Research groups

The Center for Geometry and Physics is loosely organized into multiple research groups, each of which comprises a senior scholar who leads the group and several researchers whose areas of expertise and interest overlap synergistically. A brief description of each group’s areas of focus, research goals, and members can be seen below.

The current status of symplectic topology resembles that of classical topology in the middle of the twentieth century. Over time, a systematic algebraic language was developed to describe problems in classical topology. Similarly, a language for symplectic topology is emerging, but has yet to be fully developed. The development of this language is much more challenging both algebraically and analytically than in the case of classical topology. The relevant homological algebra of $A_\infty$ structures is harder to implement in the geometric situation due to the analytical complications present in the study of pseudo-holomorphic curves or "instantons" in physical terms. Homological mirror symmetry concerns a certain duality between categories of symplectic manifolds and complex algebraic varieties. The symplectic side of the story involves an $A_\infty$ category, called the Fukaya category, which is the categorified version of Lagrangian Floer homology theory. In the meantime, recent developments in the area of dynamical systems have revealed that the symplectic aspect of area preserving dynamics in two dimensions has the potential to further understanding of these systems in deep and important ways.

Research themes and research members

  • Symplectic and contact topology:
    • Yong-Geun Oh (symplectic topology, Hamiltonian dynamics and mirror symmetry)
    • Byunghee An (algebraic topology, geometric group theory)
    • Eunjeng Lee (Toric topology, Newton-Okounkov bodies, representation theory, and algebraic combinatorics)
    • Yat-Hin Suen (Complex geometry, Symplectic Geometry, SYZ Mirror Symmetry, Homological Mirror Symmetry, Mathematical Physics)
    • Minkyoung Song (Geometric topology and related topics in group theory)
    • Jongmyeong Kim (Homological mirror symmetry)
    • Taesu Kim (Homotopy theoretic aspects of symplectic geometry)
    • Dogancan Karabas (Symplectic topology, microlocal sheaves, mirror symmetry, low-dimensional topology)
    • Sangjin Lee (Lagrangian foliations, Symplectic mapping class group, Fukaya category)
    • Seungwon Kim (Topology and geometry)
    • Hongtaek Jung (Symplectic structures of Hitchin components and Anosov representations)
  • Hamiltonian dynamics and dynamical systems:
    • Jin Woo Jang (Partial differential equations, gas dynamics, fluid dynamics, harmonic analysis)

The mathematical foundations of field and string theories remain poorly understood. As a consequence, many mathematical theories which are intimately related to the quantization of such systems are not yet subsumed into a unifying framework that could guide their further development. In particular developing a general theory of B-type Landau-Ginzburg models, as required by a deeper understanding of mirror symmetry, and developing a general global mathematical formulation of supergravity theories, which could afford a deeper understanding of the Mathematics behind “supersymmetric geometry”. Physical and mathematico-physical approaches to string theory and quantum eld theory use homotopical methods in various ways, depending on the formalizations at play. Many basic tools used in algebraic geometry are refinements of constructions from algebraic topology. In order to ensure that these tools are employed within the proper framework, with attention to contemporary developments, to facilitate their use, and to help bridge the gaps between participants in different research programs, it is useful to have an active research group focused on the fundamentals of homotopy algebra itself.

Research themes and research members

  • Calin Lazaroiu (Landau-Ginzburg models and supergravity)
  • Chang-Yeon Chough (Algebraic geometry, homotopy theory, infinity category theory)
  • Damien Lejay (Mathematical physics, vertex algebras, factorisation algebras, higher toposes)
  • Anna Cepek (Manifold topology)
  • Yifan Li (Algebraic geometry, algebraic topology and mathematical physics)
  • Mathematical Physics:
    • Alexander Aleksandrov (Mathematical physics, random matrix models, integrable systems, enumerative geometry)
  • Fano varieties are algebraic varieties whose anticanonical classes are ample. They are classical and fundamental varieties that play many significant roles in contemporary geometry. Verified or expected geometric and algebraic properties of Fano varieties have attracted attentions from many geometers and physicists. In spite of extensive studies on Fano varieties for more than one centuries, numerous features of Fano varieties are still shrouded in a veil of mist. Contemporary geometry however requires more comprehensive understanding of Fano varieties.

    Research themes and research members

    • Jihun Park (Arithmetic, birational and complex geometry of Fano varieties)
    • Jun-Yong Park (Geometry & topology of 4-manifolds and the arithmetic of moduli of surfaces)
    • Yuto Yamamoto (Tropical geometry)
    • Kyeong-Dong Park (Geometry of complex Fano manifolds and geometric structures)
    • Sungmin Yoo (Complex geometry and geometric analysis)
    • Shinyoung Kim (Complex geometry)