John Layne

About

I am a master's student at the Einstein Institute of Mathematics advised by Tamar Ziegler. I recently graduated from the University of Virginia, where my undergraduate advisor was Peter Humphries. My interests lie in analytic number theory, additive combinatorics, and ergodic theory.

I wrote my undergraduate thesis on the ergodic proof of Szemerédi's Theorem, the Green–Tao Theorem, and gave a survey/introduction to some of the results surrounding convergence of nonconventional ergodic averages, the generalized Hardy–Littlewood Conjecture, higher-order Fourier analysis, and Sarnak's Möbius disjointness Conjecture. You can find my thesis alongside some of my other projects below.

Outside of math, I'm an avid rock climber and enjoy hiking. I also enjoy explaining ergodic theory in terms of stirring a bowl of oatmeal to anyone who will listen.

My email is bhj3pr@virginia.edu.

Publications

Layne, J., Marshall, S., Sadowski, C., Shambaugh, E. Ghost series and a motivated proof of the Bressoud–Göllnitz–Gordon identities. The Ramanujan Journal 65, (2024), 593–635.

Research Projects

Shifted convolutions of generalized divisor functions (Summer 2026, in progress)
Group skew-products and strong approximation in the special linear group (Einstein Institute of Mathematics REU Summer 2025, poster)
An upper bound on newform Dedekind sums (Texas A&M Math REU Summer 2024, poster/slides)
Ghost series and a motivated proof of the Bressoud-Göllnitz-Gordon identities (Ursinus College Math REU Summer 2023, poster/slides)

Expository projects

(In progress) The Kuznetsov trace formula and the binary additive divisor problem (Summer 2026, paper)
(In progress) Four proofs of Roth's Theorem (Summer 2026, paper)
(244 pages) Linear patterns in subsets of the integers via ergodic theory, hypergraph methods, and higher-order Fourier analysis (2026, undergraduate thesis, slides)
(6 pages) Existence of bipartite Ramanujan graphs of every size (Spring 2026, paper)
(7 pages) Dirichlet's theorem and Chebotarev's density theorem (Spring 2026, paper)
Pseudorandomness (Spring 2026, slides)
(9 pages) Ergodic methods for Duke's theorem on the equidistribution of closed geodesics (Fall 2025, paper)
The ergodic proof of Szemerédi's theorem & the Green–Tao theorem (Fall 2025, poster)
(6 pages) Ergodic theory: An introduction with applications to Szemerédi's theorem & group skew products (Fall 2025, paper)
(14 pages) Introduction to abstract harmonic analysis and Tate's thesis over Q (Spring 2025, paper)
(33 pages, in progress) Prime number theorem with error term (Summer 2024, paper)
(11 pages) Burnside's p^aq^b theorem (Spring 2024, paper)
Dirichlet's theorem for arithmetic progressions and the prime number theorem (Fall 2023, slides)
(38 pages) Results in analytic number theory (Fall 2023, paper)
(4 pages) Naturality of the double dual evaluation map (Fall 2022, paper/slides)

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