I am interested in problems in contact and symplectic topology and geometry, low dimensional smooth topology, and knot theory.

Papers and preprints

1. (2022) Embeddings of Rational Homology Balls and Lens Spaces in Complex Projective Space, joint with John Etnyre, Hyunki Min, Lisa Piccirillo. (in preparation) (slides)

Abstract: Given a symplectic 4 manifold and a contact 3 manifold, it is natural to ask whether the latter embeds in the former as a contact type hypersurface. We explore this question for CP^2 and lens spaces. We will discuss a construction of small symplectic caps, using ideas first laid out by Gay in 2002, for rational homology balls bounded by lens spaces. This allows us to explicitly understand embeddings of these rational balls in CP2 that were earlier understood only through almost toric fibrations. This also gives an interesting application of understanding non-loose knots in overtwisted contact structures.

2. (2022) Constructions and Isotopies of Higher Dimensional Legendrian spheres,, (slides)

Abstract: This article explores the construction of Legendrian spheres in higher dimensional contact manifolds, from stabilizing a supporting open book along Lagrangian disks in the page. Through a careful analysis of the stabilizing procedure, we are able to show that these Legendrians are isotopic to the standard Legendrian unknot. We show that this recovers and generalises a result of Courte and Ekholm for Legendrians constructed by a doubling procedure in the standard contact Euclidean space.

3. (2021) Symplectic Fillings and Cobordisms of Lens Spaces, joint with John Etnyre, Transactions of the AMS. arxiv link

Abstract: We complete the classification of symplectic fillings of tight contact structures on lens spaces. In particular, we show that any symplectic filling X of a virtually overtwisted contact structure on L(p,q) has another symplectic structure that fills the universally tight contact structure on L(p,q). Moreover, we show that the Stein filling of L(p,q) with maximal second homology is given by the plumbing of disk bundles. We also consider the question of constructing symplectic cobordisms between lens spaces and report some partial results.

My talks

  1. Talks on Symplectic Fillings and Cobordisms of lens spaces at NCNGT 2021

Part 2 of the talk available here.

2. A talk at the Symplectic Zoominar hosted between CRM-Montreal, IAS-Princeton, Tel Aviv, and Paris, on Constructions and Isotopies of High Dimensional Legendrian Spheres:


This quote I found recently resonated with how I think of math research and how I have managed to do research till now. This is from the book “Catching the Big Fish: Meditation, Consciousness, and Creativity” by David Lynch, which is now on my to-read list thanks to how much I liked this quote.

An idea is a thought. It’s a thought that holds more than you think it does when you receive it. But in that first moment there is a spark. In a comic strip, if someone gets an idea, a lightbulb goes on. It happens in an instant, just as in life.
It would be great if the entire film came all at once. But it comes, for me, in fragments. That first fragment is like the Rosetta stone. It’s the piece of the puzzle that indicates the rest. It’s a hopeful puzzle piece.

You fall in love with the first idea, that little tiny piece. And once you’ve got it, the rest will come in time.

(David Lynch)

Mathematics graduate student at Georgia Tech