# 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 Integer partitions, q-series, and Modular forms 서울과학기술 대학 김병찬
수학강연회 Infinite order rationally slice knots 카이스트 수리과학과 박정환
수학강연회 Ill-posedness for incompressible Euler equations at critical regularit 서울대학교 정인지
수학강연회 Idempotents and topologies University of Waterloo Nico Spronk
수학강연회 Hybrid discontinuous Galerkin methods in computational science and engineering 연세대 박은재
수학강연회 How to solve linear systems in practice 이화여대 수학과 민조홍
수학강연회 High dimensional nonlinear dynamics 경북대학교 도영해
수학강연회 Heavy-tailed large deviations and deep learning's generalization mystery Northwestern University 이창한
특별강연 Harmonic bundles and Toda lattices with opposite sign RIMS, Kyoto Univ. Takuro Mochizuki
수학강연회 Hamiltonian dynamics, Floer theory and symplectic topology University of Wisconsin 오용근
수학강연회 Gromov-Witten-Floer theory and Lagrangian intersections in symplectic topology IBS, 포항공과대학교 오용근
수학강연회 Green’s function for initial-boundary value problem National Univ. of Singapore Shih-Hsien Yu
수학강연회 Global result for multiple positive radial solutions of p-Laplacian system on exterior domain 부산대학교 이용훈
수학강연회 Geometry, algebra and computation in moduli theory 서울대 현동훈
수학강연회 Geometric structures and representation spaces 서울대학교 이계선
수학강연회 Geometric Langlands theory: A bridge between number theory and physics 서울대학교 유필상
수학강연회 Generalized multiscale HDG (hybridizable discontinuous Galerkin) methods for flows in highly heterogeneous porous media 육군사관학교 문미남
수학강연회 Gaussian free field and conformal field theory 서울대학교 강남규
수학강연회 Freudenthal medal, Klein medal 수상자의 수학교육이론 서울대 수학교육과 권오남
수학강연회 Free boundary problems arising from mathematical finance 경희대학교 전준기
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