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 Geometry, algebra and computation in moduli theory file 서울대 현동훈
Math Colloquia Weyl character formula and Kac-Wakimoto conjecture file 서울대 권재훈
Math Colloquia <학부생을 위한 ε 강연> 압축센싱과 행렬완성 file 서울대 심병효
Math Colloquia Theory and applications of partial differential equations file 서울대 변순식
Math Colloquia <학부생을 위한 ε 강연> 동형암호와 근사정수론 file 서울대 천정희
Math Colloquia Periodic orbits in symplectic geometry file 서울대 강정수
Math Colloquia Mixing time of random processes file 서울대 서인석
Math Colloquia Alice and Bob meet Banach and von Neumann file 서울대 이훈희
Math Colloquia <학부생을 위한 ɛ 강연> Convergence of Fourier series and integrals in Lebesgue spaces file 서울대 이상혁
Math Colloquia Arithmetic of elliptic curves file 서울대 김도형
Math Colloquia On the distributions of partition ranks and cranks file 서울과학기술대학교 김병찬
Math Colloquia Integer partitions, q-series, and Modular forms file 서울과학기술 대학 김병찬
Math Colloquia Averaging formula for Nielsen numbers file 서강대학교 이종범
Math Colloquia Symmetry Breaking in Quasi-1D Coulomb Systems file 서강대학교 Paul Jung
Math Colloquia Noncommutative Surfaces file 서강대학교 Jens Hoppe
Math Colloquia Regularity theory for non-autonomous elliptic equations in divergence form file 서강대학교 옥지훈
Math Colloquia Elliptic equations with singular drifts in critical spaces file 서강대학교 김현석
Math Colloquia Descent in derived algebraic geometry file 서강대학교 조창연
Math Colloquia Entropy of symplectic automorphisms file 서강대학교 김준태
Math Colloquia Q-curvature in conformal geometry file 서강대 Pak Tung Ho
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