## 偏执的完美

You must have chaos within you to give birth to a dancing star.
Friedrich Nietzsche（尼采）

## 倾听大自然的箫声

One cannot invent the structure of an object. The most we can do is to patiently bring it to the light of day, with humility – in making it known it is “discovered”. If there is some sort of inventiveness in this work, and if it happens that we find ourselves the maker or indefatigable builder, we aren’t in any sense “making” or “building” these structures. They hardly waited for us to find them in order to exist, exactly as they are! But it is in order to express, as faithfully as possible, the things that we’ve been detecting or discovering, to deliver up that reticent structure, which we can only grasp at, perhaps with a language no better than babbling. Thereby are we constantly driven to invent the language most appropriate to express, with increasing refinement, the intimate structure of the mathematical object, and to “construct” with the help of this language, bit by bit, those “theories” which claim to give a fair account of what has been apprehended and seen. There is a continual coming and going, uninterrupted, between the apprehension of things, and the means of expressing them, by a language in a constant state improvement, and constantly in a process of recreation, under the pressure of immediate necessity.

As the reader must have realized by now, these “theories”, “constructed out of whole cloth”, are nothing less than the “stately mansions” treated in previous sections: those which we inherit from our predecessors, and those which we are led to build with our own hands, in response to the way things develop. When I refer to “inventiveness” ( or imagination) of the maker and the builder, I am obliged to adjoin to that what really constitutes it soul or secret nerve. It does not refer in any way to the arrogance of someone who says “This is the way I want things to be!” and ask that they attend him at his leisure, the kind of lousy architect who has all of his plans ready made in his head without having scouted the terrain, investigated the possibilities and all that is required.

The sole thing that constitutes the true “inventiveness” and imagination of the researcher is the quality of his attention as he listens to the voices of things. For nothing in the Universe speaks on its own or reveals itself just because someone is listening to it. And the most beautiful mansion, the one that best reflects the love of the true workman, is not the one that is bigger or higher than all the others. The most beautiful mansion is that which is a faithful reflection of the structure and beauty concealed within things.

## 左岸的莎士比亚书店和老George

—- 2006年秋于巴黎南郊的法国高等研究院

## 《阳光灿烂的日子？》－周平

［孔良注：这是我在1996年的春天读到的第一篇有关文革中的科大的文字。姜文的电影《阳光灿烂的日子》在1995年9月1日上映，周平的文章大概也是随后出现在网络上的。］

１９６４年，我怀着当居里夫人的梦想跨进了科大的校门，正是风华正茂的年

１９６７年元月一日出现在西单墙上和玉泉路科大校园里的署名“科大雄师战

１９６６年十二月初，我从外地串联回到北京，听说北京的一些高校和中学的

“给江青同志的一封信”等等。觉得这些年青人很有思想，他们的大字报讲得挺有

“中央文革向何处去？”这张大字报是由近代物理系青年教师朱ＸＸ，近代物

ＸＸ逮捕。科大雄师战斗队只有半个月的寿命就垮台了。

，要把我们的后台揪出来，审查来审查去，才发现我们只不过是一群不知天高地厚

。他们还问我是否讲过“江青是小资产阶级感情，爱哭。”这是我写在日记上的。

？今后该怎么办？我找不到答案，觉得好委屈，我想躺在爸爸妈妈的怀里大哭一场
，但他们远在天边，自身难保，那时除了我，我的一家都在新疆农场，爸爸在农场

。我心里一惊，很想到炼焦厂去看她最后一眼，但工宣队讲，谁也不许去。那天早

，也没听出她有什么了不起的大问题，只不过平常聊天时，说了江青几句话。而且

，要不是看你们是青年学生，你早就该坐大牢判刑了，你还嘴硬，你还想翻案，你

，他回答得简单而实在，他说，我真的觉得她很委屈，我不忍心看她一辈子受苦，

），在一打三反运动中，他因为和几个朋友在一起议论过江青而被列为全校第一号

１９７０年分配在河南的大学生都到沉湖去劳动。在沉湖农场，所有的人都知

，但我的心在流血。我是多么卑谦，我没有羞耻感，没有自尊心。只有一个强烈的

１９７２年元月，他听说我要到驻马店报到，就冒着大雪，从杨集步行了五十

。我感到有了依靠，象回到了家。我们终于可以在一起了。经过了那些恶梦般的日

，科大的学生是我国科学技术的生力军，好好学吧，将来有许多工作等着你们呢。

？哎，都是我不好，拖累了祥，连累了孩子。

１９７９年的某一天，我收到科大党委的通知去参加平反大会。在会场上见到

１９６６年底至１９６７年初，周平同志参加“雄师”群众组织，并贴出了“

１９６６年底至１９６７年初，我校教师ＸＸＸ、学生周平、ＸＸＸ等二十多

７年元月被拘留，批斗，因参加过“雄师”群众组织或同情“雄师”观点的ＸＸＸ

１９６７年元月十四日，那是一个多么寒冷的夜。那天夜里，一辆车子到科大

## 《数学的纯粹》葛力明

《数学的纯粹》葛力明

１９６５年， 我出生在江南溧阳一个农民家庭， 母亲没有上过学， 父亲只读过四年小学。 记忆中家里很穷， 小时候有时吃不饱饭， 还三天两头地生病。 １９７３年在医院里住过数月， 半年多没上学，还好在那个年代没有耽误多少学业。１９７８年，恢复高考的第二年，我参 加了中专入学统考。在当地，对一个农村孩子来说，能上中专也就算出头了。中专没念上， 却阴差阳错地被县城的江苏省溧阳中学录取。 两年的中学生活对不少人来说也许仅是两年的 学习经历而已，但对一个体弱、贫穷的孩子意义却是非凡的，它完全是世界的改变。我许多 的人生第一从此开始： 第一次离家独立生活， 要计划如何从每月三元钱的生活费里省出期末 回家需要的八角钱路费；第一次接触到英语、生物、物理、化学等课程，知道除了语文、数 学，学校里还有那么多的课程；第一次体会到什么是学习，什么是无知；也第一次遇到了许 多优秀的同学和老师，其中的一些人对我一生都产生了重要影响。我与数学的缘，最初就是 来自于一位普通而可敬的老师－－王荣章。 王老师是我们农村班的班主任，也是我们的数学老师。他教了我们两年数学，也做了两 年父母。在生活上，他关心我们到每一个细节，他得知我的生活拮据，特地找我父母谈，要 他们给我加点生活费，家里也想方设法地把我的生活费增加到了每月五元。在学习上，他不 仅尽心地教给我们数学知识， 而且对我们学的每门课程都要操心。 王老师为了给我们班争取 最好的老师， 引起了一些个人恩怨并给他的生活带来了许多不便， 但他在学生面前从来没有 表现出一点不愉快，也从来没有和我们谈过他生活上的事。 直到现在，我每次回溧阳都去看他，一见面就谈数学，他很关心我做的数学、思考的数 学问题，而且每次他都准备了很多数学问题问我，还经常要我给他寄些相关资料，他对数学 界的动态比我清楚得多， 我也很看重他的一些建议。 在我心目中， 他永远是一个合格的老师， 因为他很敬业，总是把数学教学和学生放在第一位。多年来，他一直在辅导中学生数学，虽 然在别人看来他打游击似地被各种学校临时聘用着， 但他生活得很快乐， 因为有数学陪伴着 他。他是我遇到的第一个纯粹地迷恋着数学，从不计较个人得失的人。 那两年中，要学的东西实在太多，没等我把周围的人和事弄清楚，中学时代就结束了。１９８０年高考后， 第一志愿我选择了北京大学数学系并如愿以偿。 是王老师把我领进了数 学的大门，我的人生从此又翻开了新的一页。

## 1+1= 2 ?

－－伟大的数学家Alexander Grothendieck

1＋1＝2 是一个很难的问题。我们真正理解了吗？

（1）O+O=OO,

OO （左）
|  |
OO  （右）

OO  （左）
X
OO  （右）

（2）J+J=JJ

OO
|  |
J J

O O
X
J  J

OOOOOOOOOOOOOOOOOOOOOOOOOOO 和
JJJJJJJJJJJJJJJJJJJJJJJJJJJJJJJ。

1。我们其实顶多能够把“＝”理解成一种对应。
2。而这种对应往往还不唯一。两个东西可以以两种不同的方式“一样”，还是一样吗？ What does this mean?
3。这两条其实在说，我们原来以为的“＝”是一种“绝对的一样”（我也不知道我指什么，你们知道吗？）可能是一种误解。

1. equivalence relation （等价关系）
2. equivalent class （等价类）
3. equipollence

OO，JJ，AA，PP，木木，aa，   ， 。。。。

OO，JJ，AA，PP，木木，aa，   ， 。。。。

“equivalent class” is in general not a set! It is a class. Oh, my god? What is a class?

1。最终我们是不是能完全理解1＋1＝2？Very unlikely！公理是人写出来的，不是God given的。即使我们建立了集合论的公理系统，而且完全定义了1＋1＝2。因为达到同样目的的公理系统不是唯一的，最终我们仍然需要选择和设计我们的公 理，要做一些人为的和不一定自然的选择。这种不唯一性是会令人不安的。

2。更有甚者：数学的基础也不一定要建立在集合论上。还有别的可能。比如也可以建立在范畴学（category theory）里的elementary topos理论上，最近一些年有新的一些趋势，就是在higher category的框架下建立一个新的数学基础。

## A visit to Stony Brook

s 3/1/2010:

Without being informed before it actually happened, I found myself sitting with K. Costello, M. Douglas, J. Morgan, D. Sullivan, A.J. Tolland, B. Vallette etc. in the common room of the Math department at Stony Brook. It was supposed to be one of a series of dialogues between physicists and mathematicians. During the conversation Douglas proposed a conjecture of Kontsevich-Soibelman:

Conjecture (Kontsevich-Soibelman):

A CFT satisfies gapped positive energy condition if all the eigenvalues $E_i$ of the Hamiltonian ($H=L_0 + \bar{L}_0$) are non-negative and $E_i=0$ or $E_i \geq h$ for some $h> 0$. The conjecture is that the space of all CFTs with a fixed central charge satisfying gaped positive energy condition is precompact.

The conjecture is an analogue of a theorem in Riemannian geometry (I don’t know if the following statement of the theorem is correct. I just copy it from Douglas’ handwriting on blackboard. The precise statement is not very important to us.) :

Theorem (Cheeger-Gromov):

The space of Riemannian manfolds with the diameter of $M$ smaller than $\frac{1}{n}$ and satisfying $|K| < 1$, $\mathrm{vol} M \geq \epsilon$ is compact.

Kevin Costello was asked if it is true in TCFT context. Kevin replied no. It quickly became clear that it is perhaps a good sign because the states in a TCFT are all vacuum states. Maybe it can be viewed as an evidence of the necessity of the gapped positive energy condition.

One might feel weird when he/she see this conjecture for the first time. The conjecture is not so well-defined. We don’t even know what the definition of CFT is, needless to mention the topology on the space of CFTs. However, this conjecture make perfect sense to me!

After Michael Douglas wrote down the conjecture, Kevin asked immediately how they come up with such conjecture. Michael answered “well, you had better ask themselves.” Although this conjecture make a lot of sense to me immediately, only when I was preparing my talk which was delivered today, I found more to say about this conjecture. So I added it to my today’s talk as the last part.

After nearly an hour introduction to my own proposal that a CFT should be viewed as a stringy algebraic geometry or a 2-spectral geometry, which should recover Riemannian geometry in certain classical limit, I put down the following derivation of K.-S. conjecture:

a 2-spectral geometry = a CFT

$\mathrm{diam}(M) < \frac{1}{n}$ $\Longleftrightarrow$ energy is gaped.

$|K| \epsilon \Longleftrightarrow$ ??

By adding the natural positive energy condition and ignoring ??, we arrive at the K.S. conjecture.

It is already quite interesting. But one should not stop there. It is natural to ask if we can do better?

If you are familiar with the classification of open-closed rational CFTs, you definitely can say more. Although the classification result is only available for the rational cases, it indeed suggests a lot for the irrational case as well. For example, it is reasonable to believe that the closed algebra in an irrational CFT should also be a commutative symmetric Frobenius algebra in a braided tensor category. So let $\mathrm{Vir}_c$ be a VOA generated by its Virasoro element with central charge $c$. It is natural to expect that the category of $\mathrm{Vir}_c$-modules, satisfying gapped positive energy condition plus some other natural conditions, gives a braided Frobenius tensor category as some kind of a categorification of a commutative Frobenius algebra. Then we can reformulate K.-S. conjecture and propose the following three potentially different conjectures.

Conjecture 1: The space of isomorphic classes of commutative symmetric Frobenius algebras in $Z(\mathcal{C}_{\mathrm{Vir}_c})$ + some conditions (modular invariance condition) is pre-compact.

Conjecture 2: The space of the Morita classes of simple symmetric Frobenius algebras in $\mathcal{C}_{\mathrm{Vir}_c}$ is pre-compact.

Conjecture 3: The space of the equivalent classes of indecomposable module categories over $\mathcal{C}_{\mathrm{Vir}_c}$ is pre-compact.

Look, these conjectures are still not well-defined because we don’t know what topology to choose if there is any interesting topology at all. But let us leave such problem aside for now. Conjecture 3 reminds me of a conjecture by Ostrik:

Conjecture (Ostrik): For a given rigid monoidal category $\mathcal{C}$ with finitely many irreducible objects there exists only finitely many inequivalent indecomposable $\mathcal{C}$-modules.

Motivated by Ostrik’s conjecture, we would like to give a continuous version of it.

Definitions:
1. A topological Frobenius tensor category $\mathcal{C}$ is a Frobenius tensor category endowed with a proper topology.

2. $\mathcal{C}$ is called compact if the space of 0-dimensional objects (whatever it means, D0-branes?) is compact.

Then we can propose the following conjecture:

If a topological Frobenius tensor category $\mathcal{C}$ is compact, so is the space of equivalent classes of indecomposable $\mathcal{C}$-modules.

I don’t know how to make the space of equivalent classes of indecomposable $\mathcal{C}$-modules into a topological space. But I hope that readers will find this endeavor interesting.

I don’t remember how the dialogue was ended. I forgot to ask them if such a dialogue is a regular event at Simons Center. But my impression is that it is certainly not the first one nor the last one. I expect that it will happen very often in the future. I think that Stony Brook is a very good place to study mathematical physics.

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