# 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 <청암상 수상 기념 특별강연> 동형암호, 기계학습, 근사정수론 서울대학교 수리과학부 천정희
수학강연회 <정년퇴임 기념강연> 수학의 시대정신(?) 서울대학교 수리과학부 김명환
수학강연회 학부생을 위한 강연: 브라질과 프랑스는 왜 축구를 잘 할까? - 경제와 수학과 축구와 법률 서울대학교 법과대학 김화진
수학강연회 Gaussian free field and conformal field theory 서울대학교 강남규
수학강연회 학부생을 위한 강연: Introduction to partial differential equations 서울대학교 변순식
수학강연회 곡선의 정의란 무엇인가? 서울대학교 김영훈
수학강연회 Fano manifolds of Calabi-Yau Type 서울대학교 Atanas Iliev
수학강연회 젊은과학자상 수상기념강연: From particle to kinetic and hydrodynamic descriptions to flocking and synchronization 서울대학교 하승열
수학강연회 정년퇴임 기념강연회: 숙제 서울대학교 지동표
수학강연회 Noncommutative Geometry. Quantum Space-Time and Diffeomorphism Invariant Geometry 서울대학교 Raphael Ponge
수학강연회 학부생을 위한 ε 강연회: Sir Isaac Newton and scientific computing 서울대학교 신동우
특별강연 A New Approach to Discrete Logarithm with Auxiliary Inputs 서울대학교 천정희
수학강연회 정년퇴임 기념강연: Volume Conjecture 서울대학교 김혁
특별강연 Irreducible Plane Curve Singularities 서울대학교 강정혁
수학강연회 Subgroups of Mapping Class Groups 서울대학교 김상현
수학강연회 Randomness of prime numbers 서울대학교 임선희
수학강연회 Non-commutative Lp-spaces and analysis on quantum spaces 서울대학교 이훈희
수학강연회 학부생을 위한 ε 강연회: Mathematics from the theory of entanglement 서울대학교 계승혁
수학강연회 Brownian motion and energy minimizing measure in negative curvature 서울대학교 임선희
수학강연회 Combinatorial Laplacians on Acyclic Complexes 서울대학교 국웅
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