The 1st Thematic Year

Symplectic Geometry and Mirror Symmetry

July 2013 – August 2014

Since the advent of Hamiltonian mechanics in the end of 19th century, Hamiltonian dynamics has been an area of fundamental research both in physics and in mathematics. Symplectic geometry emerged as the geometric structure governing Hamiltonian mechanics and its quantization.

With the advent of the method of pseudo-holomorphic curves developed by Gromov in the 80's and the subsequent Floer's invention of elliptic Morse theory resulted in Floer cohomology, the landscape of symplectic geometry has changed drastically. In the last two decades of development in symplectic topology, the interplay between Hamiltonian dynamics and (pseudo)holomorphic curves has revealed many genuinely symplectic phenomena in which Floer homology and its cousins sit at the center of the unifying force of dynamics and geometry. This progress was accompanied by parallel developments in physics in closed string theory.

In his 1994 ICM talk in Z├╝rich, Kontsevich proposed the celebrated homological mirror symmetry relating the derived category of coherent sheaves in complex geometry and the Fukaya category in symplectic geometry. Enhanced by the later development in open string theory of D-branes, this homological mirror symmetry has been a source of many new insights and progresses in both algebraic geometry and symplectic geometry as well as in physics. Symplectic Algebraic Topology ia also gradually taking its shape.

More recently there are several fronts of this area that see much progress:

  • New applications of Floer theory to symplectic topology and Hamiltonian dynamics
  • Proofs of homological mirror symmetry conjecture in various special cases
  • New applications of the Floer theoretic ideas to low dimensional topology and contact topology
  • Establishment of $A_{\infty}$-structures and other higher algebraic structures such as the algebraic structure needed in the rigorous formulation of mirror symmetry and symplectic algebraic topology
  • Much interaction with symplectic geometry, algebraic geometry and string theory in physics

Symplectic geometry and mirror symmetry are some of the most actively researched areas in the field of mathematics currently. However, they are still comparatively new and unexplored in Korea, especially to students in graduate programs. Only a few young researchers in these areas have recently been stationed in mathematics institutes in Korea. The purpose of this thematic program is to encourage and promote research in these areas and to educate future mathematicians.