강연영상 2 페이지 > 서울대학교 수리과학부

Hamiltonian dynamics, originated in classical mechanics, lies at the root of symplectic geometry and provides a wealth of examples. One of the central topics is the study of periodic orbits, and over the years many tools have been developed to investigate them. Among the most interesting and historically important examples is the three-body problem. Despite centuries of research, however, a complete picture of its periodic orbits is still lacking, with only partial results available. In this talk, I will present the basic ideas and techniques of Hamiltonian dynamics and symplectic geometry including Floer theory, and introduce the three-body problem and related systems. I will also briefly describe some of my recent results related with the rotating Kepler problem and three-body problem.

Diophantine approximation is the study of approximating real numbers by rational numbers. For example, one can ask whether a real number x is ψ-approximable; that is, whether there are infinitely many rationals p/q satisfying |x-p/q|<ψ(q)/q for a given monotonic function ψ. A century ago, Khintchine discovered a remarkable dichotomy for the Lebesgue measure of the set of ψ-approximable numbers. Since then, Khintchine’s theorem has been extended in various directions, including inhomogeneous approximation and higher-dimensional generalizations. In this talk, I will introduce the Allen-Ramírez conjecture on removing the monotonicity condition from the inhomogeneous Khintchine-Groshev theorem and discuss a recent proof of the conjecture in the case (n,m)=(2,1).

An algebraic function field is an algebraic extension of finite degree over the rational function field. This field was first introduced by Dedekind, Kronecker, and Weber in the nineteenth century and was further developed by Artin, Hasse, Schmidt, and Weil in the first half of the twentieth century. In this talk, we discuss various function field extensions over a finite field and examine important invariants such as the number of rational places and the class numbers.

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Hermite functions play an important role in many areas, including quantum mechanics, partial differential equations, and probability theory. In this talk, I will introduce two problems about Hermite functions that have recently attracted much attention in harmonic analysis. To help the audience follow along, I will begin with some background.

The moduli spaces of vector bundles on curves lie at the crossroads of geometry, topology, and representation theory. In this talk, I will introduce these moduli spaces with a focus on their topological aspects. In particular, I will present formulas for their Betti numbers and describe their intersection theory, along with some applications. If time permits, I will also discuss some recent developments about these classical moduli spaces.