Research

I am interested in symplectic geometry, gauge theory, Floer homology theories, and their applications to problems in low-dimensional topology.

Publications and Preprints:

1. Seiberg-Witten knot homology via holonomy filtrationGiven a null-homologous knot inside a compact oriented 3-manifold, we define a filtered chain complex via the Seiberg-Witten equations akin to the knot Floer complex of Ozsváth and Szabó. (in preparation)

2. (with Gordana Matic, Jeremy Van Horn-Morris, and Andy Wand) Detecting tightness via spectral order.We present a method to detect non-zero spectral order, hence tightness, of a contact structure and apply it to detect tightness of families of contact structures with vanishing Heegaard Floer contact invariant. (in preparation)

3. (with Gordana Matic, Jeremy Van Horn-Morris, and Andy Wand) Filtering the Heegaard Floer contact invariant.We define an invariant of contact structures in dimension three from Heegaard Floer homology. This invariant takes values in the set $\mathbb{Z}_{\geq 0}\cup\{\infty\}$. It is zero for overtwisted contact structures, $\infty$ for Stein fillable contact structures, non-decreasing under Legendrian surgery, and computable from any supporting open book decomposition. As an application, we obstruct Stein fillability on contact 3-manifolds with non-vanishing Ozsv\'ath-Szab\'o contact class. (arXiv)

4. (with Jonathan Hanselman and Tye Lidman) A remark on the geography problem in Heegaard Floer homology.We give new restrictions on Heegaard Floer homologies of homology spheres with reduced rank 1. As an application we show that a certain filtered chain complex with vanishing $\epsilon$-invariant and non-vanishing $\Upsilon$-invariant cannot be realized as a knot Floer complex. (arXiv)

5. (with Gordana Matic, Jeremy Van Horn-Morris, and Andy Wand) Algebraic torsion via Heegaard Floer homology.Motivated by Hutchings's ECH formulation, we define an analog of Wendl and Latschev's algebraic torsion in the context of Heegaard Floer homology. (arXiv)

6. (with Steven Sivek) Sutured ECH is a natural invariantWe prove that Colin-Ghiggini-Hutchings-Honda's sutured ECH associates to a sutured contact 3-manifold a well-defined group depending only on the isotopy class of the contact structure relative to the boundary and a fixed contact 1-form defining this contact structure on the boundary. Moreover, there exists a contact class which we show exhibits several expected properties., with an appendix by C. H. Taubes. (arXiv)

7. (with Yi-Jen Lee and Clifford Henry Taubes) HF=HM V: Seiberg-Witten Floer homology and handle addition.This is the final paper in a five-part series that proves the equivalence of Heegaard Floer and Seiberg-Witten Floer homologies. We prove that the filtered version of Seiberg-Witten Floer homology on $Y$, built from a compact oriented 3-manifold $M$, is equivalent to the Seiberg-Witten Floer homology of $M$. (arXiv)

8. (with Yi-Jen Lee and Clifford Henry Taubes) HF=HM IV: The Seiberg-Witten Floer homology and ech correspondence.This is the fourth paper in a five-part series that proves the equivalence of Heegaard Floer and Seiberg-Witten Floer homologies. We define a filtered version of Seiberg-Witten Floer homology on $Y$, built from a compact oriented 3-manifold $M$, and show that it is isomorphic to ech. (arXiv)

9. (with Yi-Jen Lee and Clifford Henry Taubes) HF=HM III: Holomorphic curves and the differential for the ech/ Heegaard Floer homology correspondence.This is the third paper in a five-part series that proves the equivalence of Heegaard Floer and Seiberg-Witten Floer homologies. We investigate the relationship between moduli spaces of pseudo-holomorphic curves in $\mathbb{R}\times Y$, built from a compact oriented 3-manifold $M$, and moduli spaces of pseudo-holomorphic curves that appear in Lipshitz's reformulation of Heegaard Floer homology of $M$, and prove the equivalence of ech and Heegaard Floer homology. (arXiv)

10. (with Yi-Jen Lee and Clifford Henry Taubes) HF=HM II: Reeb orbits and holomorphic curves for the ech/ Heegaard Floer homology correspondence.This is the second paper in a five-part series that proves the equivalence of Heegaard Floer and Seiberg-Witten Floer homologies. Given a compact oriented 3-manifold $M$ together with a certain Morse function, a pseudo-gradient vector field, and a second cohomology class, we construct an auxiliary manifold $Y$ equipped with a stable Hamiltonian structure. We then define a filtered variant of Hutchings's embedded contact homology (ECH), which we denote by ech, for this stable Hamiltonian structure. (arXiv)

11. (with Yi-Jen Lee and Clifford Henry Taubes) HF=HM I: Heegaard Floer homology and Seiberg-Witten Floer homology.This is an outline of our proof of the isomorphism between Heegaard Floer and Seiberg-Witten Floer homologies. (arXiv)

12. Lectures on the equivalence of Heegaard Floer and Seiberg-Witten Floer homologiesThis is a detailed account of a series of lectures on the equivalence of Heegaard Floer and Seiberg-Witten Floer homologies., Proceedings of the Gokova Geometry-Topology Conference 2012, 1-42, Int. Press, Somerville, MA, 2013. (Published version)

13. (with Clifford Henry Taubes) Seiberg-Witten Floer homology and symplectic forms on $S^1\times M^3$,We prove necessary conditions on the Seiberg-Witten Floer homology of a compact oriented 3-manifold $M$ for $S^1\times M$ to admit a symplectic form. In particular, we conclude that when $M$ is the result of longitudinal surgery on a knot in $S^3$, $S^1\times M$ admits a symplectic form if and only if the knot is fibered. Geom. Topol. 13 (2009), no. 1, 493--525. (Published version, arXiv)

Acknowledgements: My work is supported by a Simons Foundation Grant Award No. 519352. In the past, I was supported in part by an NSF MSPRF Award No. DMS-1103795 and an NSF Grant Award No. DMS-1360293.

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