Advising and undergraduate research
PhD advising
Yikai Teng, Rutgers University-Newark (2023 - present)
Undergraduate research
During the past few years, I have had the opportunity to advise undergraduate research and independent studies in several ways, including two mentees from Columbia's Science Research Fellow program, a large REU project, and a supervised reading course.
Dehn surgery versus branched covers
Throughout 2020-2021, I have been working with Erica Choi (Columbia University '23) on a project that compares two fundamental constructions of 3-manifolds based on knots: Dehn surgery and branched covers. In particular, we focused on the question "When is the double branched cover of S³ along a knot K also obtained by Dehn surgery on S³ along the same knot K?" We show that the answer is "almost never", ruling this phenomena out for large classes of knots, including work in progress addressing all knots with ≤15 crossings. However, we show that counterexamples exist, and these counterexamples exhibit many special properties.
Erica and I are writing up our work to be submitted for publication. I also presented on our work at the AMS Spring Southeastern Sectional meeting in March 2021.
Cobordism maps on Khovanov homology
In Summer 2021, I advised Alan Du (Columbia University '22) on a project concerning cobordism maps on Khovanov homology. Alan began by writing software to compute these maps in the special case of "braided cobordisms". Next, he conducted an in-depth analysis of the maps associated to two slice disks bounded by the stevedore knot and the slice disks obtained by mirroring, and investigated the relationship between these maps and an involution on the stevedore knot.
I am currently supervising Alan's senior thesis during the 2021-2022 academic year. Alan is writing up his work from this past summer and extending it to study other interesting examples of surfaces bounded by knots. Alan will be submitting this work for publication.
Knots with the same integral Dehn surgery
In Summer 2019, Ali Daemi and I ran an REU project with seven undergraduates: Gabriel Agostini, Sophia Chen, Christian Serio, Lizka Vaintrob, Cecilia Wang, Anton Wu, and Kexin Wu.
We investigated when a knot shares one or more integral Dehn surgeries with other knots. Here's a video of their final presentation.
Knot theory and contact geometry
In Spring 2019, I supervised a reading course in knot theory and contact geometry for Yi Wang (Columbia '19). The reading course included regular meetings and drew on sources including Geiges' introductory textbook and Etnyre's notes on open book decompositions.
Yi is now a PhD student at the University of Pennsylvania. And he's proven cool things about contact geometry! Check out this recent paper that Yi coauthored.