莫爾斯理論入門 [An Invitation to Morse Theory] 下載 mobi epub pdf 電子書 2024

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內容簡介

As the the title suggests， the goal of this book is to give the reader a taste of the “unreasonable effectiveness” of Morse theory. The main idea behind thistechnique can be easily visualized.
Suppose M is a smooth， compact manifold， which for simplicity we as-sume is embedded in a Euclidean space E. We would like to understand basictopological invariants of M such as its homology， and we attempt a “slicing” technique.

目錄

Preface
Notations and conventions
1 Morse Functions
1.1 The Local Structure of Morse Functions
1.2 Existence of Morse Functions

2 The Topology of Morse Functions
2.1 Surgery，Handle Attachment.and Cobordisms
2.2 The Topology of Sublevel Sets
2.3 Morse Inequalities
2.4 Morse-Smale Dynamics
2.5 Morse-Floer Homology
2.6 Morse-Bott Functions
2.7 Min-Max Theory

3 Applications
3.1 The Cohomology of Complex Grassmannians
3.2 Lefschetz Hyperplane Theorem
3.3 Symplectic Manifolds and Hamiltonian Flows
3.4 Morse Theory of Moment Maps
3.5 S1-Equivariant Localization

4 Basics of Comple X Morse Theory
4.1 Some Fundamental Constructions
4.2 Topological Applications of Lefschetz Pencils
4.3 The Hard Lefschetz Theorem
4.4 Vanishing Cycles and Local Monodromy
4.5 Proofofthe Picard Lefschetz formula
4.6 Global Picard-Lefschetz Formulae

5 Exercises and Solutions
5.1 Exercises
5.2 Solutions to Selected Exercises
References
Index

前言/序言

　　As the the title suggests， the goal of this book is to give the reader a taste of the “unreasonable effectiveness” of Morse theory. The main idea behind thistechnique can be easily visualized.
　　Suppose M is a smooth， compact manifold， which for simplicity we as-sume is embedded in a Euclidean space E. We would like to understand basictopological invariants of M such as its homology， and we attempt a “slicing” technique.
　　We fix a unit vector u in E and we start slicing M with the family of hyperplanes perpendicular to u. Such a hyperplane will in general intersectM along a submanifold （slice）. The manifold can be recovered by continuouslystacking the slices on top of each other in the same order as they were cut out of M.
　　Think of the collection of slices as a deck of cards of various shapes. If welet these slices continuously pile up in the order they were produced， we noticean increasing stack of slices. As this stack grows， we observe that there aremoments of time when its shape suffers a qualitative change. Morse theoryis about extracting quantifiable information by studying the evolution of theshape of this growing stack of slices.

莫爾斯理論入門 [An Invitation to Morse Theory] 下載 mobi epub pdf txt 電子書 格式

評分☆☆☆☆☆

很不錯的書，內容豐富

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質量不錯，以後還會購買。

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應該是比較適閤初學者的書，而且也不是很厚，值得一看。

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莫爾斯

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的分支。它是H.M.莫爾斯在20世紀30年代創立的。由莫爾斯理論得知 ，微分流形與其上的光滑函數緊密相關，利用光滑函數不僅能研究微分流形的局部性質，而且某些光滑函數例如莫爾斯函數包含瞭刻劃流形整體性質的豐富信息。莫爾斯理論主要分兩部分，一是臨界點理論，一是它在大範圍變分問題上的應用。一個莫爾斯函數也是一個非簡諧振子的一種錶達法。

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印刷很好，封皮像後來粘上去的

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數學類基礎用書，值得參考

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1.1 相關定理及推論

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微分拓撲學的一個重要分支。通常指兩部分內容：一是微分流形上可微函數的莫爾斯理論，即臨界點理論；二是變分問題的莫爾斯理論，即大範圍變分法。

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