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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.
Atachment
Attachment '1'
List of Articles
Category Subject Dept. Lecturer
Special Colloquia 허준이 교수 호암상 수상 기념 강연 (Lorentzian Polynomials) file Professor, Stanford University 허준이 교수
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Math Colloquia 학부생을위한ε강연: 수학자는 왜 선망되는 직업일까? file KAIST 김동수
Math Colloquia 학부생을 위한 강연회: 통신의 New Trend, 그리고 Big Data file KT 전무 양현미
Math Colloquia 학부생을 위한 강연회: What is the algebraic number theory? file KAIST 구자경
Math Colloquia 학부생을 위한 강연회: Tipping Point Analysis and Influence Maximization in Social Networks file KAIST 정교민
Math Colloquia 학부생을 위한 강연: 브라질과 프랑스는 왜 축구를 잘 할까? - 경제와 수학과 축구와 법률 file 서울대학교 법과대학 김화진
Math Colloquia 학부생을 위한 강연: 건축과 수학 file UI 건축사무소 위진복
Math Colloquia 학부생을 위한 강연: Introduction to partial differential equations file 서울대학교 변순식
Math Colloquia 학부생을 위한 강연: Choi's orthogonal Latin Squares is at least 61 years earlier than Euler's file 연세대학교 송홍엽
Math Colloquia 학부생을 위한 강연: A COMBINATORIAL FORMULA FOR INFORMATION FLOW IN A NETWORK file Univ. of Rhode Island/서울대학교 국웅
Math Colloquia 학부생을 위한 ε 강연회: Sir Isaac Newton and scientific computing file 서울대학교 신동우
Math Colloquia 학부생을 위한 ε 강연회: Mathematics from the theory of entanglement file 서울대학교 계승혁
Math Colloquia 학부생을 위한 ε 강연회: Constructions by ruler and compass together with a conic file 건국대/서울대 최인송
Special Colloquia 최고과학기술인상수상 기념강연: On the wild world of 4-manifolds file 서울대학교 박종일
Math Colloquia 정년퇴임 기념강연회: 숙제 file 서울대학교 지동표
Math Colloquia 정년퇴임 기념강연: 회고 file 서울대 김도한
Math Colloquia 정년퇴임 기념강연: Volume Conjecture file 서울대학교 김혁
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