顺流而下

Light of Times

This time, I do not talk about mathematics

Recently, I was watching a Japanese movie which describe a new graduate student from Law schools who entered into an enterprise of laws. This is her first job. Her dream is to use the law as the weapon to protect the weak people in society. However, when she met some people who is not good like she imagined, for example,they used law just as the tools to kill or to hurt other people. They are not honest,they told lies. She found it is really a heavy blow. She was confused that what law really was!

This movie let me remember something. At the beginning, I thought this lawyer was really naïve. Just like some students I met in my original undergraduate university. They did not care about GPA, the attendence of the course. They just did everything they wanted to do. They believed that their effort will be paid back in the end. Yes, when I was undergraduate, especially at my first year in university, I did not care anything but mathematics, I seldom attended course in computer science which makes my GPA very low. Because my attitude to these stuffs. I almost can not pass some examination in computer science. I have to retake some courses. However, I won this game at last minute because I am really lucky and met a lot of very good teacher who were willing to help me. In this case, I have a lot of chance to establish my ability. But can anyone have this luck, I am not sure. So everytime when they asked me something like this, I will tell them GPA is important, attendence of the course is important. What else can I tell them? Taught them to skip all the classes like me? Impossible, it is not my future. So it is always the case I feel some of them are really naïve.

Back to the movie. Although this new lawyer met a lot of things like that, she insists on doing the right thing which she believed. At final, she is full of confidence. I was really impressed by the last section of this movie. There is a dialog between her and the master of the Law firm.

Mater:

It seems that you insists this case to the final. Great New Lawyer!

New Lawyer:

Thank you, it is not that good.

Master:

What ever the final result will be, You win or us win. I believe that you are going to walk on a very difficult and tiring road.

New Lawyer:

Yes, I have such a consciousness

Master:

I will focus on you, focus on you from now on.

Smile.

It is really a shake for me.

At the beginning I was very admired another one in this movie, a very nice guy who was very similar to this New Lawyer when he started his life as lawyer, pursuit the truth, saving weak people, insists on justice. A typical idealist. But when he met something in real word, he grew up. He have learnt how to protect himself, however, he is very proud in his own heart. He helped the new lawyer to deal with crisis each time, giving her advice. I feel in some sense, I am similar to this guy. I am proud to be this kind of man. However, I am really gratified to see there are so many guys like that New Lawyer, though did not know any cruel things in this real world, but firmly believe the truth, firmly pursuit their dream whatever difficult will arise.

The producer of this movie name that Lawyer as

教会は、光の

Its first name means light. Yes, the LIGHT OF TIMES

I was pleased to know there are some guys in my original university because I believe, they are LIGHT OF TIMES, LIGHT OF NATION.

Motivation

上Rosenberg的课,正如我一直所觉得的,很抽象,很容易在上课10分钟后就陷入范畴以及代数计算中,到下课前10分钟才又能follow idea. 前几天找Rosenberg聊了聊非交换的idea, application. 以及他定义的Universal K theory的motivation.这几天心里塌实多了.

从表示论的观点来说,传统的K理论并不能给出很多valuable information, 例如,if g is a finite dimensional Lie algebra over a field k, then K*(U(g))=K*(k),where U(g) is the enveloping algebra of g, similarly, we have K*(An(k))=K*(k),where An(k) is the n th weyl algebra over k. 很显然,从Standard K theory, 这三个东西都是一样的,无法区分,因此我们就很自然地想研究subcategory of category of U(g)-module, 因此就有了所谓BGG category, 进一步就有Kazhdan-Lusztig conjecture.而且最重要的是,"its prove led to the reformulation of representation theory of reductive group in terms of D-modules and D-schemes making it a part of noncommutative algebraic geometry!"

而这种reduced,是基于以下一个著名的事实: The main basic facts which allowed to reduce the representation theory to the study of D-modules on flag variety is the "Beilinson-Bernstein localization theorem" which says the category of D-modules on flag variety of a reductive Lie algebra g over a field k with chark=0 and the category of U(g)-module with trivial central character is EQUIVALENCE induced by the Global section functor!!

里面一个主要的研究对象就是所谓holonomic module.

众所周知的是,Kazhdan-Lusztig conjecture,除了有reductive group version,Lie algebra version,还有Quantum group version. 并且还有chark !=0的版本,而Rosenberg的非交换代数几何就可以直接应用到Quantinized version上.

Rosenberg在和V.Luntz的一篇文章Localization for Quantum group的文章中定义了所谓Quantum flag variety(这是一种典型的noncommutative space),然后就比照Beilinson Burnstein localization建立了所谓Quantized version of this localization.试图建立Uq(g)-module category 和twisted D-module on quantum flag variety范畴的等价关系! 这样我们就可以把representation theory of quantized enveloping algebra的问题转化到D-module的研究上! 而原来Rosenberg证明了这个意义下的Global section functor是exact,所以他并不满意,而且给出了几个猜想,都是关于这个functor,当然我们希望这个函子是Equivalence!而随后Tanasaki在Math.Z上的一篇文章就证明了这个functor是fully faithful的,这样就几乎离equivalence很近了,只差dense了! 几乎就可以说明在quantized的情形下,这种reduced也是很成功的.

当然Rosenberg只是把这种在表示论中的应用当成他的非交换代数几何中的一个很小的部分,但是从这里我却看到了这个理论实在是很powerful.

现在他正在讲他和Kontscevich的一篇Spectra, associate point and representation theory的文章,里面研究(quantized)enveloping algebra of Lie algebra,或者更一般的非交换代数(比如differential algebra, Rota-Baxter algebra,in fact,they are almost commutative algebra)的enveloping algebra even quantized case就都可以用quantized flag variety来研究. 很期待他讲后面的东西!

一个问题

有这样一个性质

G is a topological group and it is a monogenic group(which means there is an element of G, say g, such that cyclic group generated by g, i.e <g>, whose closure is G). Then, G is abelian group.

也就是说 G 是这样一个群, 它有一个循环子群<g>是dense的,也就是说这个循环群的闭包就是G,现在要证明它是一个交换群.

in fact, the prove is not difficult.

Prove: for arbitrary element in G, say distinct element x and y, because there is a dense cyclic group <g>, so there is a seq g_k belongs to <g>, such that g_k--->x,and another seq h_k which belongs to <g> such that h_k--->y. because G is topological group, so the multiplication is continous, then (g_k * h_k)--->x * y. Similarly, we have (h_k * g_k)--->y*x. but cyclic group is abelian.(which means h_k*g_k=g_k*h_k for all k) Therefore, lim(g_k*h_k)=lim(h_k*g_k), so x*y=y*x. Then G is abelian.

看起来证明好象没什么问题,但是中间用了一个性质是lim(g_k*h_k)=lim(h_k*g_k), 也就是两个序列的极限相等,或者说一个序列至多只有一个极限(因为极限存在,所以变成一个序列只有唯一的极限).

所以如果这个topological group is Hausdorff的,那么极限的唯一性就被确保了.证明就works. 但是问了下别人,得知topological group 不一定要求Hausdorff,那怎么修改这个证明呢?

从wiki看, 如果要求Topological group是Hausdorff的,那么closure of Identity subgroup must be closed, 因此如果take G/K ,where K is closure of identity,那么得到的这个H=G/K就是Hausdorff的了，而如果最开始G的underlying space又是T0的,那么G/K和G又是同胚的.

问题是这个性质能干脆不要Hausdorff这个条件么?

清醒

看了几个朋友的"表白". 加上前段时间和法国学生,Rosenberg以及老板的几次聊天. 感觉"刨根问底"不见得是好的,也会把自己引上歧路. "凡事老问个目的"也不往往是好的.自己被这种东西束缚太久,还好在一事无成之前清醒了.希望能一直排除浮躁吧.

原先在国内的时候,就是说大一到大二,对代数表示论产生兴趣,然后一直就追寻Auslander and Ringel的足迹,念了不少这方面的书,尽管Ringel的东西直到他做出Ringel Hall algebra之前都不是很受美国重视,但是我依然觉得,I should follow his way to do research.因此那时,直到现在,我依然对一些不太重视(主要是太难了)的问题,比如representation of wild algebra比较有兴趣,甚至一度想去University of Sherbrooke跟Thomas Brustler做这方面的东西. 因为当时的兴趣还没有完全形成,那时候也对几何中出现的representation theory of algebra比较有兴趣.

感觉有时候,过于执着的追求"真实,正确"的信念和行动不见得在任何时候都是好的. 尤其在大三下学期的时候,老琢磨"我做这个东西要干吗?" "这个东西我能做得过那些传统的牛人么?" "做这个东西会和什么东西有联系? 重要么?" 这种现在看来无聊的"哲学"问题. 申到一些学校,就是权衡来权衡去的.我觉得自己不是一个功利的人,但最后却走向了功利的深渊.老觉得"我做这个能做出来么? 我自己能力够不够? 那些大牛们比我学的多多了,我跟他们比无疑就是炮灰"诸如此类."什么是正确的?" "什么是好的?" "什么样的数学是好的? 有前途的?"

对于这些问题,每个人都会有自己不同的答案,肖老师,刘老师,我本科导师的观点都不一样,我自己也有一些看法,那我信谁的? 取中间的? 曾经为这些也苦恼过.

对自己的兴趣的"对错,好坏"的纠缠,一直持续在选offer期间,不过幸运的是,其实也是必然,反复的选择落到了我的"最初的第一感觉"的学校上. 我自己都觉得很可笑,我这是在干什么? 其实早就已经有答案了. 只是我一直在想:"做这些,我能拼得过法国那些有传统的人么?"."几何表示论,一帮子大牛在做,我算哪根葱?" "这个东西好fancy啊" "这个名词好吸引人啊,要不要去了解了解"

把我惊醒的第一个事情是,X老师给我写了一封很长的信,说与其这样反复想,不如foucus到一个具体的问题上,如果要有一些成绩,赶紧收心,打好基础,做具体的问题!

到了Kansas后,尤其在上Rosenberg的课后,由于我觉得他那工作抽象无比,对于他来说脑子里就是几何的东西,对于我们来说,整天就是范畴论,连个category of sheaf都是一个很一般的abelian category甚至是triangualted category.然后就老问Rosenberg:"这个东西有什么用? 你这个概形Algebraic Universal K理论能用到什么东西上?你这个东西到底要干吗?"

然后在跟老板的几次聊天中,我就越来越发现这个我这个严重的坏习惯."这个代数群表示论和代数几何,数论,Lie代数表示论都有关系,很好啊!" "这个Toen的文章用了好多homotopical algebraic geometry,好有意思啊!"老板就问:"你觉得Rosenberg的Lecture是不是很fancy啊,你觉得Toen的文章是不是很fancy啊???" 老板继续说:"我知道你要说fancy,我告诉你,其实一点都不fancy!" 老板继续说:"你给我讲讲什么是model category?" 我说:"*****" 老板继续问:"看样子你很懂嘛.你给我举出两个不同的model category" 我顿时哑口,我不知道例子. 老板继续说:"你知道为什么有exact category么?" 我哑口.老板就说"****" 我说:"这个拓扑背景我怎么一开始学就会知道呢?" "既然这样,那是啊!" 我恍然大悟!! 老板接着说:"你不要老想着这个理论有什么用,去干吗??" "现在好多研究,在做之前,谁知道可以用来干吗?? 如果你都知道了可以干吗,那就不是做Research!! 即使这个结果感觉上来很好,那也是非常trivial的,谁都可以做!这样也是拿不到NSF的,你这不是研究!" 你应该focus到concrete example!!" "不过没关系,我跟你师兄也磨合了2年多才好的" "你师兄跟我一起做research时,至少算了100个Hall algebra的例子"

"你可能会觉得肖杰发在Invention上和Duke上的文章很fancy很power,那也是他们算了好多好多例子才出来的!""肖杰原来到我这里来的时候,我们做affine canonical base,我们几个月算了很多很多例子!"

"做研究,要培养自己的Curiosity,Curisosity is the most important! working on concrete example! 我要你算一个东西,不是说我知道结果是什么,让你算,在你算出来之前我也不知道会是什么.这才是研究啊!!! 只有你自己算你才能找到感觉,找到什么是有意思的,发现新的数学!! 培养自己的taste!!" 做数学不仅仅是去解决别人的问题,关键的是自己要学会提出新的问题! 自己去算例子,自己找出新的数学,这样以后才有独立研究的能力!"

"有件事没跟你说,原来不是让你跟Lusztig聊么? 后来Lusztig跟我说:'he know much things, but I do not know what he can do for me,first he should calculate something!'"

"做数学为什么要和几何有关系,那做几何的是不是为了跟物理扯上关系?! It is not right!"

"你为什么要问这些问题? 是不是怕拿不到奖?!"....

惊醒了我这个梦中人,我一直认为自己不功利,要不然也不会来学纯数,完全是自己的兴趣,但是最后却依然鬼使神差般回到了功利的道路上.

老板继续说:"你看看你现在很熟悉的一些理论都是你曾经具体计算过的,比如倾斜理论,Quiver表示,都是你原来具体计算过的! 你自己都忘了么?"

自从这次谈话之后,我就跟老板要了一个很具体的题目,也不知道能不能做出来,就自己算,这段时间发现自己的兴趣明显stable focus了,也没有象原来那样,见到什么"fancy"就有想学它的冲动,最开始觉得它fancy,就是完全不了解它! 培养自己的curiosity! 天天这样对自己说,不要管他做出来是什么样,他与别的有什么关系,do not care! First work on it!!"

需要什么再去念什么.

我想在成为大家笑柄之前,这是有利的转变.

感谢Xuanyu,Fan.

Check notes

发现Rosenberg定义的几个specturm实在很有意思,由于前段时间没有check他的notes,导致现在他讲的一些很技术性的细节已经完全听不懂了,而下个星期他要讲这个Machine如何attacking 表示论的问题了,所以需要赶快check完第2章.

下课后跟他聊了一会,其实一个很核心的思想就是cover and localization. 一个方向是affine cover, morphism contious and conservative, inverse image functor就是localization of category.跟交换的情形类似,如果取特殊的open cover,say topological bases D(f), then O(D(f))就是R localization at f. 这是一种特殊的Grothendieck Topology.As is well know to all,对于一般的open cover就不一定对了,而是要从topological base上取inductive limit.而对于非交换的情形也是一样,对于好如conservative的cover,那么localization过去就正好是cover对应的范畴. 后来的章节,对于如果cover is not the case of conservative也做了详细讨论,也定义了类似Topological base的子范畴.Serre subcategory 实际上也是一个"open set"in CX.所以在Serre子范畴上的局部化实际上就是用这个"开集"去cover. 而实际上从presheaf到sheaf也是一种exact localization functor! (从交换的情形也好理解)

Grothendieck在Tohoku paper以及后续工作里证明category of coherent sheaves有足够的injective objects,从而可以构造sheaf cohomology.而在非交换情形下,也可以证明对于Cx这个范畴里,也有足够的injective objects, 从而用Ff构造injective resolution,从而也可以有sheaf cohomology! 真是漂亮!

下个星期就要用这个machine attacking Quantum D-module,weyl algebra了,虽然我完全不懂,但还是很期待.这种感觉就类似小学的时候期待星期五,妈妈给我买棒棒冰一样~

下个学期更另人期待,这个学期他讲的是Space represented by Abelian category,事实上这还是Grothendieck的思想. 而下学期就要讲Kontsevich和Manin关于space represented by triangulated category了. 而且会working on differential graded category and A-infinity categroy.这样里面的几何就会很丰富了!

这段日子

就是象这样,没有牵挂的感觉. 见了林老师,他对我很好,当然指出了我一些缺点了,就象在国内肖老师指出的一样: 看书的时候,喜欢今天觉得这个好,明天觉得那个好. 恩,可能有点这种倾向吧.

选了代数拓扑和代数的课. 同时sitting in 代数几何以及非交换代数几何. 林老师用Ueno的书讲AG,很流畅. 而Rosenberg用自己在ICTP和MPIM的讲义讲的非交换代数几何,我对这个十分有兴趣. 尽管我基本只能听懂30%. 但是我决心好好把他的两本讲义读读. 而且他也说在后半学期会主要讲讲他的这套machine怎么在表示论里works.俄国人就是厉害,都是自己建立一套理论体系,把整个理论框架搞出来.

最老板给了一个Topic,最近念点Model category和Dg category的讲义和论文,生活还是很惬意的.

Brief Summary on quiver interpretation on Cluster algebra

I recognized that recent researches on Cluster algebras and representation theory are mostly linked with the methods arising from geometry and other field, such as Cluster algebra and Ringel Hall algebra, the coordinate rings of double Bruhat cells and Cluster algebra, relation between cluster algebra and higher Techimuller space, Cluster algebras and representation of preprojective algebra, semicanonical bases of Luzstig, etc. I am quite interested in part of these researches, especially the Ringel Hall algebra and Cluster algebra. I recalled Claus Michael Ringel ever used Hall algebra to realize positive part of Quantum Groups and J.Xiao realized negative part in the similar way. A natural question arose motivated by these works. Can we realize a cluster algebra of finite type as a “Hall algebra” of the corresponding Cluster category? Caldero and Chapoton obtained a cluster variable formula which gave an explicit expression for the cluster variable associated with a positive root corresponding to an indecomposable module. Then B.Keller gave a natural basis for Cluster algebra labeled by the set of exceptional objects of corresponding Cluster category. Recently, J.Xiao deduced projective version of Green’s formula and applied it to prove that Caldero-Keller’s multiplication formula holds for acyclic cluster algebra of arbitrary type. Now, I am reading and checking this paper. Meanwhile, Buan, Marsh and Reiten gave a quiver interpretation of mutation between Cluster algebra, a one to one corresponding between tilting seeds and initial seeds for an acyclic Cluster algebra was derived. Thus, it seems that Cluster category is a successful model for Cluster algebra and back in turn, Cluster algebra is combinatorial invariant of Cluster category which was proved by B.Keller. On learning these exciting results, I am anxious to enter this field and decide to choose Cluster algebra and related topics for my major research area. As mentioned above, Cluster algebra is invariant of Cluster categories. In another view, we can obtain this result by categorifying Cluster algebra, isomorphism between two algebras becomes triangulated equivalence between two Cluster categories. Maybe another hopeful research direction is generalizing Cluster category to higher dimension for professor Iyama developed higher dimensional Auslander Reiten theory, it gave some interesting combinatorial structures for Higher Auslander Reiten quiver.

挫折

只是个挫折没什么大不了发现自己的问题并改正.

开始申请

Back to Ground

现在安下心来了,我发现我还是在读书,看论文,写数学的时候心里特别宁静. 随身带着某个我特别尊敬的老师的教诲和提醒,时刻检查自己的行为,古人说:"吾日三醒吾身" 现在我也类似. 这两天一直在读[BMRRT]的一篇经典文章,讲Cluster范畴,细节check还算顺利,目前已经读完第2节 Ext-Configuration. 没遇到什么障碍,下个星期打算讲这篇文章. 这里不得不提一个结果[Happel]如果H是一个遗传代数,那么H=KQ ,并且Db(mod-KQ)和Q的Z copy是同构的,这个结果我觉得很好,首先它清楚的展现了遗传代数导出范畴的结构,第二,对讨论班的好处是如果承认这个结果就可以在很大程度上回避导出范畴的知识,以便在讨论班上让没有学过DC的人能听懂如何定义Cluster范畴. 呵呵:)

感觉学数学还是简单最好,每天只想着读书,读论文,讲文章,做习题这几件事,这样一天就过去了,非常地愉快.现在的主要任务就是讲好Semicanonical Bases and Preprojective Algeras的文章,讲好Cluster范畴和倾斜理论的讨论班,学好表示论.

随身带着一位老师的提醒:

学数学需要一步一步来,如果想有所成绩赶快收起心来,打好基础,用最笨的方法多做习题.数学需要从基础做起,并长期坚持.

心平气和,学数学就象小溪,细水长流.

所想

symmetry resulted in groups, another way of saying this is that group is automorphism.For finite groups and symmetric groups, I think it is obvious. Symmetric group is permutation group of N letters. Every element is automorphism. So I consider for fundamental group and homotopy groups. What kind of symmetric lead to these kinds of groups. Finally, I found that the homotopy group of a topological space X is the symmetry of the spaces of all fibers over X. （Kanex）That is the symmetry of the Universal cover of X. Then the fundamental group is the special case of them.Why to choose the group as invariant of Topological space? I am always considering this problem. Just because it is simple? I think the main reason in the symmetry. Say if we have a category C, we want to find some invariant of it in category D, then the functor is all functor from C to D modulo the automorphism of C(Kanex) 环： Ring can be also taken to be an invariant of Topological space. Actually because of more structure comparing to groups. After introducing Cup product,the cohomological group becomes ring. It carries more information. A natural generalization is K group. (Actually K ring). Rings plays important roles in Algebraic Geometry. 模： What is module. I think we should take it in two ways: 1. Module is an action of Ring on solution space. 2. Module ,The action of Ring on Aut(solution space). Considering some generalization of solution space. If we replace solution space by variety.(like elliptic curves,algebraic surface), due 2, we obtain :Some Ring--àAut(S)(Aut(E)). I think this is the better opinion to module. Module is solving the algebraic equation! Of course, why we consider module in Algebra? The reason is that we want to know ring and algebra. We can consider the module category over different rings. We can consider various Derived category of module category to build some connection between the Ring