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 A new view of Fokker-Planck equations in finite and Infinite dimensional spaces file Bielefeld Univ./Purdue Univ. Michael Roeckner
BK21 FOUR Rookies Pitch 2021-2 Rookies Pitch: Geometric Topology (김경로) file BK21 김경로
BK21 FOUR Rookies Pitch 2022-1 Rookies Pitch: Harmonic Analysis (Kalachand Shuin) file BK21 Kalachand Shuin
BK21 FOUR Rookies Pitch 2023-1 Probabilistic Potential Theroy (강재훈) file BK21 강재훈
BK21 FOUR Rookies Pitch 2023-1 Geometric Toplology (정홍택) file BK21 정홍택
BK21 FOUR Rookies Pitch 2023-1 Algebraic Combinatorics (김동현) file BK21 김동현
BK21 FOUR Rookies Pitch 2021-1 Rookies Pitch: Financial Mathematics(전재기), PDE, Kinetic Equation(배기찬) file BK21 FOUR 전재기, 배기찬
Math Colloquia Counting circles in Apollonian circle packings and beyond file Brown Univ. 오희
Math Colloquia Entropies on covers of compact manifolds file CNRS (France) François Ledrappier
Special Colloquia Mathematical Analysis Models and Siumlations file Collège de France Pierre-Louis Lions
Special Colloquia Regularization by noise in nonlinear evolution equations file Dep. Math., Kyoto Univ. Yoshio Tsutsumi
Math Colloquia Quantum Dynamics in the Mean-Field and Semiclassical Regime file Ecole Polytechnique Francoise Golse
Special Colloquia Regularity of solutions of Hamilton-Jacobi equation on a domain file ENS-Lyon Albert Fathi
Special Colloquia What is Weak KAM Theory? file ENS-Lyon Albert Fathi
Math Colloquia Topological aspects in the theory of aperiodic solids and tiling spaces file Georgia Institute of Technology, School of Mathematics and School of Physics Jean V. Bellissard
Math Colloquia Creation of concepts for prediction models and quantitative trading file Haafor 이승환
Math Colloquia Convex and non-convex optimization methods in image processing file Hong Kong Baptist University Michael Ng
Math Colloquia The phase retrieval problem file Hong Kong University of Science and Technology Yang Wang
Math Colloquia Gromov-Witten-Floer theory and Lagrangian intersections in symplectic topology file IBS, 포항공과대학교 오용근
BK21 FOUR Rookies Pitch 2023-1 Dynamics and Number Theory (이슬비) file IBS-CGP 이슬비
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