l*******n
发帖数: 33 | 1 抱着学习数学知识的态度查看了一下她/他的豆瓣
看到有同学/朋友在其下留言讨论这学期上课的教师优劣，提到的名字都是中科大的老师
所以大约她/他应该是在中科大读phd，而不是在香港 | y**k
发帖数: 222 | | i****g
发帖数: 3896 | 3 她/他又不是做数论的 反而是最鄙视做数论的 呵呵
【在 y**k 的大作中提到】
: 中科大在余红兵离开后，已经没人做数论了
| S*****a
发帖数: 190 | 4 你这是老黄历了。科大现在有欧阳毅，还有几位年轻人。
【在 y**k 的大作中提到】
: 中科大在余红兵离开后，已经没人做数论了
| R******9
发帖数: 39 | 5 看了她的一些留言，她自己提到以前在南京一所学校上本科，但不在数学系，后来自己
用功，在本科期间学习了很多，我觉得她掌握得还是挺系统深入的，后来申请去了香港
中文大学，现在做代数几何。她自己眼光比较高，而且确实勤奋聪明，是个很优秀的学
生。但是对于很多数学家的评价太偏激，对于很多数学领域持一种偏见，表现在语
言上是很鄙视，有时候会骂其他人。
老师
【在 l*******n 的大作中提到】
: 抱着学习数学知识的态度查看了一下她/他的豆瓣
: 看到有同学/朋友在其下留言讨论这学期上课的教师优劣，提到的名字都是中科大的老师
: 所以大约她/他应该是在中科大读phd，而不是在香港
| D**o
发帖数: 2653 | 6 这是她写的
我想简单讲一下为什么念书不需要做题。假设题目你会做，那你做它干嘛，没必要炫耀
自己，应该踏踏实实多学东西。假设你不会做，反正你也做不出来，浪费那时间做什么
？应该抓紧这些时间提高自己的水平，这样今后你就会做了，否则你永远不会做。综上
，读书不应该做题。
【在 R******9 的大作中提到】
: 看了她的一些留言，她自己提到以前在南京一所学校上本科，但不在数学系，后来自己
: 用功，在本科期间学习了很多，我觉得她掌握得还是挺系统深入的，后来申请去了香港
: 中文大学，现在做代数几何。她自己眼光比较高，而且确实勤奋聪明，是个很优秀的学
: 生。但是对于很多数学家的评价太偏激，对于很多数学领域持一种偏见，表现在语
: 言上是很鄙视，有时候会骂其他人。
:
: 老师
| D**o
发帖数: 2653 | 7 注意作者 \author{YHBKJ}
Atiyah-Bott Localization 1
2012-09-05 09:24:19
\documentclass[a4paper,12pt]{article}
\usepackage{amsfonts}
\usepackage{amsmath,amsthm,amssymb}
\usepackage{CJK,graphicx}
\usepackage{amscd}
\usepackage{amssymb}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{corollary}{Corollary}[section]
\newtheorem{definition}{Definition}[section]
\newtheorem{lemma}{Lemma}[section]
\begin{document}
\title{\textbf{\Huge{Atiyah-Bott Localization 1}}}\author{YHBKJ}\date{}\
maketitle
\begin{abstract}
{This is the first one of a series of notes which will be devoted to
equivariant cohomology theory. We give here a detailed derivation of the
abstract localization theorem due to Atiyah and Bott.}
\end{abstract}
\tableofcontents
\section{Introduction}
Let $M$ be a compact differentiable manifold and $X$ a vector field with
isolated zeros $\{x_i\}$ on $M$, we recall the Poincar\'{e}-Hopf index
theorem:
\begin{equation}\sum_i\textrm{ind}_X(x_i)=\chi(M),\end{equation}
where the index is defined to be the mapping degree in the local sense. This
gives the simplest example of a localization theorem.\\
In general, localization theorems aim to express the topological invariants
of $M$ in terms of the local information at the zero set of certain vector
fields. By elementary differential geometry, the space of vector fields on $
M$ form a Lie algebra of the diffeomorphism group $\textrm{Diff}(M)$. Taking
into account the definition of a Lie derivative, one sees immediately that
the zero set of a vector field corresponds exactly to the fixed points of a
Lie subgroup $G\subset\textrm{Diff}(M)$. This shows the motivation for
considering the Lie group action and using the equivariant cohomological
methods in the modern investigation of localization theorems.\\
In section 2 we consider the smooth action of a compact Lie group $G$ on a
manifold $M$, this leads to the slice theorem, which provides a nice local
description of the $G$-actions. From this we then deduce the finiteness of
the isotropy subgroups when $M$ is compact, a fact which will be used to
establish the Atiyah-Bott localization theorem.\\
Section 3 is a brief account of the equivariant cohomology theory. We begin
with Borel's fundamental construction and then introduce the Cartan model
for equivariant differential forms. The Cartan model enables us to do
explicit calculations, and more importantly, by regarding it as a double
complex, the basic spectral sequence techniques can be applied. One of the
most natural examples of a free $G$-space is the principal bundle. This fact
provides us a simple and natural approach to characteristic classes, and we
will describe it in $\S$3.2. Applying essentially the same method on a $K\
times G$-space with a free $K$ action, we then define the equivariant
characteristic classes. The equivariant Euler class will play a central role
in the Atiyah-Bott localization theorem.\\
We present the essential algebraic preparations for proving the Atiyah-Bott
localization theorem in Section 4. Although these materials are contained in
every introductory level course of commutative algebra, the method of using
the Zariski topology on $\textrm{Spec}A$ of a ring $A$ to visualize the
torsion of an $A$-module may not be familiar to less-advanced readers.\\
The derivation in detail of the Atiyah-Bott localization theorem will be
presented in Section 5. Before this, we shall prove a theorem which allows
us to focus our attention on torus actions without much loss of generality.
We should remark that this is the only proof in this article which is not so
elementary and the less-experienced reader may skip it in their first
reading.
\section{Slice Theorem}
Let $M$ be a smooth manifold and $G$ a compact Lie group acts smoothly on $M
$. It's well-known that the tubular neighborhood of a submanifold $S\subset
M$ can be identified with the normal bundle $NS$ of $S$ in $M$.\\
Now suppose $S$ is $G$-invariant, the above fact can be modified by
constructing a $G$-invariant tubular neighborhood of $S$. First notice that
the action of $G$ on $TM$ can be reduced to a subgroup action of $O(n)$. In
fact, endow $M$ with a Riemannian metric $g_{ij}$, since $G$ is compact, we
can average $g_{ij}$ on $G$ to get a $G$-invariant metric. From this we
deduce the existence of a $G$-equivariant diffeomorphism $\rho$ between $NS$
and a neighborhood of its zero section. In fact, recall that the
diffeomorphism between a vector bundle $V$ and its unit ball bundle $BV$ is
given by
\begin{equation}\upsilon:BV\rightarrow V,\textrm{ }y\mapsto\frac{y}{\big(1-\
|y\|^2\big)^\frac{1}{2}}\end{equation}
on each fiber, since $G$ acts isometricly, the $G$-equivariance of $\upsilon
$ follows easily. For every $p\in S$, we may attach an $\epsilon>0$ such
that the exponential map restricts to a diffeomorphism on the ball of radius
$\epsilon$ in $N_pS$. We get from this a neighborhood $U$ of $S$ in $NS$.
The desired $G$-equivariant diffeomorphism is given by $\exp\circ\rho$. We
have proved:
\begin{theorem}
Suppose a compact Lie group $G$ acts smoothly on $M$. Any $G$-invariant
submanifold $S$ has a $G$-invariant tubular neighborhood.
\end{theorem}
Two most important examples of a $G$-invariant submanifold are the fixed
point set $F$ and the orbits $Gp$. In fact, applying Theorem 2.1 to every $x
\in F$ shows that $F\subset M$ is a submanifold, while the fact that $Gp$ is
a submanifold of $M$ lies in the following theorem:
\begin{theorem}[Slice theorem]
Suppose a compact Lie group $G$ acts smoothly on a manifold $M$. Let $p\in M
$. Then there is a vector space $V_p$ on which the isotropy group $G_p$ acts
linearly and a $G$-equivariant embedding $G\times_{G_p}V_p\rightarrow M$
onto an open set which sends $(g,0)$ to $gp$.
\end{theorem}
\begin{proof}
The map $g\mapsto gp$ gives an embedding $G/G_p\rightarrow M$, therefore $Gp
$ carries the structure of a $G$-invariant submanifold of $M$. The theorem
then follows easily from Theorem 2.1.
\end{proof}
See \cite{gb} for a general version of this theorem. Assume $M$ is compact,
then it can be covered by a finite number of $U_p$, where $U_p$ is a $G$-
invariant tubular neighborhood of some $Gp$. By slice theorem, $G_q$ acts
linearly on $V_q$ for every $q\in Gp$, therefore there are only finitely
many possibilities for $G_q$. Using the $G$-equivariance of the projection $
U_p\rightarrow G/G_p$, it follows immediately that $G_x\subset G_p$ for
every $x\in U_p$. That is to say there are only finitely many choices of $G_
x$ for $x\in U_p$. These observations lead to the following important
corollary:
\begin{corollary}
For $M$ a compact manifold and $G$ a compact Lie group acts smoothly on $M$,
only a finite number of subgroups of $G$ can occur as isotropy groups of
points of $M$.
\end{corollary}
\section{Equivariant Cohomology}
\subsection{Basic Constructions}
The theory of equivariant cohomology aims to describe the invariants of a
topological space $M$ with a group action, therefore a prime candidate to
consider is $H^\ast(M/G)$, where $G$ is a topological group. However, the
shortcoming of this construction is easily seen, namely it overlooks the
isotropy groups by considering only the cohomology of the orbit space.
Therefore we would like to consider a free $G$-space which preserves full
information about the $G$-action on $M$. The trick, which is called the
Borel construction, is simply to use $M_G\triangleq EG\times_GM$ to replace
$M/G$, where $EG\rightarrow BG$ is the universal $G$-bundle and $EG\times_GM
$ is the quotient of $EG\times M$ by the diagonal action of $G$. Notice that
since $G$ acts on $EG$ freely, $M_G$ is a free $G$-space. Note also that
since $EG$ is contractible, whenever $M$ is a free $G$-space, we have the
isomorphism $H^\ast(M_G)\simeq H^\ast(M/G)$. We call $H^\ast(M_G)$ the \
textit{equivariant cohomology} of the $G$-space $M$, and denote it by $H_G^\
ast(M)$.\\
It follows directly from the above definition that $H_G^\ast$ constitutes a
contravariant functor from the category of $G$-spaces to the category of
modules over the base ring $H_G^\ast(pt)$. In fact, such a module structure
is given by the natural projection $\pi:M_G\rightarrow BG$, which is easily
seen to be a bundle over $BG$ with fiber $M$.\\
We now restrict ourselves to the case when $M$ is a smooth manifold and $G$
is a compact Lie group. To do explicit computations it is very convenient to
use an earlier construction due to H. Cartan. The idea is to use de Rham
cohomology $H_{DR}^\ast(M,\mathbb{C})$ instead of singular cohomology $H^\
ast(M,\mathbb{C})$, thus to store the information of the $G$-action on
differential forms. This leads naturally to the concept of equivariant
differential forms.\\
In fact, recall that for a principal $G$-bundle $P$, the Maurer-Cartan form
is a map $\theta:C^\infty(P,TP)\rightarrow\mathfrak{g}$ such that $\textrm{
Ad}(g)(r_g^\ast\theta)=\theta$, where $r_g$ is the right translation. This
can of course be identified with a map $\theta:\mathfrak{g}\rightarrow\Omega
^1(P)$ and can be easily generalized to define $G$-equivariant differential
forms on an arbitrary $G$-manifold $M$.
\begin{definition}
An equivariant differential form $\theta$ is a map $\theta:\mathfrak{g}\
rightarrow\Omega^\ast(M)$ such that the following diagram commutes:
\[\begin{CD}\mathfrak{g} @> \theta >> \Omega^\ast(M)\\
@V Ad(g)VV @VV g V\\
\mathfrak{g} @>>\theta > \Omega^\ast(M)
\end{CD}\]
i.e. $\theta$ is $G$-invariant.
\end{definition}
The $G$-equivariant differential forms form a graded algebra $\big(C^\infty(
\mathfrak{g}^\ast)\otimes\Omega^\ast(M)\big)^G$ which inherits the grading
of $\Omega^\ast(M)$. Since $d$ commutes with pullbacks, the exterior
differential of a $G$-equivariant form must be $G$-equivariant, therefore $\
Big(\big(C^\infty(\mathfrak{g}^\ast)\otimes\Omega^\ast(M)\big)^G,d\Big)$ can
be identified with a subcomplex of the de Rham complex. The cohomology of
this complex is thus a module over $\mathbb{C}$. On the other hand, by the
Chern-Weil theorem, $S(\mathfrak{g}^\ast)^G\simeq H^\ast(BG)$, where $S$
denotes the symmetric algebra. This means that $H_G^\ast(M)$ is a $S(\
mathfrak{g}^\ast)^G$-module. To obtain a de Rham model for equivariant
cohomology, it's then natural to consider the subalgebra $\big(S(\mathfrak{g
}^\ast)\otimes\Omega^\ast(M)\big)^G$. This subalgebra can be made into a $\
mathbb{Z}$-graded algebra by defining the degree of its element $\theta=f\
otimes\omega$ as
\begin{equation}\deg\theta=\deg\omega+2\deg f,f\in S(\mathfrak{g}^\ast)^G,\
omega\in\Omega^\ast(M)^G.\end{equation}
Replacing $d$ by $d_G=d-\iota_X$, where $\iota_X$ denotes the contraction
with $X\in\mathfrak{g}$, we finally arrived at a graded complex $\Big(\big(S
(\mathfrak{g}^\ast)\otimes\Omega^\ast(M)\big)^G,d_G\Big)$. Note that the
operator $\iota_X$ will increase the polynomial degree by 1 while at the
same time decrease the differential form degree by 1, this justifies the $2$
appeared before $\deg f$ in the above definition. With the homotopy formula
of E. Cartan, it's easy to check that $d_G$ is well-defined.
\begin{theorem}[H. Cartan]
Let $G$ be a compact Lie group and $M$ a compact $G$-manifold, the following
isomorphism holds:
\begin{equation}H_G^\ast(M,\mathbb{C})\simeq H^\ast\Big(\big(S(\mathfrak{g}^
\ast)\otimes\Omega^\ast(M)\big)^G,d_G\Big).\end{equation}
\end{theorem}
There is in fact a third model for equivariant cohomology, which is named
after Weil. The isomorphism between the Cartan model and the Weil model was
established by Mathai and Quillen. We postpone these materials to subsequent
notes.
\subsection{Characteristic Classes}
We take the equivariant cohomological point of view which was described in \
cite{gs}, see also \cite{bgv} for a differential geometric approach. For a
principal $G$-bundle $P$ over a smooth manifold $M$, consider the $G$-
projection $P\rightarrow pt$. Since $H_G^\ast$ is a contravariant functor,
it induces a ring homomorphism
\begin{equation}w:S(\mathfrak{g}^\ast)^G\rightarrow H^\ast(M),\end{equation}
which is called the \textit{Chern-Weil homomorphism}.\\
Suppose $G\subset GL(n,\mathbb{C})$, then we can consider its associated
vector bundle $E$. If $E$ admits a complex structure, the structure group $G
$ can be reduced to $U(n)$ by endowing $E$ with a hermitian metric. It can
be shown by a simple homotopical argument that different choices of
hermitian metrics actually induce the same Chern-Weil map, therefore the
Chern-Weil homomorphism determine exactly the topological invariants of $P$.
Consider the characteristic polynomial $\det(\lambda-A)$ of $A\in\mathfrak{
u}(n)$, we get a sequence of $U(n)$-invariant polynomials $\big\{c_i(A)\big\
}$, their images under the Chern-Weil homomorphism give the Chern classes $c
_i(E)$.\\
Similarly, endowing $E$ with a Riemannian metric reduces the structure group
$G$ to $O(n)$. The characteristic polynomial of $A\in\mathfrak{o}(n)$
induces the Pontryagin polynomials $\big\{p_i(A)\big\}$, whose image under
the Chern-Weil homomorphism are Pontryagin classes $p_i(E)$.\\
When $E$ is even-dimensional and orientable, $G$ can be reduced to $SO(2n)$.
In this case, apart from the Pontryagin polynomials there is one additional
invariant \textit{Pf} called the Pfaffian. The image of \textit{Pf} in the
cohomology ring gives the Euler class $e(E)$.\\
The following relations between characteristic classes are well-known:
\begin{equation}\label{eq:rel}e(E)\smile e(E)=p_{\frac{n}{2}}(E),e(E)=c_n(E)
.\end{equation}
With a similar method we introduce the equivariant characteristic classes.
Let $K$ and $G$ be compact Lie groups and $P$ a $K\times G$-manifold with a
free $K$-action, i.e. a free-$K$ manifold on which $G$ acts, such that the
actions of $K$ and $G$ commute. We can therefore identify $P$ with a
principal $K$-bundle over a $G$-manifold $M$. This time the $K$-projection $
P_G\rightarrow pt$ induces the ring homomorphism
\begin{equation}w_G:S(\mathfrak{k}^\ast)^K\rightarrow H_K^\ast(P_G)\simeq H_
G^\ast(M),\end{equation}
which we shall call the \textit{equivariant Chern-Weil homomorphism}.\\
For a complex vector bundle $E\rightarrow M$, we can always endow $E$ with a
$G$-invariant hermitian metric due to the compactness of $G$, this reduces
the structure group $K$ to $U(n)$. The image of the Chern polynomials $c_i(A
)$ under this map will be called equivariant Chern classes. Similarly we can
define equivariant Pontryagin classes and the equivariant Euler class. It
follows directly from our definition that the relations (\ref{eq:rel}) still
hold in the equivariant settings.\\
It's not hard to see the Whitney product formulas also hold in this setting.
Consider a connected component $F_i$ of the fixed point set $F$ of a torus
action $T$ on $M$. Since $F_i$ is connected, the isotropy representation of
$T$ on each fiber $N_p$ of the normal bundle $NF_i$ stay the same. For a
point $p\in F_i$, $H_T^2(pt,\mathbb{Z})=H^2(BT,\mathbb{Z})$, but $H^2(BT,\
mathbb{Z})$ can be identified with the line bundles on $BT$ by a simple
sheaf cohomological argument. Since a line bundle on $BT$ can in turn be
identified with a 1-dimensional complex representation of $T$, we get an
identification between the characters of $T$ and $H_T^2(pt,\mathbb{Z})$. By
applying the Whitney product formula we get the following result:
\begin{theorem}
The equivariant Euler class $e_T(N_p)$ equals the the product of weights of
the isotropy representation of $T$ for all $p\in F_i$.
\end{theorem}
For an alternative proof of Theorem 3.2, see \cite{gs}. This fact will be
used to establish the Atiyah-Bott localization formula in $\S$5.2.
\section{Localization of Rings and Modules}
In this section, $A$ will always be understood as a commutative ring with
unit. Let $S$ be a \textit{multiplicative set} in $A$, i.e., a set contains
$1$ such that whenever $x,y\in S$, $xy\in S$.
\begin{definition}
Suppose $f:A\rightarrow B$ is a ring homomorphism satisfying the following
two conditions:
\begin{itemize}
\item[(1)] $f(x)$ is a unit of $B$ for all $x\in S$;
\item[(2)] if $g:A\rightarrow C$ is a ring homomorphism satisfying (1), then
there exists a unique homomorphism $h:B\rightarrow C$ such that $g=hf$;
\end{itemize}
from (2) one sees immediately that $B$ is determined uniquely up to
isomorphism, it is called the localization of $A$ with respect to $S$. We
write $A_S$ for $B$ and $f:A\rightarrow A_S$ will be referred to as the
canonical map.
\end{definition}
We give a constructive proof of the existence of $B$. In fact, one can
define a equivalence relation $\sim$ on $A\times S$ by $(a,s)\sim (b,s')$ if
and only if $t(s'a-sb)=0$ for some $t\in S$. Then $A\times S/\sim$ is the
required ring $B$ with sums and products defined by
\begin{equation}a/s+b/s'=(as'+bs)/ss',(a/s)\cdot(b/s')=ab/ss',\end{equation}
where we have written $a/s$ for the equivalence class of $(a,s)$.\\
From the above construction it's easy to see
\begin{equation}\ker f=\{a\in A|sa=0\textrm{ for some }s\in S\},\end{
equation}
i.e. $f$ is injective if and only if $S$ contains no zero divisor of $A$.
Note that the set of all non-zero-divisors of $A$ is a multiplicative set,
and localization with respect to this set is in some sense the most thorough
one, which is called the \textit{total ring of fractions} of $A$.\\
The importance of localization lies in the fact that if $A$ is Noetherian or
Artinian, so is $A_S$.\\
We now describe the sheaf on the spectrum of $A$ raised by its localizations
. Recall that the \textit{spectrum} of $A$ is the collection of its prime
ideals. Let $I$ be an ideal of $A$, set
\begin{equation}V(I)=\{\mathfrak{p}\in\textrm{Spec}A|\mathfrak{p}\supset I\}
.\end{equation}
One verifies easily that
\begin{equation}V(I_1)\cup V(I_2)=V(I_1I_2),\bigcap_\alpha V(I_\alpha)=V(\
sum_\alpha I_\alpha).\end{equation}
From this we see there is a topology on $\textrm{Spec}A$ with $\mathcal{F}=\
big\{V(I)\big\}$ as the collection of its closed sets. We call this topology
the \textit{Zariski topology}. It's easy to see that the family of $U_a=\{\
mathfrak{p}\in\mathrm{Spec}A|a\notin\mathfrak{p}\}$ form a basis of the
Zariski topology. Consider the localization of $A$ with respect to the
multiplicative set $S_a=\{1,a,a^2,\cdot,\cdot,\cdot\}$, a sheaf $\mathcal{A}
$ can be defined on $\textrm{Spec}A$ by the assignment $\mathcal{A}(U_a)=A_{
S_a}$.\\
Such a sheaf is most useful in detecting the torsion of an $A$-module. In
fact, the localization of an $A$-module $H$ can be defined in essentially
the same way as $A$, i.e. $H_S=\{h/s|h\in H,s\in S\}$, where $S\subset A$ is
a multiplicative subset and $(h,s)\sim (h',s')$ if and only if $t(s'h-sh')=
0$ for some $t\in S$. One is able to check directly from the definition that
$H_S\simeq H\otimes_AA_S$, which shows that the localization of an $A$-
module can be passed to the localization of $A$. Note that in this case, the
kernel of the canonical map $f:H\rightarrow H_A$ is precisely the torsion
submodule $\mathrm{tor}(H)$ of $H$, therefore $H_S$ keeps track of the free
part of $H$ while at the same time killing $\mathrm{tor}(H)$. Similarly, A
Sheaf $\mathcal{H}$ can be defined for $H$ on $\textrm{Spec}A$ by setting $\
mathcal{H}(U_a)=H\otimes_AA_{S_a}$. We define the \textit{support} of $H$ to
be the support of its associated sheaf $\mathcal{H}$, i.e., the complement
of the largest open set $U\subset\textrm{Spec}A$ such that $\mathcal{H}|_U=\
emptyset$. By introducing the annihilating ideal $\textrm{Ann}(H)=\{a\in A|
aH=0\}$, we can write $\textrm{Supp}H=V\big(\textrm{Ann}(H)\big)$. Therefore
we conclude from the above discussions that: A free $A$-module has $\textrm
{Spec}A$ as its support, while the support of a torsion module is a proper
subset of $\textrm{Spec}A$.\\
Consider an exact sequence $H'\rightarrow H\rightarrow H''$ of $A$-modules,
then it's clear that the sequence $H'_{S_a}\rightarrow H_{S_a}\rightarrow H'
'_{S_a}$ is also exact. Therefore we get an exact sequence of sheaves $\
mathcal{H}'\rightarrow\mathcal{H}\rightarrow\mathcal{H}''$. In particular we
have the following lemma:
\begin{lemma}
For an exact sequence of $A$-modules $H'\rightarrow H\rightarrow H''$, we
have $\mathrm{Supp}H\subset\mathrm{Supp}H'\cup\mathrm{Supp}H''$.
\end{lemma}
\section{Abstract Localization}
\subsection{Reduction to Torus Actions}
We assume further that $G$ is compact connected and $M$ is compact with a
smooth $G$-action. Recall that the study of a compact connected Lie group $G
$ reduces to the behavior of the Weyl group $W$ acting on its maximal torus
$T$, actually this principle also applies to equivariant cohomology theory.
\begin{theorem}
With the assumptions above we have the isomorphism
\begin{equation}H_G^\ast(M,\mathbb{Q})\simeq H_T^\ast(M,\mathbb{Q})^W.\end{
equation}
\end{theorem}
\begin{proof}
Let $G_\mathbb{C}$ be the complexification of $G$, then $G_\mathbb{C}$ is a
complex connected reductive Lie group. Let $B$ be a Borel subgroup of $G_\
mathbb{C}$ containing $T$. By the Iwasawa decomposition $G_\mathbb{C}=GB$
and $G\cap B=T$, therefore we get a homeomorphism $G/T\rightarrow G_\mathbb{
C}/B$. By the Bruhat decomposition, the flag manifold $G_\mathbb{C}/B$ has a
stratification by $|W|$ strata, each of them being isomorphic to a complex
affine space. It follows that $H^\ast(G/T,\mathbb{Q})$ vanishes in odd
degrees, and the Euler characteristic $\chi(G/T)$ is equal to $|W|$. Because
the finite group $W$ acts freely on $G/T$ with quotient $G/N$, we have an
isomorphism $H^\ast(G/N,\mathbb{Q})\simeq H^\ast(G/T,\mathbb{Q})^W$.
Moreover, $\chi(G/N)=\frac{1}{|W|}\chi(G/T)=1$. It follows that $H^\ast(G/N,
\mathbb{Q})$ vanishes in odd degrees and is one-dimensional. In other words,
$G/N$ is $\mathbb{Q}$-acyclic. It means that the fibration $M\times_NEG\
rightarrow M_G$ induces an isomorphism $H_G^\ast(M,\mathbb{Q})\simeq H_N^\
ast(M,\mathbb{Q})$. This combined with the covering $M\times_TEG\rightarrow
M\times_NEG$ gives the required isomorphism.
\end{proof}
The above proof is taken from \cite{wyh}. Refer to \cite{gs} for a more
elementary proof using the spectral sequence of the Cartan complex $\Big(\
big(S(\mathfrak{g}^\ast)\otimes\Omega^\ast(M)\big)^G,d_G\Big)$ regarded as a
double complex.
\subsection{Atiyah-Bott Localization Formula}
According to Theorem 5.1, from now on we work with the torus action $T$. We
also choose to work with the coefficient ring $\mathbb{C}$ in the
equivariant cohomology since it is more convenient for us to do localization
. With these assumptions, one deduces by a direct computation that $H_T^\ast
(pt)=\mathbb{C}[X_1,\cdot,\cdot,\cdot,X_n]$, where $\deg X_i=2$. We shall
abbreviate $\mathbb{C}[X_1,\cdot,\cdot,\cdot,X_n]$ by $\mathbb{C}[\mathfrak{
t}]$, therefore $H_T^\ast(M)$ is a $\mathbb{C}[\mathfrak{t}]$-module for a
compact manifold $M$. Since $\mathbb{C}[\mathfrak{t}]$ is an integral domain
, we may take its field of fractions $\mathbb{C}(\mathfrak{t})$ so that the
corresponding localization $H_T^\ast(M)\otimes_{\mathbb{C}[\mathfrak{t}]}\
mathbb{C}(\mathfrak{t})$ provides the free part of $H_T^\ast(M)$. We remark
that with these restrictions $\mathrm{Spec}\mathbb{C}[\mathfrak{t}]$ can be
identified with $\mathbb{C}^n$.\\
Let $F$ be the set of fixed points of the torus action on $M$. The inclusion
$i:F\hookrightarrow M$ induces naturally a map $i^\ast:H_T^\ast(M)\
rightarrow H_T^\ast(F)$. One version of the Atiyah-Bott localization theorem
assets that $i^\ast$ is an isomorphism modulo torsion.\\
To prove this requires a further study on the localization of $\mathbb{C}[\
mathfrak{t}]$-modules. Let $H$ be a $\mathbb{Z}$-graded $\mathbb{C}[\
mathfrak{t}]$-module, we define its \textit{rank} to be $\dim H\otimes_{\
mathbb{C}[\mathfrak{t}]}\mathbb{C}(\mathfrak{t})$. An easy observation shows
that the affine variety $\mathrm{Supp}H$ is conic, i.e., invariant under
the scaling action of $\mathbb{C}^\times$. In fact, $\mathbb{C}^\times$ acts
on $H$ by $\lambda h=\lambda^{2q}h$ for $h\in H^q$, which is compatible
with the localization process since $\lambda X=\lambda^2 X$ for $X\in\mathbb
{C}^n$ with $\deg X=2$. Note that when $T=S^1$, $\mathrm{tor}(H)$ is
supported in $\{0\}$, therefore $H$ becomes free if we localize to $\mathbb{
C}\setminus 0$. These observations based on the grading structure of $H$
will be generalized in Theorem 5.2. As a first step we have:
\begin{lemma}
If there is a $T$-equivariant map $M\rightarrow T/K$, where $K$ is a closed
subgroup of $T$, then $\mathrm{Supp}H_T^\ast(M)\subset\mathfrak{k}_\mathbb{C
}$, where $\mathfrak{k}_\mathbb{C}$ denotes the complexification of $\
mathfrak{k}$.
\end{lemma}
\begin{proof}
The $T$-equivariant maps $M\rightarrow T/K\rightarrow pt$ give rise to ring
homomorphisms $H_T^\ast(M)\leftarrow H_T^\ast(T/K)\leftarrow H_T^\ast(pt)$.
One verifies easily be definition that $H_T^\ast(T/K)\simeq H_K^\ast(pt)$.
Let $K_0$ be the connected component containing the identity of $K$, since
we are working with complex coefficients, finite groups can be ignored and
we finally arrived at the isomorphism $H_T^\ast(T/K)\simeq H_{K_0}^\ast(pt)$
. Thus $H_T^\ast(M)$ is effectively a module over $H_{K_0}^\ast(pt)$ which
becomes a module over $H_T^\ast(pt)$ by restriction from $T$ to the sub-
torus $K_0$. Now it's obvious that for any $f\in\mathbb{C}[\mathfrak{t}]$
with $f|_{\mathfrak{k}_\mathbb{C}}=0$, $f\cdot H_T^\ast(M)=0$ and $\mathrm{
Supp}H_T^\ast(M)\subset\mathfrak{k}_\mathbb{C}$.
\end{proof}
There is a familiar situation which the above lemma applies. Let $Tp\subset
M$ be the orbit of $p$ and $U$ its $T$-invariant neighborhood. Recall the $T
$-equivariant diffeomorphism $NTp\rightarrow U$, where $NTp$ denotes the
normal bundle of $Tp$ in $M$. By slice theorem, $Tp$ can be identified with
$T/T_p$, and we obtain the $T$-equivariant map $\tau:U\rightarrow T/T_p$.
Moreover, for any $T$-invariant open set $V\subset U$, we can get a $T$-
equivariant map $V\rightarrow T/T_p$ by restricting $\tau$ to $V$. We have
proved:
\begin{lemma}
There exists a $T$-invariant neighborhood $U$ of $p$ such that for any $T$-
invariant neighborhood $V$ of $p$, $\mathrm{Supp}H_T^\ast(U\cap V)\subset\
mathfrak{k}_\mathbb{C}$.
\end{lemma}
For $S\subset M$ a submanifold, we define the \textit{relative equivariant
cohomology} $H_G^\ast(M,S)$ as $H^\ast(M_G,S_G)$, the ordinary relative
cohomology of $S_G\hookrightarrow M_G$. Our main lemma is as follows:
\begin{lemma}
Let $T$ act smoothly on the compact manifold $M$ and $S$ a $T$-invariant
submanifold. Then the supports of the modules $H_T^\ast(M\setminus S)$ and $
H_T^\ast(M,S)$ are contained in $\bigcup_{K\subset T}\mathfrak{k}_\mathbb{C}
$, where $K$ runs over the finite set of subgroups of $T$ which occur as
isotropy groups of $M\setminus S$.
\end{lemma}
\begin{proof}
Let $U$ be a $T$-invariant tubular neighborhood of $S$. By the homotopy
axiom the isomorphisms $H_T^\ast(M\setminus S)\simeq H_T^\ast(M\setminus\bar
{U})$ and $H_T^\ast(M,S)\simeq H_T^\ast(M,\bar{U})$ are easily seen.
Therefore it suffices to prove the assertion for $H_T^\ast(M\setminus\bar{U}
)$ and $H_T^\ast(M,\bar{U})$. Since $M\setminus U$ is compact, one can find
$T$-invariant open sets $U_i,1\leq i\leq k$ covering $M\setminus\bar{U}$ and
equivariant maps $U_i\rightarrow T/K_i$, each $K_i$ being the isotropy
subgroup of some point in $M\setminus S$. Set $V_r=U_1\cup\cdot\cdot\cdot\
cup U_{r-1}$, we have the following equivariant Mayer-Vietoris sequence
\begin{equation}H_T^\bullet(U_r\cap V_r)\rightarrow H_T^{\bullet+1}(V_{r+1})
\rightarrow H_T^{\bullet+1}(U_r)\oplus H_T^{\bullet+1}(V_r).\end{equation}
By Lemma 5.1 and 5.2, the end terms in the above sequence are easily seen to
be supported in $\bigcup_{K\subset T}\mathfrak{k}_\mathbb{C}$ by an
induction argument. Apply Lemma 4.1 we see the middle terms are as well,
this proves the first part of the theorem.\\
Use the obvious isomorphism $H_T^\ast(M,\bar{U})\simeq H_T^\ast(M\setminus S
,\bar{U}\setminus S)$ and examining the exact sequence
\begin{equation}H_T^\bullet(\bar{U}\setminus S)\rightarrow H_T^{\bullet+1}(M
\setminus S,\bar{U}\setminus S)\rightarrow H_T^{\bullet+1}(M\setminus S)\end
{equation}
to prove the second part.
\end{proof}
\begin{corollary}
With the assumptions as in Lemma 5.3, let $j:S\hookrightarrow M$ be the
inclusion, then the kernel and cokernel of the map $j^\ast:H_T^\ast(M)\
rightarrow H_T^\ast(S)$ are supported in $\bigcup_{K\subset T}\mathfrak{k}_\
mathbb{C}$.
\end{corollary}
\begin{proof}
This follows from the exact sequence
\begin{equation}H_T^\bullet(M,S)\rightarrow H_T^\bullet(M)\rightarrow H_T^\
bullet(S)\rightarrow H_T^{\bullet+1}(M,S),\end{equation}
since the kernel and cokernel of $j^\ast$ are respectively a quotient module
and a submodule of $H_T^\ast(M,S)$.
\end{proof}
Let $S$ be the fixed point set $F$, we obtain the following version of the
abstract localization theorem.
\begin{theorem}[Atiyah-Bott]
The kernel and cokernel of $i^\ast:H_T^\ast(M)\rightarrow H_T^\ast(F)$ have
supports in $\bigcup_{K\subsetneq T}\mathfrak{k}_\mathbb{C}$, where $K$ runs
over all the isotropy subgroups which is not equal to $T$. In particular,
both modules have the same rank.
\end{theorem}
Since $H_T^\ast(F)\simeq H_T^\ast(pt)\otimes H^\ast(F)$, it is a free module
. The last part of the above theorem implies:
\begin{corollary}
$\mathrm{rank}H_T^\ast(M)=\dim H^\ast(F)$.
\end{corollary}
The theorem also implies that $H_T^\ast(M)$ becomes a free module once we
localize to an open set $U_f=\big\{x|f(x)\neq 0\big\}$ where $f$ is any
polynomial which vanishes on all $\mathfrak{k}_\mathbb{C}$. Back to our
earlier remarks, consider the simplest case when $T=S^1$, then $\mathfrak{k}
_\mathbb{C}=0$ and $H_{S^1}^\ast(M)$ becomes free on $\mathbb{C}\setminus 0$
. As we saw earlier this follows directly from the fact that the module $H_{
S^1}^\ast(M)$ is $\mathbb{Z}$-graded. In the general case however the $\
mathbb{Z}$-grading simply tells us that the support of $\mathrm{tor}\big(H_T
^\ast(M)\big)$ is a proper cone, while Theorem 5.2 is more precise.\\
Let $f:S\rightarrow M$ be a map between compact oriented manifolds,
associated to this map we can define a pushforward on singular cohomology,
which is known as the \textit{Gysin map}:
\begin{equation}f_\ast:H^\bullet(S)\rightarrow H^{\bullet+q}(M),q=\dim M-\
dim S.\end{equation}
This can be defined by first defining the pushforward on homology and then
taking the Poincar\'{e} duality. A direct verification shows the pushforward
has the following properties:
\begin{itemize}
\item It is functorial, i.e. $(f\circ g)_\ast=f_\ast\circ g_\ast$.
\item It's a homomorphism of $H^\ast(M)$-modules, i.e. $f_\ast(vf^\ast u)=(f
_\ast v)u$.
\item If $f$ is a fibering, $f_\ast$ corresponds to integration over the
fiber.
\item When $f:S\hookrightarrow M$ is an inclusion, then $f_\ast$ factors
through the Thom isomorphism, i.e. in the diagram
\[\begin{CD}H^{\bullet-1}(M\setminus S) @> \delta >> H^\bullet(M,M\
setminus S) @>j^\ast>> H^\bullet(M)\\
@. @AA \Phi A\\
@. H^{\bullet-q}(S)
\end{CD}\]
we have $f_\ast=j^\ast\circ\Phi$ with $\Phi$ the Thom isomorphism.
\end{itemize}
Let $H_c$ denote the compactly supported cohomology. Recall the well-known
isomorphism $H^\ast(M,M\setminus S)\simeq H_c^\ast(NS)$ and the well-known
fact that the pullback of the Thom class $\Phi1\in H_c^q(NS)$ by any section
of $NS$ gives the Euler class, we conclude that $f^\ast f_\ast1=e(NS)$.\\
Such a pushforward can be extended word-for-word to the equivariant
situation. To verify this recall that by the graph construction every map
can be factored into an inclusion following by a fibering projection. Then
it suffices to check the third and forth properties above in the equivariant
theory. But when $f$ is a fibering, so is the induced map $f_G:N_G\
rightarrow M_G$ and integration over the fiber is well-defined in any
fibering with an oriented compact manifold as fiber. Similarly the Thom
isomorphism, but now applied to bundles over $M_G$. We should remark that
this equivariant Thom class, whose existence was indicated in \cite{ab}, was
constructed explicitly in the Cartan model by Mathai and Quillen in a later
work using a very elegant method which is now known as the Mathai-Quillen
formalism. We will describe these materials in a subsequent note. Finally
one should notice that the pushforward $f_\ast^G$ preserves the $H_G^\ast(pt
)$-module structure and that the pushforward $\pi_\ast^G$ of the map $\pi:M\
rightarrow pt$ corresponds to integration over the fiber of the fibering $M_
G\rightarrow BG$.\\
Corresponding to Theorem 5.2, by Lemma 5.3 one can prove similarly another
version of the Atiyah-Bott localization theorem for the pushforward $i_\ast$.
\begin{theorem}[Atiyah-Bott]
Assume $M$ to be oriented. Let $q=\dim M-\dim F$, the kernel and cokernel of
the pushforward $i_\ast:H_T^\bullet(F)\rightarrow H_T^{\bullet+q}(M)$ have
supports in $\bigcup_{K\subsetneq T}\mathfrak{k}_\mathbb{C}$, where $K$ runs
over all the isotropy subgroups which is not equal to $T$.
\end{theorem}
Actually, the kernel of $i_\ast$ is trivial by the freeness of $H_T^\ast(F)$
, but we will not need this fact here.\\
We are now prepared to deduce a general integration formula for localization
. First, combining Theorem 5.2 and 5.3 we see $i^\ast i_\ast$ is an
isomorphism modulo torsion, or more explicitly, $i^\ast i_\ast u=e_T(NF)u$
for every $u\in H_T^\ast(F)$. It follows that $e_T(NF)$ must be invertible
in the localized module $H_T^\ast(pt)_S\otimes H^\ast(F)$, where $S=\{1,f,f^
2,\cdot\cdot\cdot\}$ for some suitable $f$. To see this let $F=\{F_i\}$ be
the decomposition of $F$ into its connected components. Since terms of
positive degree in $H^\ast(F_i)$ are nilpotent, an element in $H^\ast(F_i)$
is invertible if and only if its component in $H^0(F_i)$ is nonzero. Hence $
e_T(NF_i)$ will be invertible in some localization $H_T^\ast(pt)_S\otimes H^
\ast(F_i)$ provided its component in $H_T^\ast(pt)\otimes H^0(F_i)$ is a
nonzero polynomial, this polynomial can then be taken as $f_i$ for $H_T^\ast
(F_i)$. But this component is determined by restricting to any point $p\in F
_i$. In fact, since the only fixed directions are tangential to $F_i$, the
action of $T$ on the fiber $N_p$ has no fixed point, so that $N_p$
decomposes as a direct sum of nontrivial 1-dimensional complex
representations $\alpha_j:T\rightarrow U(1)$ of $T$. Suppose the weight
attached to $\alpha_j$ is $a_j$, by Theorem 3.2 we have $e_T(N_p)=\prod_ja_j
$, which is a homogeneous polynomial on $\mathfrak{t}_\mathbb{C}$ and is
exactly the polynomial we are seeking for. We abbreviate this polynomial by
$f_i$ and it follows that $i^\ast i_\ast$ becomes invertible after
localizing with respect to $f=\prod_if_i$. Working with the localized ring $
H_T^\ast(pt)_S$ we therefore see that $Q=\sum_i\frac{i_{F_i}^\ast}{e_T(NF_i)
}$ is inverse to $i_\ast:H_T^\ast(F)\rightarrow H_T^\ast(M)$. Thus for any $
\phi\in H_T^\ast(M)$ we have after localizing the base ring
\begin{equation}\phi=i_\ast Q\phi=\sum_i\frac{i_\ast^{F_i}i_{F_i}^\ast\phi}{
e_T(NF_i)}.\end{equation}
Applying the pushforward $\pi_\ast^M:H_T^\ast(M)\rightarrow H_T^\ast(pt)$
induced by the projection $M\rightarrow pt$ to both sides of the above
formula and using the functoriality of pushforwards we obtain the \textit{
Atiyah-Bott localization formula}: \begin{equation}\pi_\ast^M\phi=\sum_i\pi_
\ast^{F_i}\big\{\frac{i_{F_i}^\ast\phi}{e_T(NF_i)}\big\}.\end{equation}
\begin{thebibliography}{99}
\bibitem{ad} A. Adem, J. F. Davis, \textit{Topics in transformation groups},
Handbook of gemetric topology, 1-54, North-Holland, Amsterdam (2002).
\bibitem{ab} M. F. Atiyah and R. Bott, \textit{The moment map and
equivariant cohomology}, Topology 23 (1984), no. 1, 1-28.
\bibitem{bgv}N. Berline, E. Getzler and M. Vergne, \textit{Heat Kernels and
Dirac Operators}, Grundlehren Text Editions, Springer, 2004.
\bibitem{bt} R. Bott and L. Tu, \textit{Differential Forms in Algebraic
Topology}, Graduate Texts in Mathematics, vol. 82, Springer, 1982.
\bibitem{gb} G. Bredon, \textit{Introduction to Compact Transformation
Groups}, Academic Press (1972).
\bibitem{bd} T. Brocker and T. tom Dieck, \textit{Representations of Compact
Lie Groups}, Springer-Verlag, New York, 1985.
\bibitem{gs} V. Guillemin and S. Sternberg, \textit{Supersymmetry and
Equivariant de Rham Theory}, Springer-Verlag, 1999.
\bibitem{wyh}W. Y. Hsiang, \textit{Cohomology Theory of Topological
Transformation Groups}, Springer, New York, 1975.
\bibitem{hm} H. Matsumura, \textit{Commutative algebra}, Benjamin, N.Y.,
1970.
\end{thebibliography}
\end{document} | l****y
发帖数: 4773 | 8 看了她关心的方向，中科大没有导师能带这方向的phd. |
| i****g
发帖数: 3896 | 9 很好奇，怎么看出来她优秀的？另外学习勤奋优秀不等于学术能力强 没自己的东西永
远是奥数机器
【在 R******9 的大作中提到】
: 看了她的一些留言，她自己提到以前在南京一所学校上本科，但不在数学系，后来自己
: 用功，在本科期间学习了很多，我觉得她掌握得还是挺系统深入的，后来申请去了香港
: 中文大学，现在做代数几何。她自己眼光比较高，而且确实勤奋聪明，是个很优秀的学
: 生。但是对于很多数学家的评价太偏激，对于很多数学领域持一种偏见，表现在语
: 言上是很鄙视，有时候会骂其他人。
:
: 老师
| l****y
发帖数: 4773 | 10 另外好奇，楼上shimura和豆瓣上与此人争论的shimura是同一个人吗？ | | | n******r
发帖数: 4455 | 11 伊说搞数论的都是loser，张的结果毫无意义，陶属于三流数学家
令人很好奇口气这么大的人自己做出了什么结果
实际生活中有人这么说话而不是故意搏出位的话，只能怀疑精神有问题
【在 R******9 的大作中提到】
: 看了她的一些留言，她自己提到以前在南京一所学校上本科，但不在数学系，后来自己
: 用功，在本科期间学习了很多，我觉得她掌握得还是挺系统深入的，后来申请去了香港
: 中文大学，现在做代数几何。她自己眼光比较高，而且确实勤奋聪明，是个很优秀的学
: 生。但是对于很多数学家的评价太偏激，对于很多数学领域持一种偏见，表现在语
: 言上是很鄙视，有时候会骂其他人。
:
: 老师
| D**o
发帖数: 2653 | 12 本来就是精神有问题
【在 n******r 的大作中提到】
: 伊说搞数论的都是loser，张的结果毫无意义，陶属于三流数学家
: 令人很好奇口气这么大的人自己做出了什么结果
: 实际生活中有人这么说话而不是故意搏出位的话，只能怀疑精神有问题
| z***e
发帖数: 5600 | 13 好像她还说不求看懂证明，看懂定理叙述就成了。不做题不抠证明数学当然容易得很，
难怪口气大
【在 D**o 的大作中提到】
: 这是她写的
: 我想简单讲一下为什么念书不需要做题。假设题目你会做，那你做它干嘛，没必要炫耀
: 自己，应该踏踏实实多学东西。假设你不会做，反正你也做不出来，浪费那时间做什么
: ？应该抓紧这些时间提高自己的水平，这样今后你就会做了，否则你永远不会做。综上
: ，读书不应该做题。
| z***e
发帖数: 5600 | 14 她应该是在代数几何算术几何方向研究数论的，对解析数论很不屑
【在 i****g 的大作中提到】
: 她/他又不是做数论的 反而是最鄙视做数论的 呵呵
| S*****a
发帖数: 190 | 15 她是做复几何分析的，对代数几何一窍不通，更不用说算术几何和数论了。
【在 z***e 的大作中提到】
: 她应该是在代数几何算术几何方向研究数论的，对解析数论很不屑
| s*****V
发帖数: 21731 | 16 求老怪都是属于知其然不知其所以然的数学家
【在 n******r 的大作中提到】
: 伊说搞数论的都是loser，张的结果毫无意义，陶属于三流数学家
: 令人很好奇口气这么大的人自己做出了什么结果
: 实际生活中有人这么说话而不是故意搏出位的话，只能怀疑精神有问题
| l****y
发帖数: 4773 | 17 我感觉也是。她所提的代数几何都是复数域上的。另外厚脸皮说句，搞数学物理的人吹
代数几何多半是用AG充门面，实际一窍不通。至少我见过的搞数学物理的都这样。当然
我没见过数学物理大牛。
【在 S*****a 的大作中提到】
: 她是做复几何分析的，对代数几何一窍不通，更不用说算术几何和数论了。
| s*****V
发帖数: 21731 | 18 她为什么对ATIYAH这么推崇？
华罗庚本人的档次很清楚，他不如陈省身不是一点点，应该是二流数学家当中的佼佼者
，这个评价是公平的。即使是陈省身，在一流数学家当中也不算最好的。他跟Atiyah，
Weil，Weyl，Elie Cartan这些人的差距还是很大的，我甚至认为他的贡献只有Atiyah
三分之一。
我文中的主要内容是谈华罗庚回国以后对中国数学的影响，他的影响看看他的学生做的
东西就知道了，所以不可能不谈他的学生。华罗庚带给中国数学的主要危害也是从这些
人身上看出来的。我认为你并没有仔细读我的文章，仅仅是一知半解，就在这里毫无根
据地说我在污蔑数论，这些言论毫无疑问是不负责任的，至少比我对华罗庚的评价要更
加不负责任。
【在 l****y 的大作中提到】
: 我感觉也是。她所提的代数几何都是复数域上的。另外厚脸皮说句，搞数学物理的人吹
: 代数几何多半是用AG充门面，实际一窍不通。至少我见过的搞数学物理的都这样。当然
: 我没见过数学物理大牛。
| l****y
发帖数: 4773 | 19 她只了解atiyah，只能这么认为。见识短浅的人容易乱喷别人。 | t*****n
发帖数: 1589 | 20 不做题怎么学数学
【在 D**o 的大作中提到】
: 这是她写的
: 我想简单讲一下为什么念书不需要做题。假设题目你会做，那你做它干嘛，没必要炫耀
: 自己，应该踏踏实实多学东西。假设你不会做，反正你也做不出来，浪费那时间做什么
: ？应该抓紧这些时间提高自己的水平，这样今后你就会做了，否则你永远不会做。综上
: ，读书不应该做题。
| | | m*********s
发帖数: 368 | 21 估计就学会了bbs数学(空谈而已)
【在 t*****n 的大作中提到】
: 不做题怎么学数学
| s*****V
发帖数: 21731 | 22 楞，陶老师太凄惨了
2013-06-03 09:22:43 煙花不堪剪 ([;d_Astar F_A=0;])
陶哲轩是数学界的小丑。数学界已经浮躁不堪，但恨智术短浅，不能制之，徒为人也！
2013-06-07 10:27:27 煙花不堪剪 ([;d_A\star F_A=0;])
6月7日更新：人们现在已经把素数对间距的上界降低到了387,620 ，更多的结果见这个
wiki页面（ htt ... 大猩猩 Hψ=Eψ
世界数学的悲剧啊，这里面除了张益唐还有点贡献，其他人都是一群民工！
老师
【在 l*******n 的大作中提到】
: 抱着学习数学知识的态度查看了一下她/他的豆瓣
: 看到有同学/朋友在其下留言讨论这学期上课的教师优劣，提到的名字都是中科大的老师
: 所以大约她/他应该是在中科大读phd，而不是在香港
| s***x
发帖数: 43 | 23 说华不如陈，主要是因为过去大半个世纪陈的学问成为数学的主流之一，华的成了偏门
。现在老张的学问从买买提上的看主要是出自华那条线的。这样好歹给华陈之比减少了
点差距吧！
Atiyah
【在 s*****V 的大作中提到】
: 她为什么对ATIYAH这么推崇？
: 华罗庚本人的档次很清楚，他不如陈省身不是一点点，应该是二流数学家当中的佼佼者
: ，这个评价是公平的。即使是陈省身，在一流数学家当中也不算最好的。他跟Atiyah，
: Weil，Weyl，Elie Cartan这些人的差距还是很大的，我甚至认为他的贡献只有Atiyah
: 三分之一。
: 我文中的主要内容是谈华罗庚回国以后对中国数学的影响，他的影响看看他的学生做的
: 东西就知道了，所以不可能不谈他的学生。华罗庚带给中国数学的主要危害也是从这些
: 人身上看出来的。我认为你并没有仔细读我的文章，仅仅是一知半解，就在这里毫无根
: 据地说我在污蔑数论，这些言论毫无疑问是不负责任的，至少比我对华罗庚的评价要更
: 加不负责任。
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