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YOUR INSTITUTION · YOUR DEPARTMENT

Dr. Your Name

Postdoctoral Research Fellow in Spectral Geometry

I study the relationship between the geometry of Riemannian manifolds and the spectral properties of their associated Laplace–Beltrami operators — asking when, and how precisely, you can hear the shape of a drum.

\(\lambda_k(M) \sim C_n \!\left(\frac{k}{\mathrm{vol}(M)}\right)^{\!2/n}\)   (Weyl's law)
My Research Publications Contact

About

Background & Interests

I am a [position] in the [Department] at [Institution]. My research lies in spectral geometry — the study of how the spectrum of the Laplace–Beltrami operator encodes and reflects the geometry of Riemannian manifolds. I am particularly interested in [specific sub-topic, e.g. isospectrality, heat invariants, spectral gaps on surfaces].

Before joining [current institution], I was at [previous institution], where I worked with [supervisor name] on [brief project description]. I received my PhD from [institution] in [year], where my thesis investigated [thesis topic].

My work draws on tools from [e.g. differential geometry, functional analysis, microlocal analysis, and analytic number theory]. I am interested in both the theoretical foundations and in concrete computations — including numerical experiments that can guide intuition and conjecture.

Spectral Geometry Laplace–Beltrami Operator Heat Kernel Methods Riemannian Manifolds Isospectrality Inverse Problems Weyl Asymptotics Spectral Gaps

Research

Projects & Themes

My research centres on the spectral theory of Riemannian manifolds and related geometries. Below are the main threads I am currently pursuing.

Isospectrality & Inverse Problems

Can you hear the shape of a drum? I investigate families of isospectral manifolds — distinct geometries sharing the same Laplace spectrum — and what invariants beyond the spectrum are needed to determine shape uniquely.

\(\mathrm{Spec}(M) = \mathrm{Spec}(M') \not\Rightarrow M \cong M'\)
Isospectrality Inverse Problems

Heat Kernel Asymptotics

The short-time asymptotic expansion of the heat kernel encodes curvature invariants of the manifold. I study higher-order heat invariants and their geometric content, particularly for manifolds with boundary or cone singularities.

\(\mathrm{Tr}\,e^{-t\Delta} \sim (4\pi t)^{-n/2}\sum_{k=0}^{\infty} a_k \, t^k\)
Heat Kernel Curvature Invariants

Spectral Gaps & Cheeger Constants

The spectral gap — the first nonzero eigenvalue \(\lambda_1\) — controls mixing rates and isoperimetric properties. I study sharp lower bounds via Cheeger-type inequalities and their optimisers on surfaces and higher-dimensional manifolds.

\(\lambda_1(M) \geq \tfrac{1}{4} h(M)^2\)
Spectral Gap Cheeger Inequality Optimisation

Publications

Papers & Preprints

For a complete list see my Google Scholar or arXiv page.

2025
2024

Curriculum Vitae

Education & Positions

Download the full CV: Download PDF

Education

  1. 2024

    PhD in Mathematics

    [Your University]

    Thesis: "[Your thesis title]"
    Supervisor: Prof. [Name]

  2. 2020

    MMath / Part III Mathematics

    University of Cambridge

    Distinction. Essay: [Essay title]

  3. 2019

    BSc (Hons) Mathematics

    [Your University]

    First Class Honours

Positions & Awards

  1. 2025

    Postdoctoral Research Fellow

    [Your Institution]

    Funded by [Funding body]

  2. 2024

    [Prize / Fellowship Name]

    [Awarding body]

    Brief description of the award

  3. 2023

    Visiting Researcher

    [Institution], [Country]

    Collaboration with [Name] on [topic]

Contact

Get in Touch

I am happy to hear from prospective students, collaborators, and anyone curious about spectral geometry. Please reach out by email — I aim to respond within a few days.

Google Scholar Your Name
ORCID YOUR ORCID
arXiv arXiv author page

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