## Poincare Conjecture and Elliptization Conjecture

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.

## Principle Bundle, Associated bundle, Gauge Group and Connections

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?

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

## 标准的椭圆理论:一个能量不等式

Proposition 1. 假设$u$是方程
$$0=\Delta u-\frac{1}{2}x\cdot \nabla u.$$

$$\int_{|x|< r}e^{-\frac{|x|^2}{4}}|\nabla u|^2\rd x\leq\frac{c}{r^2}\int_{r< |x|< 2r}e^{-\frac{|x|^2}{4}}u^2\rd x,\quad\forall r >0.$$