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강연자 박종일
소속 서울대학교
date 2013-09-26
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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첨부 '1'
  1. 07Nov
    by Editor
    in 특별강연

    A New Approach to Discrete Logarithm with Auxiliary Inputs

  2. 07Nov
    by Editor
    in 수학강연회

    정년퇴임 기념강연: Volume Conjecture

  3. 28Nov
    by Editor
    in 특별강연

    Irreducible Plane Curve Singularities

  4. 18Mar
    by 김수현
    in 수학강연회

    Subgroups of Mapping Class Groups

  5. 07Nov
    by Editor
    in 수학강연회

    Randomness of prime numbers

  6. 07Nov
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    in 수학강연회

    Non-commutative Lp-spaces and analysis on quantum spaces

  7. 07Nov
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    in 수학강연회

    학부생을 위한 ε 강연회: Mathematics from the theory of entanglement

  8. 15Dec
    by 김수현
    in 수학강연회

    Brownian motion and energy minimizing measure in negative curvature

  9. 07Nov
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    in 수학강연회

    Combinatorial Laplacians on Acyclic Complexes

  10. 07Nov
    by Editor
    in 특별강연

    Contact topology and the three-body problem

  11. 08Nov
    by Editor
    in 수학강연회

    Fefferman's program and Green functions in conformal geometry

  12. 07Nov
    by Editor
    in 특별강연

    최고과학기술인상수상 기념강연: On the wild world of 4-manifolds

  13. 06Dec
    by 김수현
    in 수학강연회

    Seeded Ising Model for Human Iris Templates and Secure Distributed Iris Recognition

  14. 11Apr
    by 김수현
    in 수학강연회

    Categorification of Donaldson-Thomas invariants

  15. 18Apr
    by 김수현
    in 수학강연회

    Random conformal geometry of Coulomb gas formalism

  16. 2021-2 Rookies Pitch: Harmonic Analysis (이진봉)

  17. 2021-2 Rookies Pitch: Regularity for PDEs (수미야)

  18. 2021-2 Rookies Pitch: Representation Theory(장일승)

  19. 2023-1 Symplectic Topology (노경민)

  20. 29May
    by 김수현
    in 수학강연회

    <학부생을 위한 ɛ 강연> 기하와 대수의 거울대칭

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