# 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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List of Articles
카테고리 제목 소속 강연자
수학강연회 Study stochastic biochemical systems via their underlying network structures 포항공과대학교 김진수
수학강연회 A dissipative effect on some PDEs with physical singularity University of Wisconsin-Madison 김찬우
수학강연회 Mathemaics & Hedge Fund 지큐자산운용 김택근
수학강연회 정년퇴임 기념강연: Volume Conjecture 서울대학교 김혁
수학강연회 <학부생을 위한 ɛ 강연> 196884=196883+1 이화여대 김현규
수학강연회 행렬, 행렬함수 그리고 행렬방정식 (Matrix, Matrix Functions and Matrix Equations) 부산대학교 수학과 김현민
수학강연회 Elliptic equations with singular drifts in critical spaces 서강대학교 김현석
수학강연회 <정년퇴임 기념강연> The Elements of Euclid 서울대/광주과학기술원 김홍종
수학강연회 학부생을 위한 강연: 브라질과 프랑스는 왜 축구를 잘 할까? - 경제와 수학과 축구와 법률 서울대학교 법과대학 김화진
BK21 FOUR Rookies Pitch 2023-1 Symplectic Topology (노경민) 서울대학교 노경민
수학강연회 High dimensional nonlinear dynamics 경북대학교 도영해
수학강연회 WGAN with an Infinitely wide generator has no spurious stationary points 서울대학교 류경석
수학강연회 Generalized multiscale HDG (hybridizable discontinuous Galerkin) methods for flows in highly heterogeneous porous media 육군사관학교 문미남
BK21 FOUR Rookies Pitch 2022-1 Rookies Pitch: Symplectic Topology (문지연) 수학연구소 문지연
수학강연회 How to solve linear systems in practice 이화여대 수학과 민조홍
수학강연회 Hybrid discontinuous Galerkin methods in computational science and engineering 연세대 박은재
수학강연회 Infinite order rationally slice knots 카이스트 수리과학과 박정환
수학강연회 Restriction theorems for real and complex curves 포항공과대학교 박종국
특별강연 최고과학기술인상수상 기념강연: On the wild world of 4-manifolds 서울대학교 박종일
수학강연회 Iwasawa main conjecture and p-adic L-functions 포항공과대학교 박지훈
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