## The Peter-Weyl theorem

From Tao’s Blog: The Peter-Weyl theorem, and non-abelian Fourier analysis on compact groups.

Let $G$ be a compact group. (Throughout this post, all topological groups are assumed to be Hausdorff.) Then $G$ has a number of unitary representations, i.e. continuous homomorphisms $\rho: G \rightarrow U(H)$ to the group $U(H)$ of unitary operators on a Hilbert space $H$, equipped with the strong operator topology. In particular, one has the left-regular representation $\tau: G \rightarrow U(L^2(G))$, where we equip $G$ with its normalised Haar measure $\mu$ (and the Borel $\sigma$-algebra) to form the Hilbert space $L^2(G)$, and $\tau$ is the translation operation $$\tau(g) f(x) := f(g^{-1} x).$$ (阅读全文 …)

## Weak Convergence in Sobolev Spaces

Suppose $\Omega\subset\R^n$ and donote $W^{1,p}:=W^{1,p}(\Omega)$ be the sobolev space for some $1< p< +\infty$. Recall that $f_i\in W^{1,p}$ convergent weakly to $f\in W^{1,p}$, if for any $\phi$ in the dual space of $W^{1,p}$, we have $\inner{f_i,\phi}\to\inner{f,\phi}$, denote as $f_i\weakto f$. This is distinguished by strongly convergence, as we use the dual normal instead of $W^{1,p}$ normal.
Proposition 1.  If $f_i\weakto f$ in $W^{1,p}$, then $f_i\to f$ in $L^p$.
(阅读全文 …)

## Overview of moduli spaces,review of G-bundles and connections

All the contents are from this wikisite, which is aimed to have a E-version lecture notes of the seminar given by Prof. Mrowka. All rights are reserved by the original wikisite, any reprint should be indicate this. Main Contributor:Christian

These lecture notes are based on notes from the 18.999 geometry seminar class taught by Tomasz Mrowka and from the IAS/Park city Mathematic series book Gauge theory and the topology of four-manifolds 1. Moduli spaces A moduli space can be viewed as a geometric object which classifies the solutions of some problem. For example, there is the moduli space of $n$-tuples of points in $I=[0,1]$,that is, points in $I^{n}$ modulo symmetric transformations:$I^{n}/ Sym_{n}$. Given a complex vector space $V$ of dimension $n$, we can look at the space of endomorphisms of $V$ modulo isomorphisms: (阅读全文 …)

## 图示一个三角函数(Triangle function)和阶梯函数(Step function)的卷积

和以前一样, 你可以随意移动定位点, 对于远离原点的定位点, 你可以使用绘图区域的不同选项来观察整个图形与局部细节.

## 流形上Laplace算子局部表达式

### 一般张量场的散度

假设$(M,g)$是一个黎曼流形, 对任意的$S\in \Gamma(T_s^rM)$是一个$r$阶反变, $s$阶协变张量场. $\newcommand{\ppt}[1]{\frac{\pt}{\pt{#1}}}$ 我们可以定义$S$的散度如下: $$\div(S)=\tr(X\to\nabla_X S),$$ 其中$X\in\Gamma(TM)$. 它是$\Gamma(T^r_sM)\to\Gamma(T^{r-1}_sM)$的线性映射. (阅读全文 …)

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