A great deal of research in symplectic geometry has revolved around Gromov-Witten invariants, a mathematical model of the physics of closed strings. Recent research in string-theory indicates that there should be an open-string analog of Gromov-Witten invariants, despite certain mathematical complications. In my thesis, I introduced a working definition of open Gromov-Witten invariants for real symplectic manifolds. I intend to study the properties of these new open Gromov-Witten invariants, and investigate how to extend the definition further.
In my thesis, I established a connection between open Gromov-Witten theory and real enumerative geometry as developed in the recent seminal work of J. Y. Welschinger. Progress in open Gromov-Witten theory should shed light on a host of problems in real enumerative geometry, especially the connection with traditional complex enumerative geometry.
The study of open Gromov-Witten theory in the context of real geometry has revealed a deep and little studied relationship between real geometry and mirror symmetry. Mirror symmetry is a striking collection of conjectures originating from string-theory that predict a comprehensive duality between symplectic geometry and complex geometry. Intuition and knowledge from symplectic geometry and complex geometry can thus be combined to solve otherwise intractable problems. In recent collaborative work with R. Pandharipande and string-theorist J.Walcher we have verified an important example of mirror symmetry in the real open Gromov-Witten setting.
I believe my research, which bridges a gap between geometry and string theory, as well as my ongoing collaboration with physicists will help build a culture of interdisciplinary interaction. Moreover, I plan to convey the knowledge I have acquired in ongoing collaborative work with leading researchers in the United States to students and researchers with whom I have already developed ties in Europe.
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