Extra Form
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.
첨부 '1'
  1. Combinatorics and Hodge theory

  2. 허준이 교수 호암상 수상 기념 강연 (Lorentzian Polynomials)

  3. Algebraic surfaces with minimal topological invariants

  4. A wrapped Fukaya category of knot complement and hyperbolic knot

  5. Regularity of solutions of Hamilton-Jacobi equation on a domain

  6. What is Weak KAM Theory?

  7. Topological Mapping of Point Cloud Data

  8. Structures on Persistence Barcodes and Generalized Persistence

  9. Persistent Homology

  10. Irreducible Plane Curve Singularities

  11. 07Nov
    by Editor
    in 특별강연

    최고과학기술인상수상 기념강연: On the wild world of 4-manifolds

  12. Queer Lie Superalgebras

  13. Regularization by noise in nonlinear evolution equations

  14. A New Approach to Discrete Logarithm with Auxiliary Inputs

  15. Contact topology and the three-body problem

  16. Harmonic bundles and Toda lattices with opposite sign

  17. Mathematical Analysis Models and Siumlations

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