数学笔记

Poincare Conjecture and Elliptization Conjecture

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1. Poincare Conjecture

Theorem 1 (Poincare Conjecture). If a compact three-dimensional manifold $M^3$ has the property that every simple closed curve within the manifold can be deformed continuously to a point, does it follow that $M^3$ is homeomorphic to the sphere $S^3$?

2. Thurston Elliptization Conjecture
Theorem 2 (Thurston Elliptization Conjecture). Every closed 3-manifold with finite fundamental group has a metric of constant positive curvature and hence is homeomorphic to a quotient $S^3/\Gamma$, where $\mathrm{\Gamma} \subset \mathrm{SO(4)}$ is a finite group of rotations that acts freely on $S^3$. The Poincare Conjecture corresponds to the special case where the group $\Gamma \cong \pi_1(M^3)$ is trivial.

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Principle Bundle, Associated bundle, Gauge Group and Connections

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Let $M$ be compact riemannian manifold without boundary, and $G$ be a compact Lie group. A principle $G$-bundle over $M$, denoted as $P(M,G)$ is a manifold $P$ with a free right action $P\times G\ni(p,g)\mapsto pg\in P$ of $G$ such that $M=P/G$, and $P$ is locally trivial, i.e., for every point $x\in M$, there is a neighbourhood $U$ such that the primage $\pi^{-1}(U)$ of the canonical projection is isomorphic to $U\times G$ in the sense that it preserver the fiber and $G$-equivariant, more precisely, there is a diffeomorphism $\Phi:\pi^{-1}(U)\to U\times G$ such that $\Phi(p)=(\pi(p),\phi(p))$ and $\phi:\pi^{-1}(U)\to G$ satisfying $\phi(pg)=(\phi(p))g$ for all $p\in\pi^{-1}(U)$ and $g\in G$.

Now for any other manifold $F$ on which $G$ acts on the left $G\times F\ni(g,f)\mapsto gf\in F$, the associated fiber bundle $P\times_\rho F$ is the quotient space $P\times F/\sim$, where $[p,f]\sim[pg,g^{-1}f]$, for all $g\in G$. With the projection $\tilde\pi:[p,g]\mapsto\pi(p)$ this is a principle bundle over $M$ with typical fiber $F$. The local trivialization is induced by the one $\Phi:\pi^{-1}(U)\to U\times G$ as Continue Reading →

[转载]What’s a Gauge?

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From: Terence Tao’s blog: What’s a Gauge.

“Gauge theory” is a term which has connotations of being a fearsomely complicated part of mathematics – for instance, playing an important role in quantum field theory, general relativity, geometric PDE, and so forth. But the underlying concept is really quite simple: a gauge is nothing more than a “coordinate system” that varies depending on one’s “location” with respect to some “base space” or “parameter space”, a gauge transform is a change of coordinates applied to each such location, and a gauge theory is a model for some physical or mathematical system to which gauge transforms can be applied (and is typically gauge invariant, in that all physically meaningful quantities are left unchanged (or transform naturally) under gauge transformations). By fixing a gauge (thus breaking or spending the gauge symmetry), the model becomes something easier to analyse mathematically, such as a system of partial differential equations (in classical gauge theories) or a perturbative quantum field theory (in quantum gauge theories), though the tractability of the resulting problem can be heavily dependent on the choice of gauge that one fixed. Deciding exactly how to fix a gauge (or whether one should spend the gauge symmetry at all) is a key question in the analysis of gauge theories, and one that often requires the input of geometric ideas and intuition into that analysis. Continue Reading →

几何分析中的变分问题与方法

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这是丁伟岳院士的一个Talk, 原文链接在其主页上有: 几何分析中的变分问题与方法.

1. 历史的回顾:1960年以前 变分法有很长的历史，如果从欧拉和拉格朗日提出以他们的名字命名的变分方程算起，至今己有250年的历史。在开始的时候，变分法的创立和应用主要是围绕物理学(力学，光学，天文学等等 )中的各种变分问题。比如，与拉普拉斯方程相联系 的Dirichlet原理就是在研究引力或电场的位势时提出的。
变分法对于几何的应用在早期主要是对曲面上的测地线和欧氏空间中给定边界的极小曲面(Plateau问题 )的研究。但在很长时期内仅限于一些特殊情形，没有重要进展。
直到上世纪早期，为了研究曲面上的测地线的个 数，Morse(20-30年代)和俄国数学家(40年代)分别建立了Morse和 Ljusternik-Schnirelman理论。其中，Morse理论不仅对变分问题的解的个数估计有许多应用而且在流形的拓扑问题有重要应用。 Continue Reading →