# 최고과학기술인상수상 기념강연: On the wild world of 4-manifolds

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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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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
by Editor
in 수학강연회

Non-commutative Lp-spaces and analysis on quantum spaces

7. 07Nov
by Editor
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
by Editor
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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