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# 최고과학기술인상수상 기념강연: On the wild world of 4-manifolds

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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.
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List of Articles
Category Subject Dept. Lecturer
Special Colloquia Mathematical Analysis Models and Siumlations Collège de France Pierre-Louis Lions
Special Colloquia Regularization by noise in nonlinear evolution equations Dep. Math., Kyoto Univ. Yoshio Tsutsumi
Special Colloquia What is Weak KAM Theory? ENS-Lyon Albert Fathi
Special Colloquia Regularity of solutions of Hamilton-Jacobi equation on a domain ENS-Lyon Albert Fathi
Special Colloquia 허준이 교수 호암상 수상 기념 강연 (Lorentzian Polynomials) Professor, Stanford University 허준이 교수
Special Colloquia Harmonic bundles and Toda lattices with opposite sign RIMS, Kyoto Univ. Takuro Mochizuki
Special Colloquia Persistent Homology Stanford University Gunnar E. Carlsson
Special Colloquia Structures on Persistence Barcodes and Generalized Persistence Stanford University Gunnar E. Carlsson
Special Colloquia Topological Mapping of Point Cloud Data Stanford University Gunnar E. Carlsson
Special Colloquia Queer Lie Superalgebras Univ. of Texas, Arlington Dimitar Grantcharov
Special Colloquia Algebraic surfaces with minimal topological invariants 고등과학원 금종해
Special Colloquia Combinatorics and Hodge theory 미국 프린스턴대 교수, 한국 고등과학원 석학교수 허준이
Special Colloquia Contact topology and the three-body problem 서울대학교 Otto van Koert
Special Colloquia A New Approach to Discrete Logarithm with Auxiliary Inputs 서울대학교 천정희
Special Colloquia 최고과학기술인상수상 기념강연: On the wild world of 4-manifolds 서울대학교 박종일
Special Colloquia Irreducible Plane Curve Singularities 서울대학교 강정혁
Special Colloquia A wrapped Fukaya category of knot complement and hyperbolic knot 포항공대 오용근
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