### 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 are as follows:

Research themes and research members

• Symplectic and contact topology:
• Byunghee An (algebraic topology, geometric group theory)
• Morimichi Kawasaki (Sympletic geometry)
• Seonhwa Kim (low dimensional topology, quantum invariants, hyperbolic geometry, knots and graphs)
• Christophe Wacheux (Symplectic geometry and topology)
• Hamiltonian dynamics and dynamical systems:
• Youngjin Bae (symplectic geometry, dynamics in Hamiltonian system)
• Jin Woo Jang (Partial differential equations, gas dynamics, fluid dynamics, harmonic analysis)
• Mathematical Physics:
• Aleksandrov, Alexander (Mathematical physics, random matrix models, integrable systems, enumerative geometry)

Ideas and principles from quantum field theory have been applied to many important and active areas of mathematics in the last three decades. Typically, such an application begins with a rigorous mathematical framework for the mathematical object in question and a corresponding example of (usually topological) quantum field theory and builds an elaborate dictionary—though the latter side may lack mathematical definition. On the physical side, in all such examples, the quantum field theory comes endowed with a physically natural duality. Via the dictionary, this leads to a duality or correspondence between different mathematical objects, which is generally extremely non-trivial mathematically. Such correspondences are usually studied on a case-by-case basis, separately for each application.

Taken in toto, these phenomena seem to imply that there should be a mathematical unification of those mathematical structures related to quantum field theory. With all the current examples and evidence at hand, it is time to ask the following fundamental question:

What is quantum field theory mathematically and

when is a mathematical quantum field theory physical?

The primary goal of the group's research is to establish an algebraic and mathematically rigorous foundation for quantum field and string theory and pursue its consequences in both pure mathematics and theoretical physics.

The members of this group have diverse backgrounds and interests in mathematics and physics, and each member’s area(s) of interest are as follows:

• 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, algebraic geometry, Galois representations and modular forms)
• Cheolgyu Lee (Algebraic geometry & Euclidean conjecture)/ Military Service

For a given smooth variety with pseudo-effective canonical class, the minimal model program produces a birational model of the variety, the so called minimal model, that has mild singularities (terminal and Q-factorial) and whose canonical class is nef. This has been verified in dimension 3 and in all dimensions for varieties of general type.

Meanwhile, if the canonical class is not pseudo-effective, then the minimal model program yields a birational model, a so-called Mori fibred space. It also has terminal and Q-factorial singularities and it admits a fiber structure of relative Picard rank one such that the anticanonical class is ample on fibers. This has been proved in all dimensions. Fano varieties of Picard rank one with at most terminal Q-factorial singularities are important examples of Mori fibred spaces.

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 attention from many geometers and physicists. In spite of extensive study of Fano varieties for more than a century, numerous features of Fano varieties are still shrouded in a veil of mist. Contemporary geometry requires a more comprehensive understanding of Fano varieties. This research group is to play a role in broadening comprehensive knowledge about Fano varieties by studying them from various points of view. Currently, the members and their research areas are as follows:

• Wanmin Liu (Algebraic geometry)
• Jihun Park (algebraic geometry, fano varieties)
• Dmitrijs Sakovics (Birational algebraic geometry, low-dimensional singularities)
• Joonyeong Won (Birational geometry)
• Taekyung Kim (Number theory, arithmetic algebraic geometry, and structure theory of algebras)

Understanding the proper mathematical description of the topological B-type string in a general setting can serve as a unifying theme for problems in deformation theory, with applications to mirror symmetry. Proper construction of the corresponding string field theory is a non-trivial challenge, being a test of many ideas involved in the still poorly-understood "theory of quantization".

The primary goal of the group is to investigate the construction and properties of the topological string field theory of B-type strings, including its topological D-branes in various settings and to explore its geometric realizations and its connections with modern deformation theory.

Group members

• Elena Mirela Babalic (String theory, quantum field theory, constructive field theory, BV-BRST methods)
• Dmitry Doryn (Algebraic geometry, number theory, quantum theory)
• Mehdi Tavakol (Algebraic geometry, Gromov-Witten theory)

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, e.g. symplectic geometry, number theory, mirror symmetry, representation theory, topology, mathematical physics, string theory, etc.

The goal of our research group is to understand structures of derived categories of algebraic varieties and find their applications to algebraic, arithmetic, symplectic geometry, mathematical physics and string theory.

Group members

• Kyoung-Seog Lee (Algebraic geometry)
• Kyeong-Dong Park (Geometry of complex Fano manifolds and geometric structures)