## April 15, 2010

### Paris in the Spring

#### Posted by David Corfield

I’m just back from a couple of days of philosophy of mathematics workshops in Paris, which turns out to be one of the liveliest places on the planet for the subject. Had I realised, I might have extended my trip by a day to take in the first screening of Edward Frenkel’s Rites of Love and Math.

I was invited to contribute to the second day on the theme of the ‘complexity of proof’ by Andrew Arana. I’m rather doubtful that one can give anything resembling a measure of the complexity of a proof, although I did hear some interesting ideas from Paul-André Melliès and Alessandra Carbone who are looking to assign invariants to formal proofs, group theoretic ones for the latter.

But the gap between proof theory and real mathematics yawns rather wide, so I thought to look more closely at the combinations of ‘ideas’ which appear in a proof. As Michael Harris pointed out in his talk, when praising mathematicians’ work in a letter of recommendation, one opts for the word ‘deep’ over ‘complex’, the latter suggesting an unnecessary intricacy. He is surely right then that a treatment of mathematical depth is of central importance. Depth may indicate novel and powerful combinations of ideas.

My approach then was to consider proofs at a high level of description as a collection of ideas and their relations. For example, Tom Leinster gave us a nice sketch of the proof that the probability that a randomly dropped needle will land across a crack between boards whose width is the length of the needle is $2/\pi$. The proof can be boiled down to

Needle can only cross once. Probability = Expectation. Expectation for needle depends linearly on length, since expectation of sum = sum of expectations. Expectation for noodle (floppy needle) depends similarly linearly on length. Expectation for unit diameter circle (circumference $\pi$) is 2.

I wonder then how far one could get in sketching the gist of any proof in terms of the relations between combinations of ideas drawn from a certain stock, which includes:

expectation, sum, measure, obstruction, completion, extension/lift, descent, dimension, representation, subobject, quotient, cover, dual, identity, nilpotent, idempotent, invariant, cohomology, orientation, generator, kernel, factor, bounded, projection, convex, anomaly, ergodic, entropy, random, compact, splitting, section, linear, approximation…

This leads me to a couple of questions,

• Have I bundled together ideas of different kinds? If so, how should they be separated?
• Why do some ideas lend themselves so readily to category theoretic description, e.g., section and completion, while others, e.g., approximation, ergodic, seem not to?

On another matter, I was trying to expound James’ point from back here that Cauchy’s theorem fails in the topos of sets equipped with an automorphism, when a question revealed that while I can see how to define the order of an element, I didn’t know how to define the order of a group. Presumably, James meant us to take the order of $\mathbb{Z}/3 \mathbb{Z}$ with its nontrivial automorphism to be 3, but how do you define the order of a group in that setting when there’s only one map in the topos from the terminal object?

Posted at April 15, 2010 12:29 PM UTC

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### Re: Paris in the Spring

Regarding your last question, it seems to me that really the right thing to ask is whether the statement of Cauchy’s theorem is valid in the internal logic of the topos in question. And if that’s the question, then I don’t even think that $\mathbb{Z}/3\mathbb{Z}$ with its nontrivial automorphism counts as a finite set in that internal logic, so I wouldn’t expect Cauchy’s theorem to apply to it. The “finite sets” in $Set^{B Z}$ are the “externally” finite sets equipped with identity automorphisms only.

Posted by: Mike Shulman on April 15, 2010 2:55 PM | Permalink | Reply to this

### Re: Paris in the Spring

So one should say that the notion of order is not defined for all groups in the topos, but where it is defined, i.e., in groups with an identity automorphism, then Cauchy’s theorem holds?

Or might we just take the order of any group to be the cardinality of $Hom(1, G)$, in this case 1, so Cauchy’s theorem holds?

Posted by: David Corfield on April 15, 2010 3:18 PM | Permalink | Reply to this

### Re: Paris in the Spring

So one should say that the notion of order is not defined for all groups in the topos, but where it is defined, i.e., in groups with an identity automorphism, then Cauchy’s theorem holds?

Yes, that’s what I would say. Just as in ordinary mathematics, the notion of order is not defined for all groups, but only for finite ones, and in groups that have an order (namely finite ones), Cauchy’s theorem holds.

It’s generally the case that theorems about natural numbers and finite sets are just as valid constructively, and therefore internal to any topos, because even in constructive logic, finite things behave more or less classically. For instance, equality on finite sets is decidable, and any finite subset of a finite set has a complement.

Note, though, that constructively a finite set (or group) can have non-finite subsets (or subgroups). So Lagrange’s theorem should be valid to say that if $H$ is a finite subgroup of a finite group $G$, then $|H|$ divides $|G|$, but if $H$ is an arbitrary subgroup of $G$, then we can’t even ask what $|H|$ is.

Or might we just take the order of any group to be the cardinality of Hom(1,G), in this case 1, so Cauchy’s theorem holds?

I don’t think this is really a good context in which to say that Cauchy’s theorem “holds.” We can perform some ad hoc constructions to obtain something that looks kind of like Cauchy’s theorem, but there isn’t an internally consistent (i.e. “sound”) interpretation of (at least some fragment of) mathematics such that this is the resulting interpretation of what we usually mean by Cauchy’s theorem.

Posted by: Mike Shulman on April 15, 2010 7:27 PM | Permalink | Reply to this

### Re: Paris in the Spring

The definition I had in mind of ‘of order $n$’ is ‘locally of order n’. So an object $X$ of a topos is of order $n$ if there exists a cover $(U_i)_{i\in I}$ of the terminal object such that for each $i\in I$, the object $U_i\times X$ is isomorphic to the coproduct of $n$ copies of the terminal object with itself.

For example, if $G$ is a finite group, then a $G$-set is of order $n$ if and only if the underlying set is of order $n$.

Another way of looking at this is that we say a set $X$ is of order $n$ if there exists a bijection from $X$ to the set $\{1,\dots,n\}$. In a topos, one typically replaces existential quantifiers with local existential quantifiers, hence the definition above. I’d be interested in knowing how this point of view on being ‘of order $n$’ relates to that of internal logic (which I still don’t understand).

Posted by: James on April 18, 2010 10:20 AM | Permalink | Reply to this

### Re: Paris in the Spring

I think your definition of “being of order $n$” is equivalent to “being of order $n$” in the internal logic for any fixed, external, natural number $n$.

That means I appear to have, once again, mixed up my notions of finiteness a bit. I did this once before back when Toby and I were writing the nLab page but I apparently haven’t learned my lesson yet. A finite cardinal in a topos is an object which classified by some global element $1\to N$ of the natural numbers object. An internally finite object is one which is locally a finite cardinal, and it’s the latter which is the same as “being finite” in the internal logic. So you’re right that the $G$-sets $X$ such that “$X$ is finite” is true internally are those whose underlying set is finite. So in particular if $G=Z/2Z$, then your $Z/3Z$ with nontrivial $G$-action is in fact internally finite.

However, I stand by my assertion that Cauchy’s theorem is still true. Just as the hypothesis “being of order $n$” should be interpreted locally in the internal logic, to be consistent, the conclusion “having an element of order $p$” should also be interpreted locally. And it is true that there is a generalized element of order $3$ in your group, defined at stage $G$: namely the map $G\to Z/3Z$ that picks out a generator.

Posted by: Mike Shulman on April 19, 2010 1:40 AM | Permalink | Reply to this

### Re: Paris in the Spring

Maybe you’re right that your statement is the most natural extension of Cauchy’s theorem to toposes, but I would still say that the one I proposed is a reasonable extension. I’m not sure what effect this has on the general discussion about one statement seeming harder than a logically equivalent one.

Posted by: James on April 19, 2010 8:01 AM | Permalink | Reply to this

### Re: Paris in the Spring

Presumably Cauchy’s theorem is not constructively provable, else its truth in a topos would not even be in question. If so, then there will be a topos in which it is not true. How would one go about finding such a topos? Perhaps avoiding a Boolean topos such as $G-Set$ would be necessary?

Posted by: David Corfield on April 19, 2010 8:02 AM | Permalink | Reply to this

### Re: Paris in the Spring

I’m saying that Cauchy’s theorem is constructively provable, and therefore true in any topos (in the sense of the internal logic, which I’m advocating is the right one). James, can you say anything about why you think the version you proposed is also reasonable? I’m having trouble seeing it; it seems to me that if you “internalize” the notion of having $n$ elements by interpreting it locally, to get a reasonable statement, you should of course do the same with the notion of “having an element of order $p$.”

Posted by: Mike Shulman on April 19, 2010 3:56 PM | Permalink | Reply to this

### Re: Paris in the Spring

Well, that’s why I said your statement was probably the most natural version. But in geometry, you often have (nontrivial) statements of the form if such-and-such exists locally, then something exists globally. So I don’t think we should close the door on such statements.

Posted by: James on April 19, 2010 8:47 PM | Permalink | Reply to this

### Re: Paris in the Spring

Certainly, local-to-global statements are interesting and important! I’m not saying we shouldn’t talk about them. But they are usually significantly stronger than purely local statements, so I don’t think they should be given the same name as the latter. In particular I don’t think it’s reasonable to say that “Cauchy’s theorem fails” if what fails is really a local-to-global property, when the purely local version of Cauchy’s theorem does hold.

To simplify the issue a bit, we could consider the statement “any set with cardinality two has an element.” This is of course provable constructively, and thus true purely “locally” in any topos. But the “local-to-global” version of it fails in G-sets for $G=Z/2Z$, since the principal G-set “has cardinality two” in the local sense, but has no global elements. But I don’t think it’s reasonable to say that the statement “any set with cardinality two has an element” is false in that topos; what fails is the local-to-global principle, not the behavior of finite sets. Likewise with Cauchy’s theorem: what fails is a local-to-global principle, not the behavior of finite groups.

Posted by: Mike Shulman on April 19, 2010 10:12 PM | Permalink | Reply to this

### Re: Paris in the Spring

Hmmm. Hard to argue with that! I guess that’s kind of what I was touching on with my statement (1’) back at the original comment.

Posted by: James on April 19, 2010 11:31 PM | Permalink | Reply to this

### Re: Paris in the Spring

That is a pity, you should have told me you are in Paris…as I am near by, at l’IHES, this month. I hope we meet another time. I saw some history of math posters at l’IHP though. I also missed the above linked Rites prepremiere…I knew that there was some math art movie this month in Paris, but not that I saw both actors alive before. :)

But missing you and the movie (and all other nice things in Paris area), I am succeeding to make some (well, minor) work done in the meantime…Among the rest, Urs and I posted to the arXiv last night, a substantial expansion of our December 2008 mini-survey which appeared in some proceedings last year. The stuff toward the end of the survey is more of a research announcement of Urs’s ideas than a survey, with the first half of the article more conservative.

Posted by: Zoran Skoda on April 15, 2010 3:18 PM | Permalink | Reply to this

### Re: Paris in the Spring

Ah, a pity.

I see higher category theoretic mathematical physics will feature at the ICM this year in Anton Kapustin’s talk Topological Field Theory, Higher Categories, and Their Applications.

Posted by: David Corfield on April 15, 2010 3:45 PM | Permalink | Reply to this

### Re: Paris in the Spring

Just for info, I was in Paris on Friday the week before. There was a very lively day on Chu spaces led by Vaughan Pratt with contributions from Paul-André and myself. I have mentioned these MaMux seminars before. They are held at IRCAM next door to the Centre Pompidou (in fact it is more or less an annex). They are attended by philosophers, musicians, physicists, psychologists, … This makes for very interesting conversation at lunch and over a beer afterwards. Check out the website if you are going to be in Paris. The next meeting will be in mid May.

Posted by: Tim Porter on April 16, 2010 6:08 AM | Permalink | Reply to this

### Re: Paris in the Spring

This is a tangential comment, not in any way disagreeing with anything you’ve said but providing a possibly interesting link about the weirder corners of proofs. David C wrote

As Michael Harris pointed out in his talk, when praising mathematicians’ work in a letter of recommendation, one opts for the word ‘deep’ over ‘complex’, the latter suggesting an unnecessary intricacy.

What’s interesting is that it’s not clear whether the “unnecessary intricacy” generally comes from the “argument used in the proof” (ie, researcher X didn’t find the better proof) or the “choice of what to prove” (X picked this really nasty thing to prove)? In connection with this, I’ve just come across seL4: Formal Verification of an Operating-System Kernel, which unfortunately is a short paper which has to spend so much time explaining roughly what was chosen to be proved that it really doesn’t say very much at all about the actual proofs, or even proof mechanisms. Clearly some of the things that they’re doing are very, very weird from the viewpoint of mathematics in general (why would you modify what you’re trying to prove and in a sense “throw away” a proof rather than generalise, etc), but arguably those are unavoidable consequences of given what you’ve decided is important to prove.

In connection with

I wonder then how far one could get in sketching the gist of any proof in terms of the relations between combinations of ideas drawn from a certain stock, which includes: ….

I suspect that you could probably summarise most of the proof as basically “showing that things intricately designed to have certain properties do indeed have those properties, showing that an embedding of a simple model into a complicated model does faithfully transfer those properties one cares about, using exhaustive enumeration to validate micro parts of this embedding-is-faithful”. This is certainly accurate, yet in a sense it doesn’t say what’s different about this proof compared to other proofs (in the same way that many, many combinatorial identities can be summarised as “there’s a bijection” whereas the interesting thing is the nature of the bijection).

(Just to be clear again, I’m not suggesting that this kind of “proof” activity is in any way typical, it’s just an interesting end of the spectrum of possibilities.)

Posted by: bane on April 16, 2010 7:59 AM | Permalink | Reply to this

### Re: Paris in the Spring

Regarding the first point, it would be interesting to have a better sense of those situations where it is known that merely combining constructions from a certain repertoire won’t suffice, so that some new ideas are needed. When can you say about a claimed proof that it can’t possibly work because it only uses ideas from a certain limited collection?

Regarding the second, Michael Harris pointed out how when conveying a new proof the amount of detail required varies greatly according to the experience of the audience. As you observe, there’s another variation according to how gist-like is the given gist. Describing the proof at the level of your example gives little specific information, and presumably could be conveyed extremely briefly to a fellow expert.

Posted by: David Corfield on April 16, 2010 10:04 AM | Permalink | Reply to this

### Re: Paris in the Spring

Again, not disagreeing with anything you’ve said, but there’s also the dual perspective: once you move into more general mathematics, the question arises what means do you use to judge that a given proposition is of such a nature that a “nasty, brutish and long” proof is likely to be the best you can do? (Clearly a lot of the propositions in computer program verification are likely to be of this nasty nature, but what expectations would/should one have about any possible valid proof of, say, proving the optimal strategy for some particular game (eg, nim, other game theory things), or the classification of finite simple groups, etc?)

Posted by: bane on April 16, 2010 11:38 AM | Permalink | Reply to this

### Re: Paris in the Spring

David wrote:

…it would be interesting to have a better sense of those situations where it is known that merely combining constructions from a certain repertoire won’t suffice, so that some new ideas are needed.

I’m sure there are a lot of interesting examples to be found in the history of mathematics, but I’ll try to state one of recent interest, to the extend that I am able: The problem of the P = NP question (the millenium problem) is that we do not know how to prove that there cannot be a more efficient algorithm than those we already know about (I’m assuming P != NP, if P = NP the proof could of course be done via a construction of a polynomial algorithm for a NP-complete problem).

Let’s look at the travelling salesman problem: We have algorithms (computer programs) A and as input data weighted non-directed graphs G (at most one undirected edge connecting two vertices, no loops, every edge has a positive integer number assigned to it).

If you specify an algorithm, I can go ahead and try to find a sequence of graphs $G_n$ such that the running time of your algorithm increases exponentially.

If you specify input data I can go ahead and try to find algorithms that solve the problem in polynomial time (actually it is one research topic of both theoretical and practial interest to specify additional structures on your input data such that certain algorithms can be designed that have polynomial runtime with respect to this kind of input, as far as I know).

If you tell me that you constructed the most efficient algorithm possible and found a sequence of input data $G_n$ such that the running time increases exponentially, thereby proving that P != NP, I would answer that there is no way to tell if and why your algorithm is the most efficient possible, there is no proof strategy/pattern to conclude that. We do not even know the most efficient algorithms to multiply numbers, and therefore you cannot possibly know that the algorithm you wrote is the most efficient possible.

Posted by: Tim van Beek on April 16, 2010 12:43 PM | Permalink | Reply to this

### Re: Paris in the Spring

I wrote

…it is one research topic of both theoretical and practial interest to specify additional structures on your input data…

Clarification: What you do is restrict the set of allowed inputs to graphs that have certain additional properties, like being chordal. You can then design algorithms that use this property and achieve polynomial runtime. Unless I am mistaken then chordal graphs allow a linear solution to the traveling salesman problem - but I’m not quite sure…

In many real world problems all input data satisfy additional properties, so that this strategy is of practical importance. I remotly remember that interval graphs, which are cordal, are important in many applications like genome sequencing…

Posted by: Tim van Beek on April 16, 2010 2:21 PM | Permalink | Reply to this

### Re: Paris in the Spring

Speaking of gists of proofs, is there a better one around than David Ben-Zvi’s gentle overview of the Fundamental Lemma?

Posted by: David Corfield on April 17, 2010 11:03 AM | Permalink | Reply to this

### Re: Paris in the Spring

This is an extended comment around the fundamental lemma. My goal is to describe, in an extremely high level way, a bit of the topography of the mathematical environment in which the fundamental lemma sits. The basic fact is that this topography is rather complicated, as I will try to explain.

(a) Ngo’s proof is a masterful piece of geometry. The ingredients in the argument are substantial, and the way they are combined to reach the desired conclusion is subtle, with fairly involved arguments.

(b) The context of Ngo’s proof is quite far removed from the original interest in and motivation for the fundamental lemma. I think that Ngo’s proof is very interesting mathematics in its own right, but the path from the fundamental lemma as Langlands originally conceived it to the setting in which Ngo works is itself quite long, and somewhat winding: there is a passage from p-adic field (which have characteristic zero) to function field in characteristic p, followed by a reinterpretation of the fundamental lemma and much of the related objects (the trace formula, orbital integrals, …) in geometric terms. The geometric objects, such as the Hitchin system and various Prym-type constructions that appear in Ngo’s proof are far removed from the original conception of the fundamental lemma, which was a combinatorial statement having its roots in harmonic analysis.

(c) The fundamental lemma as stated by Langlands is itself a special case of a more general statement in harmonic analysis on p-adic groups, about the transfer of orbital integrals from a given group to its associated endoscopic groups. The definitions and various statements here are also involved and elaborate; they are motivated by global considerations related to the trace formula, itself a notoriously elaborate and technical subject.

Slightly more precisely, the (conjectural) theory of endoscopic transfer was developed as an explanation of certain phenomena in global harmonic analysis which seemed to relate automorphic forms on a given group to those on its associated endoscopic groups. A collection of mutually intertwined conjectures and implications were derived, which in the end served to reduce the verification of the whole picture to the fundamental lemma.

(d) The big automorphic picture alluded to in (c) in turn has many implications in the theory of automorphic forms and also in arithmetic geometry. For example, it (to a large extent) governs the cohomology of Shimura varieties. For number theorists, this is perhaps the main reason to care about the fundamental lemma; now that it is proved, it becomes possible to compute the Galois representations appearing in the cohomology of Shimura varieties. Thus we have returned to geometry, but now arithmetic geometry over number fields and finite fields, with a very different flavour (or at least, apparently so) to that of (a).

The implications of (a), (b), (c), and (d) are that trying to understand the proof of the fundamental lemma, trying to understand its statement in the traditional formulation (i.e. Langlands harmonic analytic formulation, rather than the geometric formulation that Ngo proves), and trying to understand the reasons that people care about it, are three separate, and almost entirely independent, projects!

The second project is probably pointless unless you intend to specialize in the theory of automorphic forms. Indeed, the precise statement of the fundamental lemma is famously unenlightening to non-experts, while being extremely complex.

The first and third are both extremely interesting, but, as I indicated, involve (apparently) very different pieces of mathematics. I’ve spend a lot of time myself over the last year or so trying to understand better the relations between endoscopy (i.e. the “big picture” of (c) above) and the cohomology of Shimura varieties. Unfortunately, I haven’t yet found any presentations of the ideas at any kind of expository level that would make them accessible to non-specialists.

I don’t have any specific suggestions for understanding more about Ngo’s proof either, but in this case one can be hopeful that a good secondary literature will develop. (Experience suggests that geometric ideas diffuse into the broader mathematical culture much more quickly than number theoretic ones do. I’m not completely sure as to why, although I have some vague ideas … .)

Posted by: Matthew Emerton on April 23, 2010 5:04 AM | Permalink | Reply to this

### Re: Paris in the Spring

Thanks very much for this. Might what you’re describing still be thought of as filling in Weil’s Rosetta stone?

The geometric objects, such as the Hitchin system and various Prym-type constructions that appear in Ngo’s proof are far removed from the original conception of the fundamental lemma, which was a combinatorial statement having its roots in harmonic analysis.

So how much of a glimmer of understanding is there as to why such constructions work here?

(Experience suggests that geometric ideas diffuse into the broader mathematical culture much more quickly than number theoretic ones do. I’m not completely sure as to why, although I have some vague ideas… .)

Do you point it down to the strength of our geometric intuition?

Posted by: David Corfield on April 23, 2010 3:30 PM | Permalink | Reply to this

### Re: Paris in the Spring

The constructions in Ngo are far removed from the original Langlands setting in two senses: one, they live in the much easier and more geometric setting of function fields; two, they are categorified. Otherwise I’d venture to say they’re broadly speaking the same - in other words we are studying harmonic analysis, not of functions but of sheaves, on a locally symmetric space, which now (like every other object in sight) has the advantage of a modular interpretation. I think it’s important to note (as I first learned from Matt..) that Ngo really is really studying the precise analogue of what people did classically in this new setting. The advantages of the geometric, categorified setting probably don’t need to be emphasized on this blog, and they enable him to go much further and work more conceptually, finding new underlying symmetries and structures that govern the situation. (This is a typical feature of the best geometric representation theory, that complicated combinatorics gets clarified upon categorification and geometrization..)

One wonderful new wrinkle is that the results in the characteristic p setting actually imply the p-adic conjecture (by a general mechanism that’s been around for a long time, but of which I think this is the most spectacular application).

Posted by: David Ben-Zvi on April 23, 2010 5:51 PM | Permalink | Reply to this

### Re: Paris in the Spring

When I wrote “far removed” what I had in mind is that the geometry of Ngo’s proof is far from the geometry of Shimura varieties (which is the geometric setting that number theorists would naturally associate with the Langlands program, and with the fundamental lemma in particular).

As David writes, Ngo’s geometry is a very faithful geometric/categorical encapsulation of the harmonic analysis of endoscopy. In some sense it is an intrinsic geometry (intrinsic to the harmonic analysis problems presented by the fundamental lemma), whereas the Shimura variety context is extrinsic: it is a place where we hope to *apply* the fundamental lemma, and related tools, to make number theoretic deductions.

Posted by: Matthew Emerton on April 24, 2010 4:50 AM | Permalink | Reply to this

### Re: Paris in the Spring

Rereading my original post, and what I just posted a moment ago, I realize that what I’ve written doesn’t seem to be completely consistent.

What I just wrote about Ngo’s intrinsic geometry being very different from the extrinsic geometry of Shimura varieties that traditionally appears in the Langlands program is true, and no doubt it did influence my original comment that Ngo’s approach is far removed from the original conception of the fundamental lemma.

But of course, as I also wrote, David is correct about the fact that Ngo’s geometric construtions give a very faithful (and very powerful) interpretation of the combinatorics of the fundamental lemma. But I think it’s true that the objects that Ngo introduced (the Hitchin system, for example) are not at all objects that the traditional researchers in endoscopy would have imagined were related to their problems. In other words, Ngo found an amazing link between the combinatorics of the fundamental lemma, and related problems in endoscopy, to the Hithcin system and related objects, which are also part of a rich mathematical tradition, but one quite different to that of endoscopy. (It is true that there were earlier ideas connecting the fundamental lemma to geometry, say those of Groseky–Kottwitz–MacPherson, but I think that the introduction of the Hitchin system was Ngo’s new contribution, and unexpected prior to him doing it; David will hopefully correct me if I’m wrong about this.) This new connection was also a large part of what I had in mind when I wrote “far removed”.

Posted by: Matthew Emerton on April 24, 2010 5:02 AM | Permalink | Reply to this

### Re: Paris in the Spring

Ngo found an amazing link between the combinatorics of the fundamental lemma, and related problems in endoscopy, to the Hitchin system and related objects…

Do people now have an idea, or better, a clear idea, as to why Hitchin systems should be relevant to the subject matter of the fundamental lemma? For example, if Laumon is right with his

Natural expectation: The Fundamental Lemma is the consequence of a (stronger) cohomological statement.
(Fundamental Lemma and Hitchin Fibration, slide 13),

might the gist of the connection be phrased in terms of cohomology?

Posted by: David Corfield on April 26, 2010 11:20 AM | Permalink | Reply to this

### Re: Paris in the Spring

Thanks! I find such “topographies” very helpfull. Which survey on (a) would you recommend for getting an idea of it?

Posted by: Thomas on April 24, 2010 10:33 AM | Permalink | Reply to this