Thomas Rüd

Research

My work explores the structure of p-adic groups and their buildings, with a particular emphasis on relative trace formulas, orbital integrals on p-adic symmetric spaces, and local–global principles in the Langlands framework. I am especially interested in refinements of the relative Langlands program and ideas inspired by Beyond Endoscopy.

Current themes

• Comparison of Orbital integrals
• Transfer of automorphic representations for spherical pairs
• Beyond Endoscopy and detection of functorial transfers
• Local–global principles and relations to cohomology of tori
• New interpolation tensor categories.

Ongoing projects

• Explicit constructions of transfer for the Arithmetic Transfer Conjecture.
• Orbital integrals on classical groups as point counts and explicit formulas.
• Oligomorphic groups and constructions of tensor categories.
• Computation of orbital integrals in symplectic groups and low-regular Shalika germs.

Awards & Fellowships

• 2025
  SPOT Award
  Prize  Awarded by MIT math department in recognition for outstanding achievements.
• 2025
  Lusztig Mentorship Award
  Prize  Awarded by MIT math department in recognition of my mentorship in the PRIMES USA Program.
• 2024
  Charles and Molly Houseman Award
  Prize  Awarded by MIT math department in recognition for excellence in undergraduate teaching.
• 2022–2024
  Postdoctoral Fellowship
  Fellowship   Funded by the Swiss National Science Foundation.
• 2020
  Graduate Research Award
  Prize  Awarded by UBC math department in recognition for thesis and research work.

Preprints

• 2025
  Jacquet—Rallis transfer for ramified quadratic extensions
  Preprint  with W. Zhang, Preprint. In late stage of preparation, pdf already available on demand.
  Arithmetic Transfer Relative Langlands Program Unitary groups
• 2025
  Orbital integrals for linear groups as local densities
  Preprint  with M.Middlezong and L.Qi, Preprint. Link
  Orbital Integrals Local Densities
• 2025
  Oligomorphic groups: Measures on colored trees and cyclic orders
  Preprint  with T. Can, Preprint. In late stage of preparation, pdf already available on demand.
  Oligomorphic Groups Fraïssé classes Measures Tensor categories
• 2025
  A comparison problem for abelian surfaces and descent for symplectic orbital integrals
  Preprint  Preprint, 82p. arXiv
  Orbital Integrals Abelian Varieties Elliptic curves
• 2024
  A Projective Twist on the Hasse Norm Theorem
  Submitted  with A. Bu, Submitted, 25 p. arXiv
  Local–Global Principle Number Theory

Selected Publications

• 2023
  Appendix of: Counting abelian varieties over finite fields via Frobenius densities by by J. Achter, S. Altug, L. Garcia, and J. Gordon.
  Published  Algebra and Number Theory, (2023)
  Galois Cohomology Algebraic Tori Local-Global Principle
• 2021
  Explicit Tamagawa numbers for certain algebraic tori over number fields
  Published  Mathematics of Computation, 143 (2022)
  Galois Cohomology Algebraic Tori Local-Global Principle
• 2019
  On uniform admissibility of unitary and smooth representations
  Published  with U. First. Archiv der Mathematik (2019)
  Representation Theory

A complete list of publications is available on my curriculum vitae and on arXiv.

Teaching & Mentoring

I enjoy teaching algebra, number theory, and representation theory but I also learn sharing my excitment in all aspects of mathematics including calculus and geometry.

Upcoming Course

• 18.704 Seminar in algebra @MIT
   (CIM course) — Seminar on fourier analysis on groups and hyperbolic surfaces.

Current course

• 18.02 Multivariable calculus (2025) @MIT

Supervision

• Eunsu Hur 2026 @MIT DRP
    Directed Reading program, topic TBA
• Michael Middlezong and Lucas Qi 2025 @MIT PRIMES
    Year long research program as part of MIT PRIMES USA, we worked on explicit computations of orbital integrals.
• Thanh Can 2025 @MIT PRIMES
    Year long research program as part of MIT PRIMES USA, we worked on classification of measures on Oligomorphic groups.
• Luis Turino and Cedric Xiao 2024 @MIT DRP
    Directed Reading program, we read structure of elliptic curves over finite fields and applications to cryptography.
• George Shaker 2023 @MIT DRP
    Directed Reading program, we read on Fourier analysis and the spectrum of the Laplacian for hyperbolic surfaces.
• Alan Bu 2023 @MIT PRIMES
    Year long research program as part of MIT PRIMES USA, we worked on new versions of local-global norm principles.

For stu

Previous courses

• 18.01 Single variable calculus (2025) @MIT
• 18.01A/02A Accelerated course in single and multi variable calculus (2024) @MIT
• 18.781 Theory of Numbers (2024) @MIT
• Lead instructor for Petey Greene's prison education program. Framingham medium security prison (2023)
• 18.01A/02A Accelerated course in single and multi variable calculus (2022) @MIT
• MATH 104: Differential calculus (2021) @UBC
• MATH 100 Vantage: Differential calculus (2021) @UBC
• MATH 184: Differential calculu (2019) @UBC
• MATH 104: Differential calculus (2018) @UBC

For students

If you are interested in a reading course or research project related to these topics, feel free to email me with a brief description of your background and mathematical interests.

Expository & Short Writing

I enjoy drawing on Tikz (as can be seen in the background images) and occasionally wrote notes as part of projects, of potential interest.

• Compilation of Building drawings
  Feel free to reach out for the LaTeX code if you wish to modify them. PDF
• Short slides outlining my research directions as of 2024
  PDF
• Slides covering my thesis work
  Colloquium slides. PDF
• Introduction to admissibility of representations of p-adic groups.
  Master Thesis. PDF

Recent & Upcoming Talks

• 2025
  Explicit construction of arithmetic transfer in the ramified unitary setting
  Number Theory Seminar, University of Maryland, Park City. Seminar
• 2025
  Relative Langlands program and Functoriality
  Conference: Celebrating 100 years of the mathematics building at UBC. Conference
• 2023
  Buildings, orbital integrals, and Harmonic Analysis
  Workshop on Beyond Endoscopy, Fields Institute Workshop

Curriculum Vitae

A CV containing details about my positions, publications, teaching, and service is available as a PDF.

Last updated: November 2025.

Miscellaneous

Outside of mathematics, some of my time goes into movement, music, and things that still have a bit of structure and rhythm.

Photos

‹

›