Home Research Seminars Teaching

Austin Christian

Georgia Tech Mathematics

REU 2023: Persistent Legendrian contact homology

Mathematicians Maya Basu, Ethan Clayton, and Fredrick Mooers standing in front of their poster 'Persistent Legendrian Contact Homology'

Mentors: Daniel Irvine and Austin Christian
TA: Weizhe Shen
Participants: Maya Basu, Ethan Clayton, and Fredrick Mooers

In this REU, we will attempt to apply some techniques from topological data analysis to a problem in contact topology. Namely, we will use the action filtration on Legendrian knots in the standard contact manifold \((\mathbb{R}^3,\xi_0)\) to compute a persistent analogue of (linearized) Legendrian contact homology.

A Legendrian knot is a knot which has a certain compatibility with the standard contact structure on \(\mathbb{R}^3\), and Legendrian contact homology is a powerful modern invariant for distinguishing Legendrian knots up to isotopy (among other applications). Persistent homology, on the other hand, is a tool of topological data analysis which allows for the study of a given space at various resolutions. The underlying space which gives rise to Legendrian contact homology admits a natural filtration, and this filtration provides the various "resolutions" at which we will study our problem.

The early stages of the REU will be used to develop the requisite background --- you're not expected to have any familiarity with contact geometry or persistence modules. A very strong understanding of undergraduate modern algebra (groups, rings, fields, and such) and multivariable calculus should provide adequate academic background. The most important expectation of students is that they are interested in challenging themselves with an exciting new topic!

Reading list

Survey papers

Research papers


Here are some notes written by Daniel and Austin for our background sections.

And here are the exercises for our problem sessions: PDF.

More information about the Georgia Tech REU