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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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첨부 '1'
List of Articles
카테고리 제목 소속 강연자
수학강연회 Conformal field theory and noncommutative geometry file 동경대학교 Kawahigashi
수학강연회 Conformal field theory in mathematics file 고등과학원 강남규
수학강연회 Congruences between modular forms file 서울대 유화종
수학강연회 Connectedness of a zero-level set as a geometric estimate for parabolic PDEs file KAIST 김용정
수학강연회 Connes's Embedding Conjecture and its equivalent file RIMS Narutaka Ozawa
수학강연회 Conservation laws and differential geometry file Univ. of Wisconsin Marshall Slemrod
수학강연회 Contact Homology and Constructions of Contact Manifolds file 서울대 Otto van Koert
수학강연회 Contact instantons and entanglement of Legendrian links file IBS-CGP /POSTECH 오용근
특별강연 Contact topology and the three-body problem file 서울대학교 Otto van Koert
수학강연회 Contact topology of singularities and symplectic fillings file 순천대학교 권명기
수학강연회 Convex and non-convex optimization methods in image processing file Hong Kong Baptist University Michael Ng
수학강연회 Counting circles in Apollonian circle packings and beyond file Brown Univ. 오희
수학강연회 Counting number fields and its applications file UNIST 조재현
수학강연회 Creation of concepts for prediction models and quantitative trading file Haafor 이승환
수학강연회 Deformation spaces of Kleinian groups and beyond file Osaka University Kenichi Ohshika
수학강연회 Descent in derived algebraic geometry file 서강대학교 조창연
수학강연회 Diophantine equations and moduli spaces with nonlinear symmetry file 서울대학교 황준호
수학강연회 Elliptic equations with singular drifts in critical spaces file 서강대학교 김현석
수학강연회 Entropies on covers of compact manifolds file CNRS (France) François Ledrappier
수학강연회 Entropy of symplectic automorphisms file 서강대학교 김준태
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