# 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 Conformal field theory and noncommutative geometry 동경대학교 Kawahigashi
수학강연회 Subword complexity, expansion of real numbers and irrationality exponents 동국대 김동한
수학강연회 The Lagrange and Markov Spectra of Pythagorean triples 동국대학교 김동한
수학강연회 <학부생을 위한 ε 강연> 수학과 예술 - 초기 컴퓨터 그래픽 동양대학교 진중권
수학강연회 <학부생을 위한 강연> 수학과 보험산업 라이나생명 유신옥
특별강연 Combinatorics and Hodge theory 미국 프린스턴대 교수, 한국 고등과학원 석학교수 허준이
수학강연회 Fixed points of symplectic/Hamiltonian circle actions 부산대 수학과 장동훈
수학강연회 Global result for multiple positive radial solutions of p-Laplacian system on exterior domain 부산대학교 이용훈
수학강연회 행렬, 행렬함수 그리고 행렬방정식 (Matrix, Matrix Functions and Matrix Equations) 부산대학교 수학과 김현민
수학강연회 Mathematics, Biology and Mathematical Biology 부산대학교 수학과 정일효
수학강연회 Symplectic topology and mirror symmetry of partial flag manifolds 부산대학교 수학과 김유식
수학강연회 Q-curvature in conformal geometry 서강대 Pak Tung Ho
수학강연회 Averaging formula for Nielsen numbers 서강대학교 이종범
수학강연회 Symmetry Breaking in Quasi-1D Coulomb Systems 서강대학교 Paul Jung
수학강연회 Noncommutative Surfaces 서강대학교 Jens Hoppe
수학강연회 Regularity theory for non-autonomous elliptic equations in divergence form 서강대학교 옥지훈
수학강연회 Elliptic equations with singular drifts in critical spaces 서강대학교 김현석
수학강연회 Integer partitions, q-series, and Modular forms 서울과학기술 대학 김병찬
수학강연회 On the distributions of partition ranks and cranks 서울과학기술대학교 김병찬
수학강연회 Volume entropy of hyperbolic buildings 서울대 임선희
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