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

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.
 제목+내용제목내용댓글이름닉네임태그
1. 19Mar
by 김수현
in 수학강연회

<학부생을 위한 ɛ 강연> 복잡한 생명현상을 위한 21세기 현미경, 수학!

2. 08Apr
by 김수현
in 수학강연회

Circular maximal functions on the Heisenberg group

3. 01Jun
by 김수현
in 수학강연회

Entropy of symplectic automorphisms

4. 2023-2 Mathematical Fluid Dynamics (김준하)

5. 2021-2 Rookies Pitch: Number Theory (김지구)

6. 28Mar
by 김수현
in 수학강연회

Study stochastic biochemical systems via their underlying network structures

7. 01Apr
by 김수현
in 수학강연회

A dissipative effect on some PDEs with physical singularity

8. 25Sep
by 김수현
in 수학강연회

Satellite operators on knot concordance

9. 02May
by 김수현
in 수학강연회

Mathemaics & Hedge Fund

10. 07Nov
by Editor
in 수학강연회

정년퇴임 기념강연: Volume Conjecture

11. 08Jun
by 김수현
in 수학강연회

<학부생을 위한 ɛ 강연> 196884=196883+1

12. 23Apr
by 김수현
in 수학강연회

행렬, 행렬함수 그리고 행렬방정식 (Matrix, Matrix Functions and Matrix Equations)

13. 13Oct
by 김수현
in 수학강연회

Elliptic equations with singular drifts in critical spaces

14. 17Oct
by 김수현
in 수학강연회

<정년퇴임 기념강연> The Elements of Euclid

15. 07Nov
by Editor
in 수학강연회

학부생을 위한 강연: 브라질과 프랑스는 왜 축구를 잘 할까? - 경제와 수학과 축구와 법률

16. 27Nov
by 김수현
in 수학강연회

Universality of log-correlated fields

17. 2023-1 Symplectic Topology (노경민)

18. 05Dec
by 김수현
in 수학강연회

High dimensional nonlinear dynamics

19. 17Oct
by 김수현
in 수학강연회

WGAN with an Infinitely wide generator has no spurious stationary points

20. 12Nov
by 김수현
in 수학강연회

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

Board Pagination Prev 1 2 3 4 5 6 7 8 9 10 11 12 13 14 15 Next
/ 15