Mathematical Physicist (PhD) at Mercor

Mathematical Physicist (PhD) | $80-110/hr | Remote

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, and the company recommends reading it before applying.

Physicists work on research-level problems in their own subfield: creating problems, solving them, reviewing completed work, or auditing it, depending on where their publication record fits. The specific assignment is agreed once matched to an area. This is research-grade work: everything you produce must 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

Applicants are matched narrowly: you must have published on one of these specific phenomena, not in mathematical physics broadly. Each area lists its required methods.

• Special functions: Gaussian hypergeometric functions, parameter differentiation, quadratic transformation identities for hypergeometric series, analytic continuation in a series index, digamma-function series, asymptotic and series expansions.
• Integrable systems: Half-wave maps equation, Lax pair formalism and isospectral flows, Hilbert transform and singular integral operators, Haldane-Shastry spin chain, Calogero-Moser systems, conserved charge hierarchies, solitons of integrable spin-field equations, numerical quadrature of singular integrals.
• Permutation combinatorics for random tensor network entanglement: 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.
• Conformal geometry: Fefferman-Graham ambient metric construction, 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.

You should be able to point to your own papers demonstrating at least one of these method families (special function analysis, integrability, combinatorics, or geometry).

Who we are looking for

• A PhD in mathematical physics, theoretical physics or mathematics. This is a hard requirement. Postdocs, research scientists and junior faculty are the strongest fit; senior PhD students with a strong first-author record are welcome.
• Published work on the specific phenomenon, not the adjacent one. Command of the methods is not enough without publications 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 preferred. Every paper will be checked against the public record.
• Working proficiency with LaTeX, Python, SymPy and Jupyter. Symbolic computation matters more in this panel than most. Familiarity with an agentic coding extension in VS Code is useful; gaps are acceptable if declared honestly.
• English at B2 or above, including written reasoning.

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.

Application process

• Apply and complete the attached form: basic information, education, research experience, method self-attestation, and up to five areas you fit best, each with an arXiv ID or DOI of your own paper, your author position, and the methods it demonstrates.
• Papers and authorship are verified against the public record.
• You are either onboarded directly or invited to a short live alignment call to agree the area and assignment. A brief 30-45 minute assessment may be added where needed, but most applicants will not see one.