My research is motivated by two questions: What is the fundamental law of Nature? Which mathematics is used to describe it? The right theoretical framework must unify general relativity and quantum mechanics. It must be described, I think, by a language which unifies mathematical areas that were born and grew up indivisibly from these two physics developments. String theory is a leading candidate for such a framework.
"Mirror symmetry" is an example that suggests "unification of mathematics". It is a phenomenon that strings moving in two different spaces lead to exactly the same physics, in a surprising way in which symplectic geometry and algebraic geometry of the two spaces are exchanged. We have shown that it can be understood using duality in two-dimensional quantum gauge theory. This work led to further development such as correspondence between D-branes.
Currently, I am studying string compactifications using various techniques including mirror symmetry. One big goal is to obtain a picture of totality of theories with minimal supersymmetry in four-dimensions. At the same time, I aim to develop the best language to describe the theory, in collaboration with mathematicians.
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