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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. 17Oct
    by 김수현
    in Special Colloquia

    A wrapped Fukaya category of knot complement and hyperbolic knot

  2. 07Nov
    by Editor
    in Special Colloquia

    Contact topology and the three-body problem

  3. 07Nov
    by Editor
    in Special Colloquia

    A New Approach to Discrete Logarithm with Auxiliary Inputs

  4. 07Nov
    by Editor
    in Special Colloquia

    최고과학기술인상수상 기념강연: On the wild world of 4-manifolds

  5. 28Nov
    by Editor
    in Special Colloquia

    Irreducible Plane Curve Singularities

  6. 29Jul
    by 김수현
    in Special Colloquia

    Combinatorics and Hodge theory

  7. 17Oct
    by 김수현
    in Special Colloquia

    Algebraic surfaces with minimal topological invariants

  8. 07Nov
    by Editor
    in Special Colloquia

    Queer Lie Superalgebras

  9. 27Mar
    by 김수현
    in Special Colloquia

    Persistent Homology

  10. 27Mar
    by 김수현
    in Special Colloquia

    Structures on Persistence Barcodes and Generalized Persistence

  11. 27Mar
    by 김수현
    in Special Colloquia

    Topological Mapping of Point Cloud Data

  12. 07Nov
    by Editor
    in Special Colloquia

    Harmonic bundles and Toda lattices with opposite sign

  13. 26May
    by Manager
    in Special Colloquia

    허준이 교수 호암상 수상 기념 강연 (Lorentzian Polynomials)

  14. 03Sep
    by 김수현
    in Special Colloquia

    What is Weak KAM Theory?

  15. 03Sep
    by 김수현
    in Special Colloquia

    Regularity of solutions of Hamilton-Jacobi equation on a domain

  16. 07Nov
    by Editor
    in Special Colloquia

    Regularization by noise in nonlinear evolution equations

  17. 07Nov
    by Editor
    in Special Colloquia

    Mathematical Analysis Models and Siumlations

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