# 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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1. ### Irreducible Plane Curve Singularities

It is very interesting to study what problems can be computed in irreducible plane curve singularities in algebraicgeometry? Then, the aim of this talk is to compute the explicit algorithm for finding the correspondence between the family of...
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2. ### Analytic torsion and mirror symmetry

In the early 90's, physicists Bershadsky-Cecotti-Ooguri-Vafa conjectured that the analytic torsion was the counterpart in complex geometry of the counting problem of elliptic curves in Calabi-Yau threefolds. It seems that this conjecture is ...
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3. ### Fefferman's program and Green functions in conformal geometry

Motivated by the analysis of the singularity of the Bergman kernel of a strictly pseudoconvex domain, Charlie Fefferman launched in the late 70s the program of determining all local biholomorphic invariants of strictly pseudoconvex domain. T...
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4. ### 최고과학기술인상수상 기념강연: On the wild world of 4-manifolds

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...
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6. ### Queer Lie Superalgebras

The Lie superalgebra q(n) is the second super-analogue of the general Lie algebra gl(n). Due to its complicated structure, q(n) is usually called “the queer superalgebra”. In this talk we will discuss certain old and new results related to t...
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7. ### Regularization by noise in nonlinear evolution equations

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8. ### A New Approach to Discrete Logarithm with Auxiliary Inputs

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9. ### Contact topology and the three-body problem

In this talk, we discuss recent work with Albers, Cieliebak, Fish, Frauenfelder, Hofer and Paternain on several aspects of the three body problem. The ultimate goal of this project is to use modern, holomorphic curve techniques to investigat...
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10. ### Harmonic bundles and Toda lattices with opposite sign

In this talk, we shall discuss the semi-infinite variation of Hodge structure associated to real valued solutions of a Toda equation. First, we describe a classification of the real valued solutions of the Toda equation in terms of their par...
Category특별강연 소속RIMS, Kyoto Univ. 강연자Takuro Mochizuki
11. ### Mathematical Analysis Models and Siumlations

In this talk, we shall first present several examples of numerical simulations of complex industrial systems. All these simulations rely upon some mathematical models involving Partial Differential Equations and we shall briefly explain the ...
Category특별강연 소속Collège de France 강연자Pierre-Louis Lions
12. ### Connes's Embedding Conjecture and its equivalent

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Category수학강연회 소속RIMS 강연자Narutaka Ozawa
13. ### Connectedness of a zero-level set as a geometric estimate for parabolic PDEs

Studies on PDEs are mostly focused on ?nding properties of PDEs within a speci?c discipline and on developing a technique specialized to them. However, ?nding a common structure over di?erent disciplines and unifying theories from di?erent s...
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14. ### Combinatorial Laplacians on Acyclic Complexes

The main topic of the talk is a determinantal formula for high dimensional tree numbers of acyclic complexes via combinatorial Laplace operators . This result is a generalization of Temperley's tree number formula for graphs, motivated by a ...
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15. ### 학부생을 위한 ε 강연회: Mathematics from the theory of entanglement

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16. ### L-function: complex vs. p-adic

Several L-functions with the names Dirichlet, Dedekind, Elliptic, and so on usually have p-adic counterparts, so called p-adic L-functions, which share many similar properties such as an evaluation formula at s=1, class number formula, and e...
Category수학강연회 소속충북대학교 강연자선해상
17. ### 학부생을 위한 ε 강연회: Sir Isaac Newton and scientific computing

아이작 뉴턴의 계산과학에 관한 업적을 소개한다.
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18. ### A brief introduction to stochastic models, stochastic integrals and stochastic PDEs

Stochastic Model, Integrals 그리고 PDE의 기본 개념과 최신 연구 동향을 소개한다.
Category수학강연회 소속고려대학교 강연자김경훈