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)
    • Seonhwa Kim (low dimensional topology, quantum invariants, hyperbolic geometry, knots and graphs)
    • Eunjeng Lee (Toric topology, Newton-Okounkov bodies, representation theory, and algebraic combinatorics)
    • Christophe Wacheux (Symplectic geometry and topology)
  • Hamiltonian dynamics and dynamical systems:
    • Jin Woo Jang (Partial differential equations, gas dynamics, fluid dynamics, harmonic analysis)
  • Mathematical Physics:
    • Alexander Aleksandrov (Mathematical physics, random matrix models, integrable systems, enumerative geometry)

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 (Homotopy and cohomology theory of algebro-geometric objects, especially algebraic stacks)
  • Gabriel C. Drummond-Cole (Operadic and homotopy algebra, quantum field theory, moduli space of Riemann surfaces)
  • Damien Lejay (Mathematical physics, vertex algebras, factorisation algebras, higher toposes)
  • Hwajong Yoo (Number theory, Galois representations and modular forms, Arithmetic Chern-Simons theory)

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)
  • Wanmin Liu (Algebraic geometry)
  • Joonyeong Won (Birational geometry)

Derived categories of coherent sheaves on algebraic varieties are important and interesting invariants of algebraic varieties. It turns out that they contain geometric, birational geometric, arithmetic information of algebraic varieties and investigation of derived categories of algebraic varieties is now one of the most important research areas in algebraic geometry. Moreover understanding derived categories of algebraic varieties are also important in many other areas of mathematics and physics such as symplectic geometry, number theory, mirror symmetry, representation theory, topology and string theory etc. The goal of the research group is to understand structures of derived caategories of algebraic varieties and find their applications to algebraic, arithemetic, symplectic geometry, mathematical physics and string theory.

Research themes and research members

  • Kyoung-Seog Lee (Derived categories of algebraic varieties)
  • Kyeong-Dong Park (Geometry of complex Fano manifolds and geometric structures)
  • Taekyung Kim (Number theory, arithmetic algebraic geometry and structure theory of algebras)