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

    • Sam Bardwell-Evans (Symplectic geometry, Moduli spaces, Pseudoholomorphic curves and discs, Mirror symmetry)
    • Volker Genz (Explicit problems in representation theory)
    • Jaekwan Jeon (Deformations of rational surface singularities and related topics in symplectic geometry)
    • Jongmyeong Kim (Homological mirror symmetry)
    • Norton Lee (Supersymmetry, Integrable Systems, Quantum Field Theories, Mathematical Physics)
    • Seul Bee Lee (Ergodic theory of dynamical systems, Number Theory and Geometric Group Theory)
    • Yong-Geun Oh (Symplectic topology, Hamiltonian dynamics and mirror symmetry)

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

  • Igor Krylov (Birational Geometry)
  • Jihun Park (Birational geometry)
  • Luca Rizzi (Complex Algebraic Geometry, families of varieties, deformations and Torelli-type problems, Hermitian metrics)

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)
  • Hisayoshi Muraki (Noncommutative geometry, nongeometric backgrounds in supergravity, discretized geometry, matrix model)
  • Dmytro Voloshyn (Mathematical physics, cluster algebras, Poisson geometry, quantum groups, integrable systems)

Quantum topology lies in the intersection of algebra, topology and mathematical physics and it is a source of knot and 3-manifold invariants (quantum invariants) giving rise to the so-called Topological quantum field theories (TQFT). The construction of such TQFTs requires techniques from representation theory, combinatorics and topology. We mainly pursue two research directions:
(a) topological “meaning” of quantum invariants, this leads to the study of classical invariants such as Milnor invariants as well as the study of the mapping class groups and their representation theory.
(b) construction of new invariants for braids, knots and 3-manifolds, in particular, we point to a development of quantum invariants for knots in thickened surfaces.

Research themes and research members

  • Anderson Arley Vera Arboleda (quantum topology, knot theory, mapping class groups)