## 《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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## Grothendieck on innocence

Let me start my blog with some powerful words by Grothendieck that I would like to share with you all.

In our acquisition of knowledge of the Universe (whether mathematical or otherwise) that which renovates the quest is nothing more nor less than complete innocence. It is in this state of complete innocence that we receive everything from the moment of our birth. Although so often the object of our contempt and of our private fears, it is always in us. It alone can unite humility with boldness so as to allow us to penetrate to the heart of things, or allow things to enter us and taken possession of us.

This unique power is in no way a privilege given to “exceptional talents” – persons of incredible brain power (for example), who are better able to manipulate, with dexterity and ease, an enormous mass of data, ideas and specialized skills. Such gifts are undeniably valuable, and certainly worthy of envy from those who (like myself) were not so “endowed at birth, far beyond the ordinary”.

Yet it is not these gifts, nor the most determined ambition combined with irresistible will-power, that enables one to surmount the “invisible yet formidable boundaries” that encircle our universe. Only innocence can surmount them, which mere knowledge doesn’t even take into account, in those moments when we find ourselves able to listen to things, totally and intensely absorbed in child’s play.

— Alexander Grothendieck

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