# 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 Vlasov-Maxwell equations and the Dynamics of Plasmas 포항공과대학교 장진우
수학강연회 Volume entropy of hyperbolic buildings 서울대 임선희
수학강연회 W-algebras and related topics 서울대학교 서의린
수학강연회 Weak and strong well-posedness of critical and supercritical SDEs with singular coefficients University of Illinois Renming Song
수학강연회 Weyl character formula and Kac-Wakimoto conjecture 서울대 권재훈
수학강연회 WGAN with an Infinitely wide generator has no spurious stationary points 서울대학교 류경석
수학강연회 What happens inside a black hole? 고등과학원 오성진
수학강연회 What is model theory? 연세대 김병한
특별강연 What is Weak KAM Theory? ENS-Lyon Albert Fathi
수학강연회 Zeros of linear combinations of zeta functions 연세대학교 기하서
수학강연회 Zeros of the derivatives of the Riemann zeta function 연세대 기하서
수학강연회 곡선의 정의란 무엇인가? 서울대학교 김영훈
수학강연회 극소곡면의 등주부등식 KIAS 최재경
수학강연회 돈은 어떻게 우리 삶에 돈며들었는가? (불확실성 시대에 부는 선형적으로 증가하는가?) 농협은행 홍순옥
수학강연회 원의 유리매개화에 관련된 수학 건국대학교 최인송
수학강연회 젊은과학자상 수상기념강연: From particle to kinetic and hydrodynamic descriptions to flocking and synchronization 서울대학교 하승열
수학강연회 정년퇴임 기념강연: Volume Conjecture 서울대학교 김혁
수학강연회 정년퇴임 기념강연: 회고 서울대 김도한
수학강연회 정년퇴임 기념강연회: 숙제 서울대학교 지동표
특별강연 최고과학기술인상수상 기념강연: On the wild world of 4-manifolds 서울대학교 박종일
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