Surveys on my work

Sous-variétés totalement géodésiques des espaces de modules de Riemann, by Elise Goujard.

PhD Students


Bradley Zykoski, Michigan

Sayantan Khan, Michigan

Chris Zhang, Michigan


Ben Dozier, co-advised with Maryam Mirzakhani, Stanford, graduated Spring 2018

Francisco Arana-Herrera, co-advised with Steven Kerckhoff, Stanford, graduated Spring 2021



Chaya Norton

Paul Apisa

Research Experience for Undergraduates

In summer 2020 I co-organised a REU with three teams, each consisting of an undergraduate student, a graduate student, and a more senior mentor. Each team produced a paper:

Periodic points on the regular and double n-gon surfaces
By Rafael Saavedra, Chris Zhang, and Paul Apisa

Strongly Obtuse Rational Lattice Triangles
By Anne Larsen, Bradley Zykoski, and Chaya Norton
Transactions of the American Mathematical Society, 2021

Pseudo-Anosov homeomorphisms of punctured non-orientable surfaces with small stretch factor
By Caleb Partin, Sayantan Khan, and Becca Winarski
Algebraic and Geometric Topology, to appear

Here the authors are listed in increasing order of seniority (undergrad, grad, mentor).


Retrospective on Maryam Mirzakhani that I wrote for Science: preprint, published version.

Quanta Magazine article on my joint work: New Shapes Solve Infinite Pool-Table Problem. Accompanying blog post: Why Mathematicians Like to Classify Things.

Not for Publication

Course notes for Math 797, on coarse geometry and Teichmuller theory, including hierarchical hyperbolicity (with Chenakkod, Haviland, Kausik, Khan, Shchetka, Yu)

Notes on Moduli of spatial polygons

Notes on Cubic curves

Course notes on Moduli spaces of Riemann surfaces

Exercises to accompany my survey "Translation surfaces and their orbit closures"

Illumination and Security (with Kathryn Mann)

Luck's Theorem on growth of Betti numbers

Notes on Filip's proof that orbit closures are algebraic

A brief summary of Otal's proof of marked length spectrum rigidity

Deligne's Theorem on the semisimplicity of variations of Hodge structures

Smillie's Theorem on closed SL_2(R) orbits of quadratic differentials (with Ilya Gekhtman)

Topic Proposal: Translation surfaces and Teichmüller theory

Old Material

Undergraduate Projects

Chicago Informal Dynamics Seminar, 2013-2014

Reading group on the Cremona group, summer 2016