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

    • Elijah Fender (The interplay of dynamics and symplectic/contact geometry)
    • Volker Genz (Explicit problems in representation theory)
    • Hongtaek Jung (Symplectic structures of Hitchin components and Anosov representations)
    • Sungkyung Kang (Heegaard Floer theory, knot theory)
    • Jongmyeong Kim (Homological mirror symmetry)
    • Seungwon Kim (Topology and geometry)
    • Taesu Kim (Homotopy theoretic aspects of symplectic geometry)
    • Sangjin Lee (Lagrangian foliations, Symplectic mapping class group, Fukaya category)
    • Yong-Geun Oh (symplectic topology, Hamiltonian dynamics and mirror symmetry)
    • Yat-Hin Suen (Complex geometry, Symplectic Geometry, SYZ Mirror Symmetry, Homological Mirror Symmetry, Mathematical Physics)

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

  • Sai Somanjana Sreedhar Bhamidi (Algebraic K-theory, algebraic cycles, algebraic stacks and derived categories)
  • Shinyoung Kim (Complex geometry)
  • Rahul Kumar (Analytic number theory, special functions, and the theory of partitions)
  • Eunjeng Lee (Toric topology, Newton-Okounkov bodies, representation theory, and algebraic combinatorics)
  • Jihun Park (Arithmetic, birational and complex geometry of Fano varieties)
  • Samarpita Ray (Category Theory, Algebraic Geometry)
  • Haowu Wang (Theory of modular forms and its applications)
  • Yuto Yamamoto (Tropical geometry)

The mathematical relevance and deep interconnections between theoretical physics and mathematics are well-established. This subject is universally appreciated for its integrative role and for being one of the most fruitful sources of new ideas, theories and methods, and have numerous powerful applications to problems in mathematics, in particular, of geometry and topology. In recent decades, there have been various developments in supersymmetric quantum field theories and string/M-theory. In this premise, matrix models, integrable systems, Chern-Simons gauge theory, Landau-Ginzburg theory and mirror symmetry, and topological quantum field theories are the main themes of research pursued in this group.

Research themes and research members

  • Alexander Aleksandrov (Mathematical physics, random matrix models, integrable systems, enumerative geometry)
  • Saswati Dhara (Theoretical high energy physics, Chern-Simons theory in knot invariants, conformal field theory, topological field theory)
  • Yifan Li (Algebraic geometry, algebraic topology and mathematical physics)
  • Hisayoshi Muraki (Noncommutative geometry, nongeometric backgrounds in supergravity, discretized geometry, matrix model)
  • Abbas Mohamed Sherif (Einstein's general relativity theory, interfacing differential geometry, geometric analysis and general relativity)