# 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 Existence of positive solutions for φ-Laplacian systems 이용훈 수학강연회,특별강연,대중강연
BK21 FOUR Rookies Pitch 2021-2 Rookies Pitch: Regularity for PDEs (수미야) 서울대학교 수미야
수학강연회 학부생을 위한 강연: Choi's orthogonal Latin Squares is at least 61 years earlier than Euler's 연세대학교 송홍엽
BK21 FOUR Rookies Pitch 2022-1 Rookies Pitch: Algebraic Topology (송종백) 고등과학원 송종백
수학강연회 행렬함수 Permanent의 극소값 결정과 미해결 문제들 제주대학교/서울대학교 송석준
수학강연회 L-function: complex vs. p-adic 충북대학교 선해상
수학강연회 Quantitative residual non-vanishing of special values of various L-functions UNIST 선해상
수학강연회 Mixing time of random processes 서울대 서인석
수학강연회 <2020년도 젊은 과학자상 수상 기념강연> Metastability of stochastic systems 서울대학교 서인석
수학강연회 W-algebras and related topics 서울대학교 서의린
BK21 FOUR Rookies Pitch 2023-2 Differential Geometry (서동휘) 수학연구소 서동휘
BK21 FOUR Rookies Pitch 2022-1 Rookies Pitch: Geometric Group Dynamics (서동균) 수학연구소 서동균
수학강연회 Variational Methods without Nondegeneracy POSTECH 변재형
수학강연회 학부생을 위한 강연: Introduction to partial differential equations 서울대학교 변순식
수학강연회 Theory and applications of partial differential equations 서울대 변순식
BK21 FOUR Rookies Pitch 2022-2 Rookies Pitch: Probability Theory (변성수) KIAS 변성수
수학강연회 1 is big enough to understand 3 카이스트 백형렬
수학강연회 Essential dimension of simple algebras KAIST 백상훈
수학강연회 <학부생을 위한 ɛ 강연> Introduction to the incompressible Navier-Stokes equations UNIST 배한택
수학강연회 Mixed type PDEs and compressible flow POSTECH 배명진
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