Prof.Cheol-Hyun Cho
최고관리자
2025-07-18
Symplectic geometry, mirror symmetry, and singularity
Symplectic geometry originated as a mathematical language to describe physics, and such symplectic structures exist in many spaces used in mathematics. In particular, the quantum structure of these spaces has been studied over the past 30 years through symplectic geometry, and has been shown to be connected to many branches of mathematics, including algebraic and complex geometry, representation theory, and singularity theory. These correspondences are called mirror symmetry.
In particular, I have been working on Fukaya categories, which provide a systematic methodology for studying these quantum structures, and in doing so, explaining the principles of the mirror symmetry conjecture, and applying it in new ways to algebra and singularity theory.
As an example of this work, given a two-variable invertible polynomial, the complex solution space of the polynomial is given by a surface. I have described a geometric principle that for every curve drawn over this space, there is a corresponding mirror matrix factorization of the mirror polynomial.
In this way, I am working on finding and analyzing relationships between seemingly different mathematical concepts such as curves and matrix factorizations by analyzing the symplectic structure of the corresponding spaces.
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