Associate Professor, Department of Mathematics, Ohio State University

Email: lipnowski "dot" 1 "at" osu "dot" edu

Office: Math Tower 412 

I am currently an Associate Professor in the Ohio State University Department of Mathematics.  My research interests include number theory, representation theory, and automorphic forms.

Here is my CV.

Papers

Completed

• Dirac spectral flow and Floer theory of hyperbolic three-manifolds, with Francesco Lin.  Submitted.

• Cohen-Lenstra heuristics and bilinear pairings in the presence of roots of unity, with Will Sawin and Jacob Tsimerman.  Submitted.

• Rigid meromorphic cocycles for orthogonal groups, with Henri Darmon and Lennart Gehrmann.  Forum of Math Sigma, accepted for publication.

• Closed geodesics and Froyshov invariants of hyperbolic three-manifolds, with Francesco Lin.  Journal of the EMS, accepted for publication.

• Towards optimal spectral gaps in large genus, with Alex Wright.  Annals of Probability (2024).

• The Seiberg-Witten equations and the length spectrum of hyperbolic 3-manifolds, with Francesco Lin.  Journal of the AMS (2022).

• Monopole Floer homology, eigenform multiplicities, and the Seifert-Weber dodecahedral space, with Francesco Lin.  IMRN (2022).

• How large is A_g(F_q)?, with Jacob Tsimerman. Duke Math Journal (2018).

• Geometry of the smallest 1-form eigenvalue on hyperbolic manifolds, with Mark Stern.  GAFA (2018).

• Cohen Lenstra Heuristics for Étale Group Schemes and Symplectic Pairings, with Jacob Tsimerman.  Compositio Mathematica (2019). 

• Detecting large simple rational Hecke modules for Γ_0(N) via congruences, with George Schaeffer.  IMRN (2020).

• On Bhargava's heuristics for GL_2(F_p)-number fields and the number of elliptic curves of bounded conductor. Experimental Mathematics (2021).

• Twisted limit formula for torsion and cyclic base change, with Nicolas Bergeron. Journal de'l École polytechnique - Mathématiques (2017).

• Equivariant torsion and base change. Algebra and Number Theory (2015).

• The equivariant Cheeger-Muller theorem on locally symmetric spaces. JIMJ (2016).

In preparation

• Grid building algorithms with applications to reduction theory and the topology of arithmetic groups, with Aurel Page. Slides 1, Slides 2, Summary.

• Polarization counting for abelian varieties over finite fields, with Jacob Tsimerman. Summary.

• Growth of torsion for weight 1 modular forms, with Gerard Freixas i Montplet and George Schaeffer. Summary.

Computer code

• Bilinear pairings on (relative) class groups in the presence of roots of unity.  Used to compute class group data as part of this paper.  Class group data here.  Full pairing data for non-trivial class groups here.

• Approximating low-lying eigenvalues for 0-forms, 1-forms, and spinors on hyperbolic 3-manifolds via the trace formula.  Used to compute eigenvalue data for this paper, this paper, and this paper.

Expository Writings

Below are informal notes written to accompany lectures that I presented for various installments of Stanford's number theory learning seminar.

• Taylor-Wiles primes

• p-divisible groups

• Faltings' finiteness theorems

• Baily-Borel compactifications