Mathematical Physicist (PhD)

About the work

CritPt is a public benchmark of research-level physics challenges, built to test whether frontier AI models can carry out genuine physics research reasoning rather than textbook problem solving. The benchmark paper is [arXiv:2509.26574](https://arxiv.org/abs/2509.26574) and we recommend reading it before applying. It will tell you quickly whether this work interests you.

We are engaging physicists to work on research-level physics problems in their own subfield. Depending on where your publication record fits, that can mean creating problems, solving them, reviewing completed work, or auditing it. We agree the specific assignment with you once you are matched to an area.

This is research-grade work rather than volume work. Whatever you produce has to be complete enough for another specialist in your subfield to follow and verify independently, so written reasoning is part of every assignment. In this panel the standard of proof sits closer to mathematics than to physics.

Research areas in this panel

Four areas. We match narrowly: you need to have published on one of these specific phenomena, not in mathematical physics broadly. Each area lists the methods it requires.

1\. Special functions: hypergeometric identities and analytic continuation: Gaussian hypergeometric functions, parameter differentiation of special functions, quadratic transformation identities for hypergeometric series, analytic continuation in a series index, digamma-function series, asymptotic and series expansions of special functions.

2\. Integrable systems: half-wave maps, Lax pairs, Haldane-Shastry and Calogero-Moser: Half-wave maps equation, Lax pair formalism and isospectral flows, Hilbert transform and singular integral operators, classical Haldane-Shastry spin chain, Calogero-Moser systems, hierarchies of conserved charges, solitons of integrable spin-field equations, numerical quadrature of singular integrals.

3\. Permutation combinatorics for random tensor network entanglement (Cayley distance, Weingarten): Cayley distance and geodesics in the symmetric group, minimal factorizations of permutations, non-crossing partitions, replica permutation spin models, random tensor network entanglement entropy, Weingarten calculus, multipartite entanglement measures, combinatorial enumeration of configurations.

4\. Conformal geometry: Fefferman-Graham ambient metric, obstruction tensors: Fefferman-Graham ambient metric construction, conformal geometry and Weyl covariance, extended obstruction tensors, Schouten and Weyl curvature tensors, order-by-order solution of Ricci-flatness conditions, poles and residues under dimensional continuation, Poincare-Einstein asymptotic expansions.

Methods we expect to find in your own publications

You should be able to point to your own papers demonstrating at least one of the following families:

• Special function analysis: Gaussian hypergeometric functions, parameter differentiation, quadratic transformation identities, analytic continuation in a series index, asymptotic expansions

• Integrability: Lax pair formalism, isospectral flows, Hilbert transform and singular integral operators, hierarchies of conserved charges, numerical quadrature of singular integrals

• Combinatorial: Cayley distance and geodesics in the symmetric group, minimal factorizations of permutations, non-crossing partitions, Weingarten calculus, multipartite entanglement measures

• Geometric: Fefferman-Graham ambient metric construction, Weyl covariance, Schouten and Weyl curvature tensors, order-by-order solution of Ricci-flatness conditions, Poincare-Einstein asymptotic expansions

Who we are looking for

A PhD in mathematical physics, theoretical physics or mathematics. This is a hard requirement. Postdoctoral researchers, research scientists and junior faculty are the strongest fit. Senior PhD students with a strong first-author record are welcome to apply.

Published work on the specific phenomenon above, not the adjacent one. This is the single most common reason we decline otherwise excellent physicists. Command of the methods is not enough if you have not published on the phenomenon itself.

A verifiable publication record. Three to five representative papers with arXiv IDs or DOIs, ideally from the last five years. First author strongly preferred. Every paper you list will be checked against the public record.

Working proficiency with LaTeX, Python, SymPy and Jupyter. Symbolic computation matters more in this panel than in most. Some familiarity with an agentic coding extension in VS Code is useful. Gaps there are acceptable if you declare them honestly.

English at B2 or above, including written reasoning. A large part of the value you add is how clearly you set out your argument.

Application steps

• Apply and complete the attached form. Basic information, education, research experience, your method self-attestation, and up to five of the areas above that you are the best fit for. For each area you select, give an arXiv ID or DOI of your own paper as proof, with your author position and the methods it demonstrates. A selection without proof is not scored.

• We verify your papers and authorship against the public record.

• Then one of two things happens. Either we onboard you directly, or we invite you to a short live alignment call to agree the area and the assignment with you.

• A brief 30 to 45 minute assessment may be added, but only where we need it. Most applicants will not see one.

Commitment and rate

10 hours per week, sustained across an 8 to 10 week window, starting immediately. Remote and asynchronous with no fixed hours.

$80 to $110 per hour, set by depth of subdomain match.