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Austin Christian

Georgia Tech Mathematics

REU 2022: Legendrian contact homology

Mathematicians Edo Biluar and Sarah Brown standing in front of their poster 'Legendrian connected sum'
Our REU group standing on steps in the Skiles courtyard
Mathematicians Bitong (Shelly) Cao, Michael Edwards, and Manyi Guo standing in front of their poster 'Product structure as an invariant of Legendrian knots'
Mentors: Austin Christian and Agniva Roy
Participants: Edo Biluar, Sarah Brown, Shelly Cao, Michael Edwards, and Manyi Guo

The goal of this REU is was to study a modern algebraic invariant of Legendrian knots. A Legendrian knot is a knot which has a certain compatibility with the contact structure on \(\mathbb{R}^3\). Natural questions to ask about Legendrian knots include:

Some keywords for the above questions include Legendrian isotopy, Lagrangian filling, and Lagrangian concordance, but we'll learn those in due time. Our primary tool for addressing these questions will be Legendrian contact homology, which associates to each Legendrian knot a differential graded algebra. This algebra can be very complicated, but, even without a full understanding of the algebra, we can make interesting conclusions about Legendrian knots.

The scariest words in the above description will be defined in the early stages of the REU --- you're not expected to have any familiarity with contact geometry. An undergraduate course in modern algebra (groups, rings, and such) and a recollection of 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

Background (which we'll cover in the REU)

Survey papers

Research papers


Here are some notes written by Agniva and me for our background sections.

And here are the exercises for our problem sessions: PDF

Final reports

Product structure on pairs in Legendrian knot atlas, Shelly Cao, Michael Edwards, Manyi Guo

Legendrian connected sums, Edo Biluar, Sarah Brown

More information about the Georgia Tech REU