Mathematics and Acoustics PhD: Hyperbolic Geometry for Acoustic Metamaterials

Mathematics and Acoustics PhD: Hyperbolic Geometry for Acoustic Metamaterials

Quick Decision

The mathematical core is to develop cut-and-project schemes in hyperbolic space and study the resulting point sets, diffraction measures and Schrödinger-operator spectra. Acoustic metamaterial design, fabrication and testing are described only as an initial exploration, not as a fully specified experimental work package. The project suits applicants with advanced mathematics preparation and an interest in acoustics; confirm the intended theory/experiment balance, lab access and training before treating the application title as a promise of device development.

Research Problem and Doctoral Work

What Is Unsolved and Why It Matters

A cut-and-project scheme selects points from a higher-dimensional lattice using a geometric window and projects them into the space of interest. The resulting point sets can be aperiodic yet structured, linking geometry, number theory and dynamical systems with diffraction and spectral questions. Such structures are interesting for wave engineering because a material's geometry can shape how sound propagates.

Earlier work by Davies, Chaplain, Starkey and Craster asked whether modified cut-and-project constructions could improve graded acoustic metamaterials. Their 2023 study used a varying projection angle to alter local spectral gaps along a quasiperiodic waveguide and demonstrated fractal rainbow trapping with theory, simulation and experiment. This supports the project's motivation; it does not show that a hyperbolic construction will produce a better-performing device. Graded quasiperiodic metamaterials perform fractal rainbow trapping.

The proposed mathematical work moves the construction into hyperbolic space: extend cut-and-project theory there, then extract spectral information from diffraction measures and associated Schrödinger operators generated by the point sets. The applied phase is explicitly to initiate investigations into building and testing metamaterials based on that theory. The project description does not specify a device design, experimental protocol, performance target or time allocation, so it should not be read as a promise of a finished prototype or extensive laboratory campaign. Official project description.

Work, Methods, and Potential Outputs

The advertised work implies three connected tasks:

1. Develop mathematical constructions and criteria for cut-and-project schemes in hyperbolic space, drawing on aperiodic order, hyperbolic geometry, number theory and ergodic theory.
2. Characterize point sets from those schemes and investigate their diffraction measures and associated Schrödinger-operator spectra.
3. Start exploring whether selected mathematical structures can be translated into acoustic metamaterial designs and tested.

The first two are substantive research aims; the third is exploratory. Plausible topics to discuss include which hyperbolic constructions are tractable, what diffraction or spectral features might be useful for design, and what physical constraints must be imposed before a mathematical point set can become a realizable structure. These are candidate research angles rather than promised thesis results.

A directly aligned 2026 preprint by Howat, Samuel and Yıltekin-Karataş studies cut-and-project schemes associated with cocompact Fuchsian groups acting on the Poincaré disc. It gives a condition for chaotic Delone sets and finds countably infinitely many tile lengths, including for triangle groups. This is a plausible mathematical foothold for the advertised direction, but it remains a preprint and does not establish an acoustic application. Cut and project schemes in the Poincaré disc.

School, Programme and Field Context

The degree-awarding institution is the University of Exeter. The project is advertised through Mathematics and Statistics and appears to fit the MPhil/PhD Mathematics route at Streatham Campus; a co-supervisor is based in Physics and Astronomy. Confirm the formal registration title and lead-supervisor location. The Mathematics programme describes strengths in pure and applied mathematics, including number theory, arithmetic geometry and dynamical systems. The programme page reports “Top 20 in the UK for Mathematics” in The Times and The Sunday Times Good University Guide 2027 and the Complete University Guide 2027. It also reports “Top 10 in the world” for Mathematics and Computer Science research by the CWTS Leiden Ranking 2024 share of articles in the top 10% most cited. These are dated subject/research indicators, not measures of this project's outcome. Mathematics PhD programme, Mathematics and Statistics research overview.

The Mathematics programme lists training in research methodology and relevant software, bibliographic resources, mathematical typesetting, desk/computer access and advanced PhD training through APTS and MAGIC. Exeter's general full-time PhD maximum is four years, while DLA support is up to 3.5 years; the University says funded students should plan submission by their funding end date. Ask what thesis schedule, fees and any writing-up arrangements apply if registration goes beyond the award period. The programme describes regular timetabled supervision as a general norm. Mathematics PhD programme, degree-duration rules.

Exeter's mathematical research includes relevant activity in dynamics, analysis, number theory and geometry. Separately, the Centre for Metamaterial Research and Innovation (CMRI) describes a community of academic, industry and government partners with capabilities spanning theory, simulation, measurement and fabrication. Its acoustic expertise page lists modelling, fabrication and measurement capabilities. This makes the local environment plausible for the proposed bridge, but it does not confirm this student's facility access or CMRI cohort membership. Dynamical Systems and Analysis, Number Theory, Algebra and Geometry, CMRI, acoustic metamaterial expertise, CMRI doctoral training.

Career Pathways and Market Boundaries

The transferable skills point toward mathematical research, applied modelling and scientific computing, and acoustics/wave or engineering R&D. Exeter's Mathematics page lists graduate destinations such as postdoctoral researcher, lecturer, quantitative analyst, senior data scientist and statistician, but notes that the job titles and employers are listed independently and are not paired. Prospects describes pure and applied mathematics research across academic, industry and government employers, while cautioning that specialist research posts are concentrated in a limited number of institutions. The 2026 UK metamaterials roadmap includes acoustic/noise/vibration control among opportunity areas but identifies scale-up, manufacturing, assurance and commercialization as continuing challenges. A 2025 parliamentary briefing reports broad STEM recruitment difficulties, while recording disagreement over the size and causes of skills gaps. These sources support plausible pathways, not a job or salary forecast for this doctorate or a confirmed market for hyperbolic acoustic structures. Mathematics PhD programme, Research scientist (maths), Prospects, UK Metamaterials Commercialisation Roadmap, UK STEM skills pipeline.

Supervisor and Lab Assessment

Supervisor Research Directions

The official project sheet names Tony Samuel as lead supervisor and Gregory Chaplain as co-supervisor. Exeter's staff lists identify Samuel as a Senior Lecturer in Mathematics and Statistics and Chaplain as a Senior Lecturer in Physics and Astronomy. Samuel's profile describes mathematical quasicrystals, fractal sets and applications of fractal geometry; Chaplain's profile emphasizes mathematical modelling, simulation and design of metamaterials for sound and vibration. Their pairing is well matched to the project's mathematical-to-acoustic scope. Tony Samuel, Exeter staff list, Gregory Chaplain, Exeter staff list, Samuel research profile, Chaplain research profile.

Recent Important Publications

• Howat, Samuel and Yıltekin-Karataş (2026), “Cut and project schemes in the Poincaré disc.” Studies hyperbolic cut-and-project schemes, chaotic Delone sets and tile lengths. Closest technical match, but still a preprint. arXiv:2603.11311.
• Davies, Chaplain, Starkey and Craster (2023), “Graded quasiperiodic metamaterials perform fractal rainbow trapping.” Combines graded cut-and-project geometry with changing local band gaps, validated for an acoustic waveguide through theory, simulation and experiment. Motivates this project but uses a non-hyperbolic construction. arXiv:2305.10520.
• Moore, Starkey and Chaplain (2024), “Acoustic metasurfaces with Frieze symmetries.” Designs, simulates and experimentally characterizes seven one-dimensional acoustic metasurfaces built around distinct frieze groups. Shows how mathematical symmetry can shape acoustic dispersion, but does not test hyperbolic cut-and-project structures. Journal of the Acoustical Society of America.
• Lawrie, Tanner and Chaplain (2024), “A Non-diffracting Resonant Angular Filter.” Uses quantum-graph theory to model a resonant interface and numerically realizes an acoustic meta-grating. Relevant adjacent mathematical acoustics, not a direct project deliverable. arXiv:2410.17329.
• Gohlke, Mitchell, Rust and Samuel (2024), “Rauzy fractals of random substitutions.” Develops fractals and measures for random substitutions and studies their geometric and dynamical properties. Relevant to Samuel's aperiodic/fractal mathematics, though not to metamaterial design. arXiv:2401.06732.

Lab, Resources, and Collaboration Environment

The supervisor pair spans Mathematics and Physics, and Exeter's CMRI describes relevant acoustic modelling, measurement and fabrication capability. The operational gap is whether this student will use the facilities directly, work with a partner group, or limit the applied phase to initial modelling and testing. Ask who will train and supervise any fabrication or measurement, what equipment is available to this project, and whether CMRI doctoral training applies.

Funding, Application, and Living Cost

The opportunity sits in Exeter's EPSRC DLA for September 2027 entry. Applications close at midday on 8 December 2026 and must use the project-specific route in the official PDF. The central call describes a project-led competition, up to 16 studentships and up to three project applications. It asks for a project-named personal statement, CV, transcripts and two references; English-language evidence may also be required. The general DLA academic baseline is usually a First or Upper Second in an appropriate science, engineering or technology subject. Lower Second applicants may be considered with a Master's degree or significant relevant experience. Residence criteria are separate and can be complex. Award page, central DLA guidance.

For successful eligible awardees, the package is up to 3.5 years full-time, tuition fees and £5,000 total Research Training Support Grant. The displayed stipend is £21,805/year: the detailed terms call it the 2026/27 rate, while the award-page summary labels the same amount as the 2027/28 rate. Treat it as a current published figure and ask for the actual 2027/28 rate and indexation. Exeter Mathematics lists international 2027/28 tuition at £27,600, while the DLA says successful international awardees receive home-rate fee funding plus a University fee waiver for the international difference. International awardees are capped at up to 30% of the cohort and pay their own visa, Immigration Health Surcharge and UK relocation costs. Award page, central DLA guidance, Mathematics programme and fees.

The project award page says references must reach PGR Admissions by 8 January 2027; central DLA guidance says 15 February 2027. Both warn that missing documents may prevent shortlisting. The project-specific page is the safer working deadline until Exeter confirms otherwise. Its interview dates are 15–24 February 2027, while central guidance still gives 2026 interview/outcome timing for the 2027 intake. Confirm the correct timeline with PGR Admissions before relying on the later date.

Exeter estimates basic living expenses at £1,300–£2,000 per month, based on average local spend in April 2026. Dividing £21,805 by 12 gives about £1,817/month: roughly £517 above the lower budget estimate but £183 below the upper estimate, before relocation and unexpected costs. The room finder lists some current postgraduate self-catered rooms at £124.74/week, but availability, bill coverage and 2027/28 prices need checking. GOV.UK's £1,171/month outside-London figure is the visa maintenance requirement for up to nine months, not a realistic spending budget. Exeter budgeting guide, room finder, Student visa money requirement.

Applicant Fit and Preparation

Required Background and Skills

Separate three layers of preparation:

• DLA baseline: normally First or Upper Second in an appropriate science, engineering or technology subject; Lower Second may be considered with a Master's degree or significant relevant experience.
• Mathematics programme entry: a mathematics, statistics or other relevant degree with a high level of mathematics, normally First or strong Upper Second equivalent.
• Project-specific preferences: the project sheet says strong MSc-level understanding of at least three areas—aperiodic order, dynamical systems, hyperbolic geometry, number theory, ergodic theory, measure theory, fractal geometry or stochastic processes—“would be desirable.” Knowledge of acoustic metamaterials and some coding are “highly advantageous.” These are not presented as separate mandatory entry rules.

A persuasive application should show proof-based mathematical maturity, independent research and relevant coding or modelling, while making a credible case for learning the acoustic side. The central trade-off is an abstract, demanding first phase and a second phase that requires willingness to work across mathematical and physical constraints. Applicants primarily seeking hands-on engineering or purely abstract mathematics should ask about the actual balance before committing.

Preferred Experience and Suggested Preparation

Before applying, read the 2026 hyperbolic preprint and the 2023 graded-quasiperiodic study. Prepare one short mathematical question and one physical constraint that would need to be bridged. Map advanced coursework to the project's preferred areas and describe a concrete example of independent problem solving. A coding example can help show readiness for mathematical exploration or simulation. The DLA personal statement should name the project and connect prior research experience and motivation to its two-stage plan; avoid promising a performance improvement the project has not established. Official project description, Prospects research scientist (maths).

Next Actions and Questions to Verify

Before Applying

1. Ask both supervisors about the intended theory/experiment split and the mathematical starting point for year one.
2. Ask PGR Admissions to confirm the governing referee deadline, corrected 2027 interview dates and application document checklist. Until clarified, plan for references by 8 January 2027.
3. Check DLA residence eligibility early, especially if residence or immigration status changed in the last three years. International applicants should budget visa, IHS, relocation and initial housing separately.
4. Request written confirmation of the 2027/28 stipend, full-period fee coverage and the permitted use of the £5,000 RTSG.

At Interview

• What are the first mathematical milestones: a new construction, criteria for aperiodic/Delone properties, diffraction results or operator spectra?
• How will the team select point sets to translate into acoustic designs?
• Does “initiate investigations on building and testing” mean simulation only, a small fabricated sample or repeated experimental characterization?
• Which facility, technician or partner will support fabrication/testing, and is the student expected to join CMRI doctoral training?
• How will the team allocate time between mathematics and applied work, and can the balance change as results develop?

Before Accepting an Offer

• Confirm the degree registration, lead-supervisor location, expected thesis deadline and support if registration extends beyond 3.5-year funding.
• Confirm stipend indexation, tuition and international fee-waiver terms, and RTSG rules in the offer documentation.
• Ask how often the two supervisors meet jointly, what milestones are used, and which group provides day-to-day mathematical and experimental support.

Evidence and Sources

Project and funding

• EPSRC DLA award page, project 6019
• Official project description PDF
• Central EPSRC DLA guidance

Programme, supervisors and research environment

• Exeter Mathematics PhD programme
• Mathematics and Statistics research overview
• Dynamical Systems and Analysis
• Number Theory, Algebra and Geometry
• Tony Samuel, Exeter staff list
• Gregory Chaplain, Exeter staff list
• Tony Samuel research profile
• Gregory Chaplain research profile
• Centre for Metamaterial Research and Innovation
• Acoustic metamaterial expertise
• CMRI doctoral training
• Research degree durations

Publications and field context

• Cut and project schemes in the Poincare disc (2026 preprint)
• Graded quasiperiodic metamaterials perform fractal rainbow trapping (2023)
• Acoustic metasurfaces with Frieze symmetries (2024)
• A Non-diffracting Resonant Angular Filter (2024 preprint)
• Rauzy fractals of random substitutions (2024 preprint)
• UK Metamaterials Commercialisation Roadmap (2026)

Cost and careers

• Exeter budgeting guide
• Exeter accommodation room finder
• Student visa financial requirement
• Research scientist (maths), Prospects
• UK STEM skills pipeline, Parliamentary Office of Science and Technology