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- Dept of Mathematics
- Faculty of Maths & Physical Sciences
- 2011 - Royal Society University Research Fellow at UCL
- 2009 - 2011 Humboldt Foundation postdoctoral fellow at HU Berlin
- 2007 - 2009 NSF postdoctoral fellow at ETH Zürich
- 2005 - 2006 Postdoc at LMU Munich
- 2004 - 2007 C.L.E. Moore Instructor at MIT (on leave 2005-06)
- 1999 - 2004 PhD student in mathematics at Courant Institute, NYU, Advisor: Helmut Hofer
- 1994 - 1997 Undergraduate in physics at Harvard University
- 1993 - 1994 Undergraduate in physics and music at Amherst College
I work mainly in the fields of Symplectic and Contact Topology, two intimately related branches of differential geometry that have close connections with topology, dynamics, PDEs and mathematical physics. Symplectic manifolds were first studied over 100 years ago as the natural geometric setting for Hamiltonian mechanics, and contact manifolds can be viewed as a special class of energy hypersurfaces in symplectic manifolds. Since Gromov's seminal work in 1985, the most powerful techniques used in this area have come from the theory of pseudoholomorphic curves: these are solutions to an elliptic PDE that generalizes the Cauchy-Riemann equations of complex analysis. By counting solutions to these equations in various settings, one can define a wealth of symplectic and contact invariants that go by names such as Gromov-Witten Theory, Floer Homology, Contact Homology and Symplectic Field Theory. The algebraic structure of these theories is a topic of considerable interest in itself, among other reasons because they have formal similarities to objects that arise naturally in Quantum Field Theory.
My own research in this field tends to focus less on algebra and more on the analysis of pseudoholomorphic curves and their applications in symplectic and contact manifolds. Here are some broad examples of the kinds of questions I spend my time on:
- How do contact structures on odd-dimensional manifolds interact with symplectic structures on manifolds of one dimension higher? For instance, which contact manifolds can be related to each other through symplectic cobordisms? How does the contact topology of the boundary influence the symplectic topology of the cobordism, or vice versa?
- What kinds of local and global structures exist naturally on moduli spaces of pseudoholomorphic curves? Under what topological circumstances is this structure especially nice, and what does that imply about the symplectic/contact topology of the ambient manifold?
- How do various types of geometric decompositions (e.g. Lefschetz fibrations, open book decompositions, fiber connected sum decompositions) help us to understand the global structure of symplectic and contact manifolds?
- What algebraic structures arise naturally from moduli spaces of holomorphic curves, and what impact does this have on Question 1?