印度行纪(四)
再来说说几个我比较感兴趣的Lectures。数论方面的有
C. Breuil, IHES, France
The emerging p-adic Langlands programme;
K. S. Kedlaya, MIT, USA
Relative p-adic Hodge Theory and Rapoport-Zink period domains;
M. Kisin, Harvard University, USA
The structure of potentially semi-stable deformation rings;
K. Soundararajan, Stanford University, USA
Quantum unique ergodicity and number theory;
R. Greenberg, University of Washington, USA
Selmer groups and congruences
A. Venkatesh, Stanford University, USA
Statistics of number fields and function fields
T. Saito, University of Tokyo, Japan
Wild ramification of schemes and sheaves
S. Morel, Harvard University, USA
The intersection complex as a weight truncation and an application to Shimura
Varieties
其中K. S. Kedlaya和K. Soundararajan都是印度人,是现在数学界的新秀,K. Soundararajan获奖的呼声很高。
在Algebraic and Complex Geometry方面,有:
R. Thomas, Imperial College, London, UK
An exercise in mirror symmetry
S. Saito, University of Tokyo, Japan
Cohomological Hasse principle and motivic cohomology
D. Kaledin, Independent University of Moscow, Russia
Motivic structures in non-commutative geometry
C-C. M. Liu, Columbia University, USA
Gromov-Witten theory of Calabi-Yau 3-folds
V. Srinivas, Tata Institute of Fundamental Research, India
Algebraic cycles on singular varieties
R. Thomas这哥们挺搞笑的,他是topology组的Chair,穿着条沙滩裤,踩着双拖鞋,在那里吊来吊去,据gowers说,这哥们相当崇敬Jacob Lurie,觉得Lurie的工作将会革新一部分数学。
Geometry的就只听了麻小南的一个报告,
X. Ma, Institut de Math´ematiques de Jussieu, Paris, France
Geometric quantization on K¨ahler and symplectic manifolds;
麻小南的报告很精彩,报告他和别人合作解决了上一届ICM上一个法国女数学家提出的一个猜想。
Topology 组的听了两个:
D. Gabai, Princeton University, USA
Hyperbolic geometry in the 2000s
J. Lurie, Harvard University, USA
Moduli problems for ring spectra;
Gabai,Princeton的教授,讲话很有特色,比较轻柔,看上去是那种大智若愚,天才型的数学家,他在低维拓扑理做了很多非常重要的工作。
Lurie曾在普林斯顿高等研究院做过maston.morse 年度演讲,看来IAS的那些大佬们很欣赏他的工作。Lurie讲了他的大框架理论里的一个小脚手架,向人们描述了他现在做的东西的一个概貌,我估计连概貌都算不上,他的语速很快,讲到最后,似乎还没有把他的准备的内容讲完。我很看好lurie的工作,给我的感觉是,他在弄一套全新的理论,而且,在某种 special case的情况下,与经典理论有很好的吻合。在与gaitsgory的一篇文章里讲了如何把他的关于环的谱理论应用于代数群的表示论。感觉他的看家本领还是代数拓扑里面的东西,他的环论是加了代数拓扑结构的,也就是附带上homotopy group的结构,这样,自然就与非交换几何联系上了,貌似可以扯的很广。下面我摘录了一段Gowers的关于Lurie话,可以当八卦看看:
That was the regular 1.45-2.45 slot. I then had to decide which talks to go to in the parallel sessions. I decided on two prodigies: Marianna Csörnyei and Jacob Lurie. Marianna I have known quite well for many years, and she works in areas that I can be expected to understand reasonably well (which is not quite the same as saying that I do understand them reasonably well, but in fact I do usually follow quite a bit of her talks). Jacob Lurie is the opposite: I had never met him, or even seen him, and had absolutely no chance of understanding anything he would say, so I was going for the sole purpose of gawping.
Why do I describe them both as prodigies? Well, in Marianna’s case there are some extraordinary anecdotes about how David Preiss, then at University College London and now at Warwick, brought her over to England at some extremely young age such as 18 (I can’t remember exactly) and set her unsolved problems with deadlines of, say, 48 hours. In case you want me to confirm what I’ve just written, I do indeed mean that he would say things like, “This is an interesting unsolved problem: you have until the day after tomorrow to tell me the answer.” And the even more extraordinary thing was that Marianna would indeed come back two days later with the answer. And I’m not talking about something that happened just once: it was a regular occurrence. The problems were things like finding sets of real numbers with extraordinary properties, and to solve them Marianna would produce incredibly delicate inductive constructions of sequences of sets that would tend to a limit with the desired properties. I think there are similar stories about Jacob Lurie — I heard about professional mathematicians consulting him when he was about 16 and getting the answers they sought the next day. And the amazing thing was that it wasn’t amazing — they consulted him because they knew he would be able to do it.
While I’m on this subject, let me counteract it with another story, that of James McKernan. He was an undergraduate at Trinity College, Cambridge in the same year that I was. I did not know him all that well (there were 40 of us), but I remember him as someone who was … I don’t know quite the right way of putting it because the truth is that I don’t remember all that well how good he was at mathematics at this stage. But the fact that I don’t remember puts upper and lower bounds on his performance: he wasn’t one of the best in the year and he wasn’t one of the worst in the year. I think the result of that was that he did not get a PhD place in Cambridge, but it’s possible that he just didn’t want to do a PhD here, and so he disappeared out of my life — I think he did his PhD at Brown — to reappear over twenty years later as the coauthor with Christopher Hacon of some astonishing papers that completed Mori’s minimal model programme. Now I don’t know what that programme is, but I do have some conception of how important it is, and have seen how excited algebraic geometers are about these developments.
Both Hacon and McKernan were invited speakers at the ICM. I was strongly tempted to go to McKernan’s talk, but, agonizingly, it clashed with Assaf Naor’s and I went for the latter. It occurs to me that, given that I tended to bump into the people I knew about two or three times a day, I may have accidentally cut McKernan dead a few times (I stupidly didn’t remind myself what he looked like until well after I had left Hyderabad). James, if I did, and if by any chance you read this, then please know that it was the opposite of what I planned. What I would have liked to do is tell him how pleased I was at how well he has done. I don’t know whether it is right to describe him as a late developer, but the evidence I have suggests that that is a reasonable description. I hope it is, because I very much like stories of late developers: I think it is important to show the world that if you are not a Marianna Csörnyei or a Jacob Lurie, then you still have a chance of proving major theorems.
Marianna’s talk was packed, partly because the room was much too small for an invited lecture at an ICM, so I ended up standing. Given my state of tiredness and the temperature in the room, this was both a good thing (it stopped me going to sleep) and a bad thing (it was pretty tiring). Marianna spoke quietly, but loudly enough to be audible in that room, at least if there wasn’t any background noise. I’ve forgotten what she said about her own work, except that it sounded amazing in the way that it always does, but let me mention a pair of results of David Preiss that she mentioned as part of her introduction. (Added later: if you want to know about more than this, here is her ICM proceedings article.)
After Marianna’s talk I allowed myself a break (as I had in the morning when I skipped Carlos Kenig’s plenary lecture — today, as on the previous day, I was sufficiently worried about overdoing it that I missed some talks that I would ordinarily have liked to go to) of just over an hour until Jacob Lurie’s talk. I thought that there were likely to be many people besides me who would be there for entirely extra-algebro-topological reasons, so I decided to turn up ten minutes early. The room was already very full, but I tried a tactic that sometimes works — to march to the front, spot one seat right in the middle of the fourth row that nobody has quite been able to face getting to, and to face getting to it. OK you have to climb over six sets of knees, but it seems that people don’t hold permanent grudges against you for this (except perhaps if you arrive late for a film and clamber over someone, blocking their view at a crucial moment, but then you’ll probably never see them again).
As I’ve already made clear, Lurie is a certified genius. I mean “certified” in the sense of “universally acknowledged” but there was just a hint of an alternative interpretation in the way that he moved his head, which seemed somehow more loosely attached to the rest of him than most people’s heads are. We sat waiting for quite a long time, partly because I had arrived early (which I was glad of, because there were plenty of people standing for this talk as well) and partly because Richard Thomas, who was chairing the session, was trying to persuade the conference organizers to remove the partition between the room we were in and the room next door, which was empty. Unfortunately, he failed to persuade them, although in another way it wasn’t unfortunate: I think it does something good for the atmosphere of a talk if the room is packed.
What can I say about the talk itself? Well, early on he put up a slide that had the following names on it: Deligne, Drinfeld, Feigin, Hinich, Kontsevitch, Milgram, Schlessinger, Soibelmann, Stasheff. I haven’t heard of all of those, but I’ve definitely heard of some of them and was duly intimidated. And Deligne-Mumford stacks made an appearance too (you may remember those from Ngo’s work). There were also some things called Artin stacks.
By the end of the talk, Lurie had leapt into first place for interesting or amusing quotations with three. (The next day David Aldous managed two, but he had an hour.) The first one was one of those bits of folk wisdom that I mentioned earlier: having pointed out that it was difficult to define (or discuss, or do something to) something or other using equations, he said, “If you can’t use equations, then what you want to do is use words.” In other words, you wanted to be more conceptual about what you were doing (where “you” means “Jacob Lurie”).
I’ve written, “Lots of mathematical structures — abstract.” What I meant was that to someone like me, who still has the temerity to think about mathematical objects rather than sticking to sets of sets of sets of objects (all very nicely structured of course), the experience was a sort of bombardment. I don’t really understand why I should be happy that we can define a canonical sequence of graded Artin stacks or whatever it might be (whatever it was, it wasn’t that, but it sort of sounded like that). But I wasn’t there to understand — just to drink in the experience while Lurie told us, “And then one can do A, and then B, and then C,” and the algebraic structures he mentioned became more and more sophisticated (or did they? I can’t claim to be sure of this).
I think Lurie is slightly sensitive to the criticism that his work is too abstract (not that I’m making that criticism myself — I’m not really in a position to judge how abstract it is). This sensitivity led to the second of his great quotes, which came about two thirds of the way through, when he said, “I don’t want you to think all this is theory for the sake of it, or rather for the sake of itself. It’s theory for the sake of other theory.” This got a good laugh, as you might imagine. He said that he would demonstrate that by giving us an example, which to my inexperienced eyes seemed to be yet another building of algebraic structures, but I suppose to be fair to him he did say at one point (or at least I wrote), “ is the moduli stack ,” where I think had been an abstract something or other about which something in one of his abstract results had been stated.
There was one bit that intrigued me, where he talked about things called algebras (which I would completely have forgotten about had I not written anything down). If I remember correctly, an algebra is associative, and as the algebras become more and more commutative in some sense. Additive combinatorialists will see why I found this idea appealing, though I think the appeal might vanish on closer investigation (or rather the reasons for it might — I wouldn’t rule out their being replaced by different and better reasons).
The third quotable sentence came after 35 minutes or so. He said, “I expected that I would be going overtime, but I think I haven’t.” And that was the end of the talk, apart from some questions that sounded frighteningly intelligent to me. I found myself wondering about a nightmarish scenario in which my brain suddenly inhabited Lurie’s body. Would I be able to answer the questions in a way that would seem genuine to most of the audience? Each time Lurie answered one, I realized that the answer was definitely no — there were little touches of a kind that I just wouldn’t have thought of, that made it clear that some kind of communication really was going on.
A short time afterwards I found myself chatting with Richard Thomas, who clearly had a very great respect for Lurie. I asked him two questions that I can now remember. The first was, roughly speaking, whether it was true that Lurie is revolutionizing mathematics. The answer is yes, apparently. Richard told me that Lurie has a huge programme and is slowly working through it, writing hundreds and hundreds of pages. (I think I had heard this from other sources too.) The second question was whether all this theory was leading to solutions of open problems that could not be solved without it. The answer to this is also yes, apparently, though it seems that the process of using the theory for applications is in its infancy. In other words, we can expect to hear a great deal more about Lurie over the next few years. He also said that Lurie has made several claims about what he will eventually be able to do, and already has an impressive record of backing those claims up with results. It’s just that there’s a lot more work to do.
Lie Theory and Generalizations组的听了一个,
E. Lapid, Hebrew University of Jerusalem, Israel
Some applications of the trace formula and the relative trace formula;
Lapid,以前在山大时就听他讲过trace formula,那时,他老师Gelbart也去了。Lapid好像信仰犹太教,吃素,而且留着一头很劲爆的头发。
Mathematical Physics组的听了三个,
K. Wendland, Augsburg University, Germany
On the geometry of singularities in quantum field theory
A. Kapustin, California Institute of Technology, USA
Topological field theory, higher categories and their applications
M. Marcolli, California Institute of Technology, USA
Noncommutative geometry and arithmetic
Kapustin 在他的lecture中也提到了Lurie的工作,higher categories原来是有物理背景的,物理学家似乎是在很自然的使用着这些高度抽象的概念。看来,懂一些物理,对自己的数学思考是绝对有帮助的。我们现在去看那些19世纪的大师们,他们几乎都懂物理,而且,代数,几何,分析(特别是PDE),拓扑,非常和谐的融合在他们的每一件工作中。难怪陈省身先生和Weil都不断强调,一定要看19世纪那些大师的文集。
M. Marcolli是A.Connes的学生,非交换几何。现在已经变得非常非常广乐,在数学版图上,几乎无孔不入,有人形容A.Connes,说他就是个卖大力丸的。M. Marcolli把Noncommutative geometry和arithmetic
挂上钩,这就有点看头了,不过,我目前暂时不懂Noncommutative geometry,仅仅只晓得她演讲中关于算术的那部分,把那些经典结果非交换化,我就晕了。
Ngo的工作真正代表了现在数学最核心的部分。这里要无可避免的谈到Langlands programme,而Langlands programme中,最重要的研究对象之一是Automorphic forms,Automorphic forms是reductive algebraic groups上一些算子的eigenforms,这些eigenforms所对应的eigenvalues蕴涵了很多算术信息,这些算术信息或者来自数论,或者来自算术几何,在Langlands最初给Weil的一封信中最早的提出了现在的所谓Langlands programme中一些最重要的猜想和哲学,Langlands的目的,是想能用一种可以操作的方法(也就是可计算)来控制这些Automorphic forms及其对应的eigenvalues,从而,也就完全掌握了所对应的一些最深刻的算术信息,他的总体哲学是,用代数群上的调和分析(也就是Harish-Chandra的那套庞大,复杂的理论)来刻画算术中最深刻的一些性质。现在,我们对数论中的很多问题的认识,还系那个当肤浅,随便给你一个丢潘图方程,你可能都会感到不知所措的,因为目前根本就没有一种统一的方法来处理这种算术方程。现在衍生出的这些大量的抽象,复杂的理论,其根本的动力,还是来自于研究这些看上去形式很简单的算术方程的解。比如,x² + ny²=p,这个方程有哪些整数解?问题看上去很简单,但是,要真正去解决它,需要一套庞大的理论,那就是class field theory,现代数论最核心的理论之一。而目前,在研究Automorphic forms时,一个最有力的工具是trace formula,而当我们要用trace formula来研究automorphic representations时,我们就必须使用fundamental lemma来保证这种使用时合理的。Ngo证明fundamental lemma,使用了一些代数几何中最深刻的工具,比如perverse sheaf,另外,他的证明的一个核心思想史,引入了新的几何思想:the Hitchin bration,the Hitchin bration最初是被用来研究一个黎曼曲面上向量丛的模空间的,怎样把这个几何的,大范围的概念改造得适应解析的trace formula,这里就必须用到scheme的一个推广概念:stack(法文是champs),具体的说,Ngo的证明中,是使用了Artin stack和Deligne-Mumford stack。可以粗略的说,Hitchin _bration 可以看成是global trace formula的一个几何化版本。有了这种看法后,思路就开阔多了,代数几何里的一些工具就可以拿来用了,比如Springer fbres,这个概念最初是被Springer用来研究weyl 群的不可约表示的分类的。而Affne Springer fbres,则是由Kazhdan 和 Lusztig引进的,他们重点研究了它的一些几何性质。而Ngo观察到,Affne Springer fbres和Hitchin fibre是相容的,并且,在证明中,他又引入了另一个概念:picard stack。Ngo最终要证明的事情就是要stabilize the anisotropic part of the traceformula,注意,这里的指的是代数几何中的Grothendieck-Lefschetz trace formula,这里Ngo借用了Goresky, Kottwitz 和 MacPherson.的一些思想,不过,他们三个处理的情况是一般的 equivariant cohomology of a local affine Springer fibre.,而Ngo处理的情况是关于 global Hitchin fibration,的perverse cohomology 。最终,Ngo通过建立一个几何化的关于稳定化的等式后,就可以推fundamental lemma了。Ngo的整个证明相当庞大,是一件杰作,应用了代数几何和自守形式理论里面一些最深刻的结论,是整个Langlands programme里面的一个很大的进展,不过,我们对Langlands programme里面的一些问题和猜想的了解,还远远没有达到我们所期望的程度,所以,Langlands programme,在将来仍然是数学的一股主要潮流。