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Extra Form
Lecturer 박종일
Dept. 서울대학교
date Sep 26, 2013
Despite of the fact that 4-dimensional manifolds together with 3-dimensional manifolds are the most fundamental and important objects in geometry and topology and topologists had great achievements in 1960's, there has been little known on 4-manifolds, in particular on smooth and symplectic 4-manifolds, until 1982. In 1982, M. Freedman classified completely simply connected topological 4-manifolds using intersection forms and S. Donaldson introduced gauge theory to show that some topological 4-manifolds do not admit a smooth structure. Since then, there has been a great progress in smooth and symplectic 4-manifolds mainly due to Donaldson invariants, Seiberg-Witten invariants and Gromov-Witten invariants. But the complete understanding of 4-manifolds is far from reach, and it is still one of the most active research areas in geometry and topology.
My main research interest in this area is the geography problems of simply connected closed smooth (symplectic, complex) 4-manifolds. The classical invariants of a simply connected closed 4-manifold are encoded by its intersection form , a unimodular symmetric bilinear pairing on H2(X : Z). M. Freedman proved that a simply connected closed 4-manifold is determined up to homeomorphism by . But it turned out that the situation is strikingly different in the smooth (symplectic, complex) category mainly due to S. Donaldson. That is, it has been known that only some unimodular symmetric bilinear integral forms are realized as the intersection form of a simply connected smooth (symplectic, complex) 4-manifold, and there are many examples of infinite classes of distinct simply connected smooth (symplectic, complex) 4-manifolds which are mutually homeomorphic. Hence it is a fundamental question in the study of 4-manifolds to determine which unimodular symmetric bilinear integral forms are realized as the intersection form of a simply connected smooth (symplectic, complex) 4-manifold - called a existence problem, and how many distinct smooth (symplectic, complex) structures exist on it - called a uniqueness problem. Geometers and topologists call these ‘geography problems of 4-manifolds’.
Since I got a Ph. D. with a thesis, Seiberg-Witten invariants of rational blow-downs and geography problems of irreducible 4-manifolds, I have contributed to the study of 4-manifolds by publishing about 30 papers - most of them are average as usual and a few of them are major breakthrough for the development of 4-manifolds theory. In this talk, I'd like to survey what I have done, what I have been doing and what I want to do in near future.
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  1. Hybrid discontinuous Galerkin methods in computational science and engineering

  2. The phase retrieval problem

  3. Theory and applications of partial differential equations

  4. Analysis and computations of stochastic optimal control problems for stochastic PDEs

  5. <학부생을 위한 ε 강연> 압축센싱과 행렬완성

  6. <학부생을 위한 ε 강연> 수학과 예술 - 초기 컴퓨터 그래픽

  7. Faithful representations of Chevalley groups over quotient rings of non-Archimedean local fields

  8. Quasi-homomorphisms into non-commutative groups

  9. Entropies on covers of compact manifolds

  10. Iwahori-Hecke algebras and beyond

  11. On the resolution of the Gibbs phenomenon

  12. <학부생을 위한 ε 강연> What mathematics can do for the real and even fake world

  13. The process of mathematical modelling for complex and stochastic biological systems

  14. Random walks in spaces of negative curvature

  15. Solver friendly finite element methods

  16. Brownian motion and energy minimizing measure in negative curvature

  17. 학부생을위한ε강연: 수학자는 왜 선망되는 직업일까?

  18. Generalized multiscale HDG (hybridizable discontinuous Galerkin) methods for flows in highly heterogeneous porous media

  19. Weyl character formula and Kac-Wakimoto conjecture

  20. Nonlocal generators of jump type Markov processes

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