收获
这几天和郝教授一起住,讨论了很多问题,他给了我很多做数学的建议,很感谢他.
首先是第一天住进来,自我介绍后我马上问他做了什么工作,他说是环的一些性质和Hopf代数,主要是矩阵环的刻画.然后他说了一个很有意思的东西.
考虑两个线性recursion序列, 然后乘积起来 问这个新的序列的recursion关系是什么 我问对于Fib序列有什么结果么,他说有专门这方面的书.问题在于怎么刻画这类序列,或者说这些序列组成的集合上有无代数结构,幸运的是乘积是封闭的,所有是个环.然后就可以用刻画环的工具来做了,比如cohomology.
前天我问,一般在代数几何的问题,引进Scheme or stack以后马上就变成交换代数的问题,有没有反过来做的,郝老师说如果把环的问题变成素谱的问题可能会变复杂,因为素理想不太清楚.但是我却又见过这么做的.比如Huzhengyu的一篇文章.
前天郝老师表达了一个观点,就是代数中只有多项式和矩阵环两个基本对象,其他的对象都是这两个对象经过twist and product生成的. 我说EndR是twist 但是他把这个仍然看成matrix ring.
续)
代数几何VS表示论
我发现我还是更喜欢代数几何.嘿嘿 看到cohomology的中间部分了. Cech上同调 检查了细节
不过还是总结一下一些我感兴趣的表示论的主题
Topic Gentle algebra Given by Surface Triangulations.
Why I was interested in this topic for following reasons.
1. I ever read the representation theory of Gentle algebra. When I first met this classes algebra, I have strong interested in its definition. Say if an algebra A is morita equivalent to some path algebra with quiver Q and admissible ideal, and satisfy 3 conditions.
So in the topic. The triangulation of an oriented surface can give the gentle algebra, I was surprised. The speaker Thomas Brustle say the {set of strings} has bijection with {indecomposable modules over gentle algebra} , I think this is a very beautiful relation. Because it is known that {set of positive roots} correspondent 1 – 1 {indecomposable modules} in hereditary algebra and Lie algebra. I think it is very similar. So I am interested in them.
After the lecture, I have contacted with the speaker,I think we will have a communication from now on.
2. This kind of gentle algebra is cluster tilted algebra, but cluster tilted algebra is very important in Tilting theory.
Topic Representation theory for vertex operator algebra.
Why I was interested in this topic?
1. In fact. I am not familiar with the vertex operator algebra at all. But the method used attracts me. The highest weight theory with application to this kind of algebra. Especially the M[0] can determine the M[i] for all I because I do not have lectures at hand. So I only describe the impressive part.
Topic Minimal Length Elements and G Stable pieces.
You ask me why I was interested in this topic. Though I did not know anything about this.
I think building the connection between the finite weyl groups and the geometry of algebraic groups is anyhow very important. And I think the Deligne-Luzstig varieties associated to minimal length elements are affine ,this result is non-trivial.
I think the idea in the abstract is enough to have my attention. Saying translating some ideas and constructions from Luzstig’s geometric setting to the combinatorial settings of finite weyl groups.
I ever read the GTM129 flavored in Combinational. The structure of combinational is more and more important now. You can consider the representation theory of Lie Groups. It induces the study in Hecke algebra. Then using a combinational settings is suitable. I think Xuhe Hua did a prominent work.
Topic Cluster Algebras of Rank 3
Obviously I am interested in this topic because it has close connection with the representation type. It shows that if cluster tilted algebra is not hereditary, then H is representation finite. And the Loewy length actually characterize some finiteness.
Topic Classification of finite dimensional basic Hopf Algebra as according to their representation type.
Obviously I will interested in this topic because I am focusing on the problem of Rep Type for a long time.
Covering theory, Bocs Differential and Subcategories are the efficient methods. Hopf algebras are the special case. I think the key is the condition “Basic”.
Topic Classification of Quantum Groups
There are many ways to classify the Quantum Groups, using the Dykin quiver and Extend Dykin quiver. So I am interested in this topic. But I am confused that there shall be some very different quantum groups with the same classical limit.
Topic Generic sheaves over elliptic curves.
It gives a beautiful result saying for every rational number q, there is a generic sheaves with slope q over E. In the lecture notes, a lot of technique in Tilting theory are used. The generic sheaves are the generalization of generic modules. And the Tilting sheaf are generalization of Tilting module as well. I saw a lot of similarity appeared,such as the definition of the Euler Form, only replace the module with sheaf. There is a good result for the Euler form as well. I will read some of papers.
Topic Conical Extension of Derived Categories.
Before the lecture, I was interested in it.
1. What is the Toroidal Lie algebra ?
2. Why can one realize the Kac-Moody Lie algebra and Lie algebra using only one derived categories, but using two realize the Toroidal Lie algebra. What is the hard point?
After the Lecture, I know the hard point that the quiver of Toroidal Lie algebra is complicated any how. So one try to “partition”it to some parts. Then using derived categories of corresponding Dykin quiver,then define the Ringel Hall algebra to realize respectively, final, using the conical extension to realize the whole Toridal Lie algebra. I think the important things are this result mentioned in the lecture: if given derived categories have Auslander Reiten translations then the conical extension derived categories has Auslander Reiten translations as well.
I was extremely interested in the picture that Pengliangang showed, it told me how to realize the Lie algebra with Affine Dykin quiver. Hmmm,It is interesting.
Duality of Arithmetic Groups Mapping class Groups and Outer automorphism of free groups.
I will check the detail of the lecture.
Because Professor Ji Lizhen lead us to a comprehensive realm of mathematics. A lot of beautiful results are given. Hehe
一点有意思的东西 Math&哲学
今天坐飞机到了拉萨,然后坐了出租到了Himalaya宾馆,在见过林教授和芮教授后便径自到了自己的房间.有点高原反应 既然是International Conference on Representation theory. 我带了一些Luzstig的文章.主要是关于Canonical Basis的,在杭州和林教授聊数学的时候,他说Intersection Cohomology是一个非常重要的工具.我对这个并不了解,但是有很强的兴趣,因为K也说有些人考虑在同调群上用Intersection建立环结构.
好多牛
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
最近的一些事情
最近数学的学习进度放慢了 主要是因为考试太多 那些课程平时都不学 都看数学了 考试前突击了5天 累死了. 可是一想到考完就可以看AG,就呵呵
最近AG进度比较慢 可能到7月16可以开始看上同调.7月20日会写一个关于代数几何前2章的讲义并且准备在开学的讨论班上讲. 最近在K的提醒下注意到一个事实就是[Ha] 2.8中Differential和Frobenius Automorphism很象,不知道有没有人专门研究过这种东西,搜了很久也是一些不相关的东西. 杭州听了几天K理论 觉得收获不小.首先是唐国平老师讲一些基本的K理论,讲到一个用配边理论和Whitehead Torsion以及K理论证明了5维Poincare猜想,这个我当时就激动了,代数拓扑的东西本来只能是粗略的处理拓扑空间的,众所周知的同调群和同伦群.一般都比同胚要弱,两个空间的同调群相等或者同伦群同构 而拓扑空间不同胚的例子很多. (当然同伦可以控制同调),什么是Poincare猜想呢,简单说来就是
连通的紧的n维流形如果同伦于n维球面 则必有他们两个同胚,而我们通常见到的3维Poincare猜想就是这个的特殊情况. Smale 在别人工作的基础上证明了当n大于等于5的情形下,这个结论是成立的. 具体的过程只能回去写了,马上就要离京了
