Extra Form
Lecturer 박종일
Dept. 서울대학교
date Sep 26, 2013
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.
Attachment '1'
List of Articles
Category Subject Dept. Lecturer
Math Colloquia It all started with Moser file Univ. of Wisconsin/포항공대 Paul Rabinowitz
Special Colloquia Irreducible Plane Curve Singularities file 서울대학교 강정혁
Math Colloquia Introduction to Non-Positively Curved Groups file KAIST 김상현
Math Colloquia Integer partitions, q-series, and Modular forms file 서울과학기술 대학 김병찬
Math Colloquia Infinite order rationally slice knots file 카이스트 수리과학과 박정환
Math Colloquia Ill-posedness for incompressible Euler equations at critical regularit file 서울대학교 정인지
Math Colloquia Idempotents and topologies file University of Waterloo Nico Spronk
Math Colloquia Hybrid discontinuous Galerkin methods in computational science and engineering file 연세대 박은재
Math Colloquia How to solve linear systems in practice file 이화여대 수학과 민조홍
Math Colloquia High dimensional nonlinear dynamics file 경북대학교 도영해
Math Colloquia Heavy-tailed large deviations and deep learning's generalization mystery file Northwestern University 이창한
Special Colloquia Harmonic bundles and Toda lattices with opposite sign file RIMS, Kyoto Univ. Takuro Mochizuki
Math Colloquia Hamiltonian dynamics, Floer theory and symplectic topology file University of Wisconsin 오용근
Math Colloquia Gromov-Witten-Floer theory and Lagrangian intersections in symplectic topology file IBS, 포항공과대학교 오용근
Math Colloquia Green’s function for initial-boundary value problem file National Univ. of Singapore Shih-Hsien Yu
Math Colloquia Global result for multiple positive radial solutions of p-Laplacian system on exterior domain file 부산대학교 이용훈
Math Colloquia Geometry, algebra and computation in moduli theory file 서울대 현동훈
Math Colloquia Geometric structures and representation spaces file 서울대학교 이계선
Math Colloquia Geometric Langlands theory: A bridge between number theory and physics file 서울대학교 유필상
Math Colloquia Generalized multiscale HDG (hybridizable discontinuous Galerkin) methods for flows in highly heterogeneous porous media file 육군사관학교 문미남
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