顺流而下 数学和我的生活 Shizhuozhang@gmail.com

一个屋子 一个外国人, 巴黎11大学的 导师是Fields Medalist

一个要去Washington的

一个已经在UCLA做数论的

在牛圈中........

终于知道什么叫stack了,原来只是知道Scheme的推广就是stack. 问了法国大牛,知道有个orbifold locally looks like the R^k/G for some group G. 对应的就是stack. as the scheme corresponding to the manifold.

orbifold的定义很形象,有个奇点.相当于是glue起来得到一个点.

法国大牛是做motivic cohomology的. 大牛推荐了几本书 一本Mumford的 一本

Andrew Kresch(???)写的.大牛对Ha的书比较熟.给我解释了好多疑问和习题.大牛肯定把Ha的习题都做了.....

敬过大牛以后回自己的宿舍.看见傅利叶大牛正在和别人讨论同伦和同调的问题.貌似是homotopy control the homology的问题.

他们讨论的热火朝天.我和K聊天,以下是让我很受启发的一些东西

if all pi0 -> pin = 0
then pi(n+1) = h(n+1)
n-th homology tell you how n-cells are glued to (n-1)-cells
n-th homotopy depends on the whole n-skeleton
when a space has easy homology, it has crazy homotopy
when a space has easy homotopy, it has crazy homology

sphere has easy homology, it has difficult homotopy

classifying space has easy homotopy and difficult homology

cohomology = dual of homology, which is abelianization of homotopy

if there is a nonabelian cohomology theory, it will be dual to homotopy in some sense.

giving a space X, get H^n(X;R) and pi_n[X]

then we have dim(h0)-dim(h1)+dim(h2)-... = chi = |pi0|/|pi1|*|pi2|/...

no proved even no verification

homology is +
homotopy is x
乘法难点。
mirror symmetry / langlands is just like this duality
Hom(S^1, X) and Hom(X, S^1)
this is beyond human reach
group is the symmetry of something. pi_1 is the symmetry of what thing?
all group is the symmetry of some the stuff
automorphism = symmetry = group