Research

I work in low-dimensional topology, which has to do with the properties of shapes we can (nearly) visualize.  My work draws on techniques from combinatorics, which has to do with the properties of discrete structures, and from symplectic geometry, which is the mathematical formalism for describing classical mechanics.

My papers on the arXiv.

I wrote a survey article on Heegaard Floer homology for the Notices of the American Mathematical Society (January 2021).  It was awarded the 2023 Levi L. Conant Prize of the AMS.

A popular article in Quanta magazine describes my joint work with Andrew Lobb on the rectangular peg problem (June 2020).

I am an editor at Journal of Topology and Studia Scientiarum Mathematicarum Hungarica: Combinatorics, Geometry and Topology.

Slides (and some videos) from talks

The square pig problem. Yellow Pig Math Days Plenary Lecture, Hampshire College. July 15, 7*17*17.

The rectangular peg problem. Princeton University Department Colloquium. September 23, 2020.  Video link.

Loops intersecting at most once. LA Joint Topology Seminar, Caltech. March 9, 2020.

Floer homology and Dehn surgery. 30 years of Floer homology for 3-manifolds, Casa Matemática Oaxaca.  August 1, 2017.  Video link.

Perspectives on Dehn surgery. Texas Geometry / Topology Conference, University of Texas, Austin. November 17, 2017. 

Trees and Numbers. Navajo Nations Teacher Workshop. March 6, 2021 (not 202, as misdated on the actual slides...).

Courses

I maintain the pages for current and past graduate courses I have taught.

Fall 2024: Topology of Lagrangian submanifolds.

Spring 2022: Knot homology theories.

Fall 2021: Algebraic topology.

Spring 2021: Symplectic geometry and pseudoholomorphic curves.

Spring 2019: Combinatorial methods in low-dimensional topology.

Spring 2018: Graduate combinatorics.

Fall 2015: Introduction to knot theory.

Spring 2013: Combinatorial methods in knot theory.

Seminars

The Boston Graduate Topology Seminar 

BC Geometry/Topology Seminar 

This google calendar reports what is happening in topology seminars across the Boston area.

Summer School and Book

I organized a graduate summer school entitled  "Perspectives on Dehn Surgery'' at ICERM during the dates of July 15-19, 2019.  The link will take you to video and other materials from the event, and a book based on the lecture series is in the works.