinity. These are foundational scope questions. Project PDF, recent shifted-convolution paper.

3. Moments of moments of characteristic polynomials. For a random matrix A from the unitary, symplectic or orthogonal groups, the target is M_N(k,β) = E[((1/2π)∫₀²π |Λ_A(θ)|^(2β) dθ)^k]. The PDF links these statistics with Painlevé equations, partition functions and conjectural L-function moments. Andrade and Best have proved asymptotics for moments of moments in symplectic and orthogonal ensembles and for joint derivative moments across several ensembles. These papers give direct methodological context but do not identify which k, β ranges the PhD will target. The PDF phrase reads “previously untreated Engineering and Physical Sciences Research Council Doctoral Landscape Award regimes of k and β.” Those funding-scheme words interrupt the mathematical syntax and appear to be an editing insertion. Do not infer a mathematical restriction from it; ask for the intended sentence and open range. Andrade and Best, moments of moments, joint moments of characteristic-polynomial derivatives.

Flexible scope and computational tools

The three strands are guides, not fixed work packages. Supervisors may select another topic within analytic number theory, function-field arithmetic or random matrix theory based on the candidate's background and interests. The project promises training in at least one of those areas, not necessarily all three. Symbolic computation and machine-learning pattern detection are described as optional aids to conjecture generation and numerical checking alongside classical analytic and algebraic methods. Python, SageMath or Magma experience is an advantage, not an entry requirement. Applicants should ask which tools are genuinely expected in year one and whether the preferred topic is already settled. Official project PDF.

School, Programme and Field Context

The named route is PhD in Mathematics at the University of Exeter, in the Department of Mathematics and Statistics. The project PDF gives the study location as Laver. Exeter's Mathematics course page lists Streatham and Penryn routes; confirm the registration campus, supervisory unit and practical implications of Laver in writing. Exeter is the advertised host and likely degree-awarding institution; no external partner or joint-degree arrangement is named. Project PDF, Mathematics PhD programme.

The department's pure-mathematics research includes number theory, arithmetic geometry, analysis and dynamical systems. Its Number Theory, Algebra and Geometry group explicitly lists analytic number theory and function-field interests, and lists Andrade and Tseng among academic staff. The project describes a supportive setting with regular seminars, workshops and outreach. Exeter's Mathematics course page also describes research-method training, computing resources and access to the APTS and MAGIC training consortia. These establish a relevant departmental environment, but the project does not specify a personal training schedule or which courses the student would attend. Department research theme, Mathematics PhD facilities and training, potential PhD advisors booklet.

For subject-level context, Exeter reports Top 20 in the UK for Mathematics in the Times and Sunday Times Good University Guide 2027 and Complete University Guide 2027, and Top 10 worldwide for Mathematics and Computer Science research under CWTS Leiden 2024, measured by the share of articles among the top 10% most cited. The latter combines two fields; neither ranking directly assesses this project niche. REF 2021's Mathematical Sciences unit profile was 42% 4-star world-leading and 52% 3-star internationally excellent. For overall reputation only, Exeter lists 136th in QS World University Rankings 2027. Mathematics programme ranking details, REF 2021 results, Exeter overall rankings.

The award lasts up to 3.5 years full-time, while Exeter gives a full-time PhD a maximum registration period of four years. The University says funded students should plan thesis submission by the funding end date. This does not mean the student must take four years, but the expected completion schedule, any extension funding and part-time option should be confirmed. Degree durations and funded-study guidance, EPSRC award terms.

Supervisor and Research Environment

The project PDF names Julio Andrade as lead and Jimmy Tseng as additional supervisor; it calls Tseng “Dr,” while Exeter's current department directory calls him “Professor.” Andrade's department research area includes analytic number theory, random matrix theory and L-functions over function fields. Tseng's record focuses on homogeneous dynamics, ergodic theory, Diophantine approximation and related analysis. The project says its team combines analytic number theory with dynamics and Diophantine approximation. This is a plausible intellectual complement, but the documents do not allocate strands, specify meeting frequency or divide supervisory duties. Official project PDF, department research group, official staff directory, Andrade and Tseng research areas, Andrade publication list, Tseng research outputs.

Recent work relevant to the advertised directions

• Andrade and Best, 2023, Random Matrices: Theory and Applications. They prove asymptotics for moments of moments of symplectic and orthogonal random-matrix characteristic polynomials, give integral forms for leading coefficients and relate L-function conjectures to shifted-moment conjectures. This directly supports strand 3 but leaves the studentship's parameter range open. Paper record.
• Andrade and Best, 2024, Journal of Physics A. They establish asymptotic formulas for joint moments of two derivatives of characteristic polynomials over symplectic and orthogonal ensembles, with leading coefficients in hypergeometric-determinant and partition-sum forms. This shows technical continuity, but treats joint derivatives rather than the exact advertised M_N(k,β). Paper record.
• Andrade, Gonek and Lee, 2025, Advances in Mathematics. They prove a universality result for Dirichlet L-functions over Fq[x] for odd q and suitable polynomial moduli. It shows current function-field L-function activity, although it concerns approximation by analytic functions rather than s=1 moments. Article.
• Andrade and MacMillan, published online 2025. Their asymptotic for twisted second moments over primitive characters modulo a polynomial provides related function-field techniques, but addresses a different family and average from quadratic class-number moments. Article.
• Tseng, 2023, Advances in Mathematics. Work on shrinking-target horospherical equidistribution develops translated Farey sequences and renormalization. It supports the additional supervisor's dynamics/Diophantine expertise, but not direct L-function specialization. Article.
• Tseng, 2026 preprint. A current preprint studies badly approximable numbers, Schmidt games and Hausdorff dimension. It is not peer reviewed and is adjacent to, rather than part of, the advertised strands. Preprint.

The relevant setting is mathematical seminars, computing and peer research rather than an advertised experimental laboratory. Ask about seminar rhythm, computational support, the supervision split and whether a preferred question has already been scoped.

Research Direction and Career Context

Recent work continues on function-field universality, twisted moments, class-number distributions and shifted correlations. In 2025, Florea, Lalín, Malik and Sahay derive an asymptotic for d(f)d(f+h) in a large-degree range and treat mixed variants. This is a concrete comparison for strand 2 but does not settle the advertised k-fold correlation. Andrade and Best's matrix-ensemble formulas similarly show progress for some derivatives and symmetry classes; remaining cases depend on k, β, ensemble and averaging convention. These are plausible research directions, but the project PDF does not name a target theorem or preliminary calculation. Shifted-convolution paper, moments of moments, joint derivative moments.

The most direct career route is mathematical academia, where progress depends on original theorems, publication development and postdoctoral openings. Broader quantitative roles may include data science, statistics, finance and software if computational and statistical skills are developed. Exeter lists examples of mathematics postgraduate destinations such as postdoctoral researcher, lecturer, quantitative analyst, senior data scientist and statistician, but warns that job titles and employers are not necessarily paired and gives no placement rate. The Academy for the Mathematical Sciences estimates 5.11 million UK mathematical-science occupation workers in 2024 and £489bn in direct GVA. This broad category includes occupations using mathematics-derived techniques; it does not forecast openings for pure number-theory PhDs. Since programming is optional in this project, applicants considering non-academic work should ask which computational skills or industry-facing activities are available and develop evidence beyond theorem training. Exeter Mathematics careers and graduate examples, Academy for the Mathematical Sciences workforce report.

Funding, Application and Living Cost

Item	Verified information	What to check
Entry	September 2027	Confirm start date and registration arrangements in the offer.
Deadline	Midday Tuesday 8 December 2026	Apply through the project-specific link in the PDF.
Stipend	Index-linked; currently £21,805/year at the 2026/27 rate; full-time award up to 3.5 years	Confirm 2027/28 rate, schedule and part-time availability.
Fees	Home-rate tuition; Exeter covers the home/international difference for successful international awardees	Obtain this in the written offer. Mathematics page lists £27,600 international tuition for 2027/28 and Home fee TBA.
Research support	£5,000 total RTSG over 3.5 years	Ask eligible expense and carry-over rules; it is not personal stipend income.
International places	Up to 30% of the cohort under scheme guidance	Fee coverage does not remove selection or residence rules.
Other costs	UKRI does not fund visa or immigration health surcharge	Budget visa, IHS and relocation separately.

Exeter's project-specific page resolves the international-fee question for successful awardees: the University covers the tuition difference. UKRI's general rule alone funds only home-fee rates, so the Exeter waiver must remain explicit. The Mathematics course page lists international fees of £26,800 for 2026/27 and £27,600 for 2027/28, with the Home fee TBA and annual fee changes expected. Do not calculate an applicant-paid tuition gap from these figures because the award page says Exeter covers it for successful international candidates. Project award page, UKRI student-support rules, Mathematics course fees, general PGR fee table.

Exeter estimates basic living costs at £1,300–£2,000 per month, based on local spend in April 2026. The £21,805 stipend averages about £1,817/month: £517 above the low estimate and £183 below the high estimate. The weekly estimate is £316–£421; self-catered halls are £150–£270/week including bills and private rooms £135–£250 before bills. This suggests a modest margin at the lower end and a shortfall at the upper end before travel, visa/IHS, deposits or initial purchases. Exeter international-student budget.

The project-specific page asks for an online application through the project link, personal statement naming the project, CV, full transcript(s), two references and English evidence when required. Documents must be in English, with certified translations of qualifications. Separate applications are needed for each of up to three DLA projects. Ref 6019 says references are due 8 January 2027. The generic scheme page instead says 15 February 2027 and includes stale 2026 interview/outcome text, while Ref 6019 lists interviews between 15 and 24 February 2027. Use the earlier referee date as the planning assumption and ask admissions to confirm. Project-specific award/application page, scheme-level application details.

Applicant Fit and Preparation

Project-specific requirements

The PDF asks for a first-class undergraduate degree (or equivalent) and a Master's degree in Mathematics. It expects substantial analytic number theory and random matrix theory preparation, usually through advanced Master's coursework, dissertation or research. Familiarity with L-functions, Dirichlet characters and polynomial rings over finite fields is expected; complex analysis and algebra are important. Candidates should demonstrate mathematical maturity, independence and strong research motivation. Python, SageMath or Magma is an advantage, not required. This is stricter than the scheme-wide baseline of First or Upper Second, with possible flexibility for some Lower Second applicants who also have a Master's or relevant experience. The project-specific rule should be treated as decisive unless the supervisor confirms an equivalent path. Official project PDF, general DLA academic criteria.

Suggested preparation, not added requirements

A focused application can show advanced mathematics transcripts, coursework/research in analytic number theory and random matrix theory, complex analysis and algebra, and any experience with L-functions, characters or finite-field polynomial arithmetic. A short statement should identify how prior work transfers to at least one strand without assuming the project has already selected it. Coding artifacts can help if available, but are not mandatory. Applicants whose background is mostly applied mathematics, statistics or machine learning without graduate-level number theory may face a substantial mismatch; a strong pure-mathematics candidate can meet the stated requirements without programming.

International applicants may need English evidence. Exeter's Mathematics course page maps Streatham to Profile A and Penryn to Profile B2. Profile A lists IELTS Academic 6.0 overall with no component below 6.0. Since the PDF names Laver but does not map that location to a language code, confirm the appropriate profile before booking a test. Mathematics entry, English and programme information, Profile A requirements.

Next Actions and Questions to Verify

Before applying

1. Email Julio Andrade a concise summary of academic preparation. Ask which strand is most likely to anchor the project and whether to frame the personal statement around one strand.
2. Ask for the intended text replacing “Engineering and Physical Sciences Research Council Doctoral Landscape Award regimes of k and β,” and the target parameter range, matrix ensemble and preliminary work.
3. Ask whether the shifted sum is over monic f, which restrictions apply to a, what tends to infinity and which d_k cases are within scope.
4. Confirm the first-class undergraduate plus Mathematics Master's condition and accepted international equivalents; check EPSRC residence eligibility and the international award cap.
5. Apply through the project-specific PDF link by midday 8 December 2026. Confirm whether references are due 8 January or 15 February; plan around 8 January until clarified.

At interview

• What would be a realistic first-year mathematical milestone and a plausible doctoral contribution in the chosen strand?
• How are responsibilities divided between Andrade and Tseng, and how often does the supervisory team meet?
• Which seminars, courses and computational resources will be available?
• Are symbolic computation and ML optional aids, and what tool training is offered?
• What thesis schedule is expected relative to the 3.5-year funding period?

Before accepting an offer

• Obtain written confirmation of tuition coverage, stipend indexation, award dates, RTSG conditions and any 2027/28 update.
• Confirm degree registration, Laver campus, English profile, supervision and part-time availability.
• Compare current housing contracts with the confirmed stipend and budget separately for visa/IHS and move-in costs.

Evidence and Sources

Project and Exeter mathematics environment

• Official EPSRC DLA project PDF
• Project-specific EPSRC award page, Ref 6019
• EPSRC DLA scheme information
• Mathematics PhD programme
• Exeter PhD degree durations
• Number Theory, Algebra and Geometry group
• Potential PhD advisors booklet
• Mathematics and Statistics staff directory
• Exeter overall rankings
• REF 2021 results

Supervisors and mathematical research

• Julio Andrade publications
• Jimmy Tseng research outputs
• Andrade and Best, moments of moments
• Andrade and Best, joint derivative moments
• Andrade, Gonek and Lee, function-field universality
• Andrade and MacMillan, twisted moments
• Tseng, shrinking-target horospherical equidistribution
• Tseng, 2026 preprint on badly approximable numbers
• Recent shifted-convolution work over function fields
• Andrade and Shamesaldeen, class-number moments

Funding, costs, careers and applicant requirements

• UKRI support for funded students
• Exeter general PGR fee table
• Exeter international-student budgeting
• Exeter English Profile A
• UK mathematical-sciences workforce report