I am interested in extensions of the Langlands program, both in the “relative” and “metaplectic” directions. Currently, I am thinking about questions on relative trace formulas and related notions of endoscopy. I also think about generalized Fourier transforms and Poisson summation formulas in the context of the Braverman-Kazhdan-Ngo-Sakellaridis conjectures with Jayce Getz, Chun-Hsien Hsu, and Pam Gu.

In Progress

Unitary Friedberg–Jacquet periods and the arithmetic fundamental lemma (joint w/ Jingwei Xiao and Wei Zhang)

The stable trace formula for SL(2)-periods (joint w/ Siddharth Mahendraker)

Preprints/Publications

• On triple product L-functions and the fiber bundle method, (joint w/ Jayce Getz, Pam Gu, and Chun-Hsien Hsu)
  • submitted, arXiv version arxiv:2503.21648 (2025).
• Unitary Friedberg-Jacquet periods and their twists: Relative trace formulas, (joint w/ Jingwei Xiao and Wei Zhang)
  • submitted, arXiv version arxiv:2503.09664 (2025).
• Unitary Friedberg-Jacquet periods and their twists: Fundamental lemmas, (joint w/ Jingwei Xiao and Wei Zhang)
  • submitted, arXiv version arxiv:2503.09500 (2025).
• Symmetric varieties for endoscopic groups,
  • submitted, arXiv version arxiv:2401.09156 (2024).
• Modular forms of half-integral weight on exceptional groups (w/ Aaron Pollack),
  • Compositio Mathematica, published version,
  • arXiv version arxiv:2205.15391 (2024).
• Harmonic analysis on certain spherical varieties (w/ Jayce Getz and Chun-Hsien Hsu),
  • J. Eur. Math. Soc., published version,
  • arXiv version arxiv:2103.10261 (2023).
• On the stabilization of relative trace formulae: descent and the fundamental lemma,
  • Advances in Mathematics (2021), published version,
  • arXiv version arxiv:2006.04993.
• The endoscopic fundamental lemma for unitary Friedberg-Jacquet periods,
  • Annals of Mathematics, published version,
  • arXiv version arXiv:1911.07907 (2024).
• A fundamental lemma for the spherical Hecke algebra: the Jacquet-Rallis case,
  • Journal of Number Theory, published version (2022),
  • preprint version
• Endoscopy for unitary symmetric spaces,
  • Results are superseded by “The endoscopic fundamental lemma for unitary FJ periods” above,
  • introduced the notions of relative endoscopy for unitary Friedberg-Jacquet periods, and contains an independent computational proof of the fundamental lemma in rank 2.
  • preprint available at arxiv:1910.09685 (2019).
• Parity Sheaves and Smith Theory (w/ Gus Lonergan),
  • J. für die Reine und Angew. Math. (2021), published version,
  • arXiv version arXiv:1708.08174.
    • Erratum: Remark 3.5.4 is not relevant, as the self-Ext-class that exists over modular coefficients does not lift to a self-extension over the Witt vectors (a sign character intervenes). In particular, the case of p=2 works just as the case of p odd for the entire paper. I want to thank Tony Feng for pointing this out to me.
• An analogue of the Grothendieck–Springer resolution for symmetric spaces,
  • Algebra Number Theory (2021), published version,
  • arXiv version arxiv:1904.09217.
• Resonant Mirkovic-‏-Vilonen Polytopes and formulas for highest-weight characters,
  • Selecta Mathematica New Ser. (2019) published version,
  • arXiv version arXiv:1808.10508.
• A Generalized Theta Lifting, CAP Representations, and Arthur Parameters,
  • Trans. Amer. Math. Soc. 372 (2019), published version,
  • arXiv version arXiv:1703.02597.

Doctoral Thesis

Theta Liftings on Higher Covers of Symplectic Groups, Ph.D. thesis (Boston College), available here (2018), contents contained in “A generalized theta lifting…” above with some additional calculations on the first occurrence of the lift.

Undergrad research that made me want to go to math grad school

Laplace equations for real semi-simple associate algebras of dimension 2, 3, or 4 (with James S. Cook, Minh L. Nguyen, and Bailu Zhang), published version, Springer conference proceedings (2013)