My work lies at the intersection between low-dimensional topology, geometry, and elliptic PDEs; most of it involves the Seiberg-Witten equations on three-manifolds. My research is supported by the NSF and the Alfred P. Sloan Foundation.
- Indefinite Stein fillings and Pin(2)-monopole Floer homology
- Monopole Floer homology and SOLV geometry
- Hyperbolic four-manifolds with vanishing Seiberg-Witten invariants
with Ian Agol, submitted.
- The Seiberg-Witten equations and the length spectrum of hyperbolic three-manifolds
with Michael Lipnowski, submitted.
- Non-formality in Pin(2)-monopole Floer homology
- Pin(2)-monopole Floer homology and the Rokhlin invariant
Compositio Mathematica 154 (2018), no. 12, 2681-2700.
- Monopole Floer homology and the spectral geometry of three-manifolds
Accepted for publication in Communications in Analysis and Geometry.
- Bar-Natan's deformation of Khovanov homology and involutive monopole Floer homology
Mathematische Annalen 373 (2019) no. 1-2, 489-516.
- Pin(2)-monopole Floer homology, higher compositions and connected sums
Journal of Topology 10 (2017), no. 4, 921-969.
- The surgery exact triangle in Pin(2)-monopole Floer homology
Algebraic & Geometric Topology 17-5 (2017), 2915--2960.
- Exact Lagrangian caps of Legendrian knots
Journal of Symplectic Geometry, Vol. 14, No. 1 (2016), pp. 269-295.
- A Morse-Bott approach to monopole Floer homology and the Triangulation conjecture
Memoirs of the AMS, 255 (2018), no. 1221. With Errata.
TeachingHere's some graduate level classes I have taught:
- Spring 2020: Algebraic Topology II.
- Spring 2019: Topics in Topology: Elliptic operators and Topology.