Knowledge, Technique,Knowledge, Technique,and Imaginationand Imagination

Kefeng LiuKefeng LiuHangzhou, April 2004Hangzhou, April 2004

I want to discuss the importance of knowledge through some of my personal experiences and thinking.

知识 , 技巧与想象力的经验与反思 .

From very beginning to now I have been working on geometric and topological problems from mathematical physics.

It is very exciting: physicists need mathematics, they made mathematical conjectures based on their physical theories. We solve these conjectures, which verify their theories, and motivate their new theories. Such interactions are most fruitful for both disciplines.Many revolutions were created this way.

God created the world according tomathematical formulas!?

Examples:Calculus and Newton’s laws.

General relativity and Riemannian geometry.

Quantum field theory and index theory: elliptic genus.

Conformal field theory and moduli spaces: Verlinde formula.

Yang-Mills and 4d topology.

Chern-Simons and 3d topology, knot theory.

String theory, mirror symmetry and Calabi-Yau: Hori-Vafa conjecture.

Chern-Simons, Calabi-Yau and Gormov-Witten: Marino-Vafa conjecture.

String theory and 3d topology, Ricci flow.Mirror symmetry and number theory....

How physicists learn mathematics so fast:　 They learn index formula first, then Riemannian geometry. (Taubes)循序渐进 ? 法乎其上 !

They always go to infinity: SL(2, Z); large N Chern-Simons; path integrals…. 妙在无穷 , 美即有用 .

Beauty ~ Infinity; Beautiful = Useful.

How should we learn mathematics?

In September 1988 I walked into the office of Yau(丘成桐 )at Harvard.

He asked me: want to start to do research or want to learn more?

I said: I want to start to do research.

He said: try to learn as much as you can. After you graduate, it is not easy to learn.

He asked me to study algebraic geometry, algebraic number theory, geometric analysis.... I learned many things that I did not understand at all at that time, many not even today.

This influenced my career and my life.

Knowledge and technique: which one is more important?

For young students: knowledge! Knowledge makes us stand high.

Mathematical competitions: emphasize too much on techniques.

Our students need more extensive knowledge starting from high school.

I will use my own experience to discuss the importance of learning more and the interactions among the different branches of mathematics and with other sciences like physics, also with friends.

Upon arriving at Harvard, what moved me most is the hard working style of the professors and the students.

What we lack in China now is nothing but such kind of style.

What drives people there to work hard is the curiosity and “love” of mathematics, and the intellectual challenge. 良性竞争 .

I felt diving into the sea of knowledge, the Sun of everyday seemed different. Very exciting years.

While in the Graduate School of Chinese Academy of Sciences, our classmates had seminars, we discussed things we did not understand at all, like Chern classes, index theory, Mordell conjecture....

This opened our eyes. We at least understoodwhat kinds of mathematics are “good”, andworth to learn. This is very important for allof us. We developed our own tastes of math.

If you are not sure what mathematics are good, then just follow the “Masters”.

Witten’s paper “Supersymmetry and Morse Theory” (乾坤大挪移 )had most influence on me, so were Bott’s style of research:厚积薄发 , 举重若轻 .

The idea of localization was reallyappreciated and understood during that period.

Bott: Never resist the main stream of mathematics. 顺流而下 , 不要逆流而上 .

All revolutions of mathematics are brought about by new ideas and methods, and the merge of existing branches of mathematics.

One needs to be creative and more learned to be part of a revolution.

The interactions of mathematics and physics are no-doubt one of the most exciting main streams in mathematics.

Physicists made conjectures which are far above our imaginations: infinity.

Most of them are proved to be correct.

Examples:1.Calculus and linear algebra: differential geometry.

2. Symmetric functions, more generally representation theory of compact groups: Chern classes, K-theory, Riemann-Roch, index theory.

3. Algebraic number theory and differential geometry: arithmetic geometry, the Arakelov theory; Faltings’ proof of the Mordell conjecture.

4. Modular forms, representation theory and topology: elliptic genus.

5. Interactions of mathematics and physics: Newton equations; Einstein’s general relativity; Maxwell equations; Yang-Mills; Witten, Vafa and duality in string theory.

6. Mathematics and life: invention of computer; Maxwell equations in life: electricity; internet....

Mathematicians have contributed so much to the society, and daily life.

I will discuss a few topics that I worked on to illustrate the importance of knowledge, and the interactions of mathematics and physics in my own researches.

Imagination is more important than Knowledge (Einstein).

Elliptic Genus

This is my thesis at Harvard. When Witten made the conjecture about rigidity of elliptic genus, based on quantum field theory. Taubes and Bott worked on it for quite a while. Professors at Harvard and MIT had seminars at Harvard, which I attended.

Elliptic genus is a combination of index theory and modular forms, it is index theory on loop space.

Rigidity needs Jacobi-theta functions and modular forms which came from number theory. The symmetry by SL(2; Z) is magical. It is the guiding principle of duality in string theory. It also appeared in representation theory of affine algebras and loop groups.

The knowledge of number theory and algebraic geometry drove me to understand the rigidity from modular surfaces. I believed that the symmetry of SL(2; Z) should imply the rigidity.I came up to the idea while on the way to Princeton and back to Harvard, for a Table Tennis tournament.

But it took me several months to finally get the proof. Several times I was so frustrated, but I strongly believed that if such a beautifulidea did not work, then mathematics would not be interesting, and I would give up doing mathematics. The final proof came out while I was watching a movie.

Having learned more knowledge helped me for this first step of my research career. I understood better those knowledge through the research.

The method went much farther than I expected. It is still useful today.

数学好玩 !?!?!?

Moduli Space and Heat Kernel

Around 90s, Verlinde formula was a very popular topic. It came from a conjecture from conformal field theory. It computes the dimensionof the holomorphic sections of a canonical line bundle on the moduli space of stable bundles on Riemann surfaces.

Such moduli spaces had been studied from many point of views in mathematics, especially algebraic geometry and topology. Thecomputations of such dimensions have resisted many approaches in mathematics. But the string theorists gave the surprisingly simple andclosed formulas.

Witten conjectured the closed formulas of intersection numbers of the moduli spaces. Such intersection numbers in principle gave the Verlinde formula, through the Riemann-Roch, or the index formula.

While at MIT, I attended many seminars on such topics, trying many different ways to understandthe formulas of Witten, which are ainfinite sum over all irreducible representations of compact Lie groups. I understood symplectic geometry during those days.

One day in MIT library, I randomly saw the expression of heat kernel on Lie groups, which is given by an infinite sum of the same type. Seeing that convinced me that Witten’s formulas could be derived from heat kernel on Lie groups.

Again this took me several months to finally get the rigorous proof. Motivated by my work, Bismut was able to use this method and these formulas to give the proof of the general Verlinde formulas.

“Open a book, you learn.” 开卷有益 .

Mirror Principle

One hour before I left Boston to Stanford, Yau called me to talk about mirror symmetry. In 1990Candelas el al made a beautiful conjecture about the numbers of rational curves in Calabi-Yau quintic. Mathematicians have tried for hundred of years to compute such numbers, they got the first3 numbers. But the Candelas formulas gavethe formulas of all such numbers, in terms ofthe solutions of a simple ordinary differentialequation, the Picard-Fuchs equations.

I did not follow the progress of mirror symmetry before 1996. But the phone callof Yau drove me into it. I read papers andthought hard. One day I realized one key pointfor the solution which is the inductive structureof the moduli spaces of stable maps. Thenwe realized the “functorial localization formula”,which was a powerful tool for several of mysubsequent works, is the key for the proof.

“Among three persons, there is a master.”三人行 , 必有我师 .

Hori-Vafa Conjecture

My years at UCLA has been the most productiveyears so far. During the work of mirror principle,we keep close eye on the related development.

String theorists also ”proved” mirror conjecture with various new conjectures. Hori-Vafa conjectured a formula to write down the explicit formula for Grassmannian manifolds. On the other hand, mathematicians also made some related conjectures.

To understand them, we found the functorial localization formula can be used to prove such conjectures, but applied to another moduli: quot-schemes. This involved difficult combinatorics and algebraic geometry. Chien-Hao Liu was able to push all of these through.

It is important to make good friends, and to talk to good and right persons.

Marino-Vafa Conjecture

Since Kontsevich’s proof of the Witten conjecture, which claimed that the generating series of integrals of certain Chern classes on the moduli spaces of curves satisfy infinite number of differential equations, I have been paying close attention to the subject. The extension of mirror principle to higher genus case also needs more general integrals of such types. Finally I came across the Marino-Vafa conjecture.

Marino and Vafa conjectured that the generating series of more general classes of integrals on the moduli spaces of curves are given by closed formulas from representations of symmetric groups, which are also the Chern-Simons knot invariants. The conjecture was based on the string theory and Chern-Simons duality.

In summer 2002 Jian Zhou and I had discussed on such topics like mirror symmetry during our several trips: Beijing to Hangzhou, to Shanghai to Beijing. After that we have many discussions through email. This is very fruitful. Zhou Jian clarified the combinatorics in the formula and realized the combinatorial cut-and-join equation, and the next step was to prove the same formula for the Hodge integrals. We tried many different ways with functorial localization.

Chiu-Chu Liu came to UCLA in April, 2003. I told her our project after she attended the graduate course I was teaching. We discussed the problem. Very quickly we three together solved the conjecture. This is a very pleasant experience.

So to be in a group of smart people, you will become smarter. 近朱者赤 , 近墨者黑 .

Yau’s Conjecture

In early 80s, after he, with S.-Y. Cheng and Mok proved the existence of the Kahler-Einstein metric on the Teichmuller space of Riemann surfaces, Yau has conjectured the equivalence between this metric and several classical metrics, including the Teichmuller metric and Kobayashi metric. Such metrics have been studied for many many years. McMullen also introduced a new metric which he used to prove thatthe moduli spaces of Riemann surfaces are Kahler hyperbolic.

I have been interested in metrics on Teichmuller spaces and moduli spaces.

When I was a student, I went to classes, trying to learn the subject. I realized that the curvature property of the Weil-Petersson metric on the moduli space implies several important results in algebraic geometry.

I wrote two short papers on the topic. This is the most effective way to learn a subject.

Discussing with Xiaofeng Sun at UC Irvine,whom I knew since he was a student at Stanford when I was teaching there, was again fruitful. We discussed the problem together with Yau. Finally we were able to understand these metrics and proved the conjecture of Yau. There are still a lot to go on.

Summary of Experiences

Knowledge is more important than techniques when we are young. 爱知识 , 爱你没商量 .

To have good friends to discuss and to learnfrom each other. Be open-minded to all kinds of knowledge. 珍惜好朋友 , 胸襟开阔 .

Imagination and strong belief can only be built on extensive knowledge. 想象力与信念依于宽广的知识 .

We all belong to “Chern Classes”! 物以类聚 , 我们都属陈类 ( 实属 Yau 言 ).

Thank You All!Thank You All!