(*1*)

Let be a discrete group, let be some element of , and let be some finite subset of . Furthermore, let denote a Hilbert space of square-integrable function on . We are interested in , where is now seen as an operator on , and is the Von Neumann dimension defined as follows: given , take orthogonal projection onto and put

,

where is a neutral element of and is a standard scalar product on .

Computing is rather difficult. The lemma relates to a (much more easily computable) standard linear dimensions of certain finite-dimensional spaces.

(*2*)

Before giving the lemma, few words on the Von Neumann dimension. It is an interesting number mostly for -equivariant spaces (right equivariant, because we assume that elements of the group act on the left, and images of such operators are right equivariant).

Suppose for a second that is a finite group. What is a (standard) dimension of a given ? To compute it you can take a projection onto and take the trace of this projection, so the (standard) dimension of is

.

But when is right invariant then so is projection (these statements are equivalent) and so every summand above is equal to . Therefore we get that

,

or in other words: is just a normalized dimension (normalized in such a way so that ).

The point is the following: in the case when is finite, the formula gives a normalized dimension. However, in the case of infinite the formula also makes sense, although the “normalized dimension” is !

So the Von Neumann dimension is potentially useful in situations where we want to say how big certain infinitely dimensional equivariant set is, when taken into account the group symmetries.

As an example consider , the group of integers. By Fourier transform we have that , and the action of on the latter is realized by a pointwise multiplication by the function . Example of an equivariant subspace of is a subspace of functions which have the support contained in some set . What is ? It is very easy to see that it is a measure of the set .

(*3*)

One of the big conjectures in the geometric group theory is, however, that when the group is torsion free then is . More generally, one considers not only acting on , but also acting on . In this case the Von Neumann dimension is defined as a sum of standard von Neumann dimensions of diagonal elements of a matrix, and the conjecture says that (for a torsion-free ) so defined dimension of the kernel of an element of is an integer.

Similarly, everything I write further can also be generalized to matrices over a group ring in a straightforward way, but to simplify the notation I will consider only .

(*4*)

One more definition: when is a finite subset of then we define of a given subspace of to be

,

where is, as usual, an orthogonal projection onto .

In general it is rather difficult to compute because it involves a projection onto potentially infinite dimensional space. However, it turns out that sometimes we can hope to approximate this situation by a finite dimensional one. This is what the lemma is about.

Lemma:Let be a discrete group, be an element of , and be a finite subset of . Then we have that,

where is a subset of consisting of those element which are mapped by outside of ; and is an operator restricted to

**Proof:** First ingredient is the following standard equality:

,

where denotes the closed image. I don’t know how to prove it, but you can find it in Wolfgan Lueck’s book on -invariants. The second ingredient is another equality:

,

which is just a standard linear algebra of finitely dimensional spaces.

Now you plug the first equation to the second one and use the following simple observation:

,

which you prove by noticing that for (this alone proves the first inequality), and that for a -equivariant . This gives the second inequality.

**Q.E.D.**

(*4*)

The lemma is valid in a general context of discrete groups. However, it’s clearly “designed” to work well for amenable groups. In the case of amenable groups, for every there exists a finite set such that is arbitrarily small. Therefore for amenable groups we get the following corollary:

Cory:Let be amenable, and let . Then the following are equivalent:

- there exists an element such that
- there exists an element such that .

This follows from the above lemma and the fact that for -equivariant space , is iff . For general discrete groups above cory is an open conjecture (maybe one should restrict atttention to torsion free groups).

In as similar (slightly more involved) fashion one gets so called strong approximation conjecture for amenable groups:

Cory:Let be a directed system of normal subgroups of , such that . Then.

(*5*)

Two things catch my attention: does the condition

For every , and there exists a finite set such that characterize amenable groups? Most likely so, but I have to think about it a trifle more.

And second, can you take some more general subspace of instead of ? The main problem here seems to be the equality

, which no longer holds (after suitable deinifition od as a normalized trace of a projection from to ) in general.

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(*1*)

First decategorification. Frankly, I don’t know what it is, but I roughly know what degroupoidification is. Suppose you have a groupoid (i.e. a category whose morphisms are invertible). Then degroupoidification of it is a vector space spanned by isomorphism classes of objects in .

Example which comes to mind is the following: Let be a topological space with a distinguished point . Let be a groupoid whose objects are loops starting and ending in , and whose morphisms are homotopy classes of homotopies between loops. Then isomorphism classes of objects in are enumerated by elements of , the fundamental group of , and so the degroupoidification is , a vector space spanned by elements of .

(*2*)

However, we know that there is a multiplication structure on . As far as I understand, groupoidification of a vector space with additional structure is a groupoid with some additional structure which “naturally” gives rise to the structure on the degroupoidified groupoid.

The aim of Alex’ talk was to show groupoidification in this sense of certain class of rings, Hecke algebras.

Why should one care about it? Alex said that in some cases one can do operations on the groupoid level and descent them to degroupoidified level, obtaining some subtle structure which was very hard to notice without passing to the higher, groupoidified, level. However, so far no new informations were obtain about Hecke algebras.

Groupoidification of a given structure doesn’t seem to be unique in any way. The “correct” groupoidification would be then the one which gives rise to some interesting new structure.

(*3*)

A remark about a decategorification. As I mentioned at the beginning, I don’t know what is a right notion of the decategorification. However, with any category one can associate a groupoid by simply forgetting all non-invertible morphisms, and then degroupoidify it.

(*4*)

Alex talked also about (one possibility for) a groupoidification of linear maps: a span in a category of groupoids (span between objects and is an object with morphisms and .

To justify it I have to explain how to get linear map out of the span.

Basic idea is seen already when working in the category of sets. When is a span between and then associated map between vector spaces generated by A and B is obtained as follows: given , associated vector goes to .

However, in this case one gets only linear maps represented by matrices with integer coefficients. On the other hand, given a groupoid , we can define to be , where the sum is over isomorphism classes of objects in . Just as in the previous paragraph we can map a vector associated to the isomorphism class of to , where is a groupoid of elements isomorphic to (“connected component of “).

This works precisely this way for the category of groupoids which have only finitely many isomorphism classes. Otherwise questions of convergence arise.

(*5*)

Lastly, I want to write about categorified vectors. These are just morphism of groupoids . To such data we associate a following vector in the degroupoidified : . Given a span and a categorified vector one gets a categorified vector over by first taking the pullback to get a vector over , and then composing with .

(*5a*)

There are at least two kinds of pullbacks of groupoids, depending on whether we work in the category whose morphisms are strict morphisms or a quotient of morphisms by natural isomorphisms between morphisms. Respective pullbacks are called a strict and a weak pullback. In the definition above we used the latter.

A weak pullback of groupoids is a groupoid whose objects are triples , with , , and , and whose morphisms between and are pairs of morphisms , , such that the obvious diagram in commutes. Given these definitions we get obvious maps of groupoids and .

Let’s check that this has the required universal property. Suppose we have a diagram

We need to show that there is a unique map making the obvious diagram commute. We define it in the following way on objects: goes to , where is a natural isomorphism given by the fact that the above diagram is commutative up to natural isomorphism. On morphisms we define it analogically.

It is clear that suitable diagram commutes. As to the uniqueness of , if there is some map which also makes this suitable diagram commute, then again by definition of morphisms in our category there exist natural isomorphisms between and and between and . Using these it is easy to define an isomorphism between and as well.

For example, take , where is a group (i.e. groupoid with only one object) and and are both the identity morphism. Then has objects ~~with morphisms only between objects corresponding to conjugate elements of the group~~ each of which has morphisms. This – as expected – is weakly isomorphic to . ~~This is not (equivalent to) so I have a problem here.~~

(*5b*)

Note that categorified vectors can be added – just take a disjoint sum of groupoids over a base groupoid. This sum commutes with decategorification, and also with the map of categorified vectors defined above. To check this last assertion it is enough to convince yourself that sum commutes (* in a strict sense!* this is important, because categorified vectors which are just weakly isomorphic don’t give the same decategorified vector) with taking a weak pullback.

With this in mind it is straightforward to check that the two described ways of getting linear maps out of spans of groupoids give the same results.

(*6*)

Final remark is that given a span one can construct two morphisms: from decategorified to decategorified and the other way around. It’s easily seen that matrices of these are transposes of each other and so the morphisms are adjoint.

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– Kahler groups. Kahler groups are groups which are fundamental groups of Kahler manifolds. The big question is obviously “which groups are Kahler groups?”. The lecture was particularly interesting, because it combined GGT techniques with algebraic/complex geometry techniques. Forunately, tomorrow or the day after I’ll have another opportunity to listen about that so there are some chances I’ll blog about it.

-Serre-Bass theory. We had a fast course in Serre-Bass theory. The concern of S-B theory are so called graphs of groups. The outcome is, informally speaking, that one can say whether a given group G is isomorphic with some nontrivial amalgamated free product. This is done by very geometric techniques – if one was able to use only algebraic techniques it would be probably a hopeless question.

-Some theorems about Simplicial Non-Positive Curvature. Notably that complexes satysfying SNPC are aspherical.

-Some dirty methods to check whether given complex is Cat(0) or not.

I fixed the link to Lieven Le Bruyn description of a healthy branch of mathematics (I think GGT fits this description) in a day 1 post.

Bonfire was a very nice social meeting, but it ended very late – I’m unable to blog anything more. Also, I still think about a suitable form for blogging semi-live from a school/conference, after yesterday sort-of-failure.

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Whole day was full of lectures: apart from coffe breakes there was only one long (2 hours) break for a lunch. First lecture was at 9:00 (which is *way* to early for me, I tried to sleep on almost every coffe break) and last question session was at 19:20. Actually, it’s a *bit* to much for me, as some lecturers are handing out some exercises and there is virtually no time to solve them. But to much is better then to little :-).

There is parallel conference taking place here on the topic of differential equations, so there’s some additional fun from talking with all these guys who know how to solve PDEs and stuff. God I wish I had this easiness with deltas and epsilons. :-)

Here are some notes from today lectures.

Two main courses at the conference are “Simplicial Non-Positive Curvature” and “Cat(0) cubical complexes”. These are supplemented with one-hour talks about some new results.

(*Simplicial Non-Positive Curvature*)

We’ll be interested in simplicial complexes. People often try to express some properties from other branches of geometry – like differential geometry – in combinatorial terms that would be applicable to general simplicial complexes. This is because simplicial complexes are easier to understand then, say, manifolds – after all a simplicial complex can be easily presented as some very finite combinatoric data. For example, there is a Gauss-Bonnet theorem for 2-dimensional simplicial complex X:

Σ (3-α(v)) + Σ (6-α(v)) = 6∙χ(X),

where first sum is over vertices v lying on the boundary of X, the second sum is over vertices lying in the interior of X, and α(v) is numberof triangles meeting in v (and χ(X) is Euler characteristic). This is, more or less, consequence of Euler’s formula. So as you see by comparison with classical Gauss-Bonet, the number of triangles meeting in a vertex say how “negatively curved” the space is in this vertex. This is quite natural when compared to imaginable examples of non-positively curved manifolds – I always imagined “negatively curved points” on such manifolds as being in a saddle (hence “having more space” around then points have on Euclidean plane)

(So simplicial complexes can be thought as a limit of Riemannian manifolds in which the curvature is “focused in single points”)

Troughout the lecture we used this Gauss-Bonnet theorem only in case X=2-disk. Then the proof of this identity is for sure just a consequence of Euler’s formula – by for example takin two copies of this disk and gluing them into a 2-sphere.

As far as I understand, one of the founding fathers of Geometric Group Theory is Mikhail Gromov. Some years ago he posed more or less the following problem: Find a combinatorial condition for a simplicial complex that would guarantee that this complex is non-positively curved in a metric sense.

Metric space is said to be non-positively curved iff it is a geodesic (that is, between any two points there exist a geodesic, and geodesic is in this context by definition a locally shortest path) space and all triangles are “thick”.

The last condition means that if we map isometrically a triangle (triangle = three points plus geodesic segments that connect them)

onto a triangle in Euclidean plane then any two points in this Euclidean triangle are nearer to each other on Euclidean plane then their counterimages are in the space under consideration.

Non positively curved metric space is also called a Hadamard space and a CAT(0) space – 0 is because we compared triangles with triangles on a Euclidean plane; if we compared them with triangles on a sphere we would get a CAT(1) space, and if we compared them with triangles on hyperbolic plane we’d get a CAT(-1) space.

A theorem by Cartan and Alexandroff says that Riemannian manifolds with non-positive sectional curvature are in fact non-positively curved spaces. Who’d thought that :-).

Simplicial Non-Positive Curvature is not really an answer to a Gromov question, but it was motivated by it. It’s a rather simple combinatorial condition for simplicial complexes. Curiously, it has quite many implications similar to CAT(0) for metric spaces (like, say, asphericality (this links to wikipedia article coauthored by me :-)). For this reason some believe (this was part of some other talk) that there should some underlying meta-theory that would contain both CAT(0) and SNPC spaces.

So let’s define this SNPC condition. Simplicial complex is *flag* iff everytime it contains a set of points joined pairwisely by 1-simplices it also contains a simplex spanned by this set of points.

Simplicial complex is *k-large*, k≥4, iff it is flag and every cycle γ in X of lenght 3<|γ|<k has a diagonal (So for k=4 we get just flag and for k=5 we get “no empty square condition”)

Simplicial complex is *k-systolic *iff it is connected, simply connected and k-large.

And finally, 6-systolicity (which is often shortened to “sistolicity”) is precisely SNPC condition. Weird as it may seem at first sight, it has some interesting properties ;-).

First, I gave perhaps the shortest of possible definitions. Now I’ll give a longer one, but one which is almost purely local – so one can perhaps see better why it might be similar in some sense to CAT(0) condition.

For this, few more definitions: for a given simplex σ of simplicial complex X we define a link X_σ to be “a sphere of radius 1 around σ”. That is X_σ is a subcomplex of X consisting of all those simplices that are disjoint with σ but which span a simplex in X with σ.

Systol sys(X) of a simplicial complex X is a lenght of a shortest cycle that doesn’t have a diagonal (for example, “minimal” homotopically non-trivial cycles don’t have diagonals, but also a boundary of clasically triangulated hexagon doesn’t have a diagonal).

Now, the theorem says that a simplicial complex X is k-systolic iff sys(X)≥k (this is “nonlocal”) and sys(X_σ)≥k for every simplex σ (this is “local”). It;s not very difficult to prove.

It’s 1AM here, so I’ll end eith giving some (non)examples of systolic complexes:

- tree
- torus with quite a tricky triangulation coming from a hexagon with identifications on boundary
- Triangulation of Euclidean plane by equilateral triangles
- Triangulation of Hyperbolic plane by equilateral traingles with angle 2π/6
- Some triangulation of Hyperbolic 3-space (there was a pictureon a blackboard but I didn’t get it
- Cartesian product of a tree and a line has a systolic triangulation, but a product of two nontrivial tree hasn’t.
- It follows easily from above Gauss-Bonnet theorem that a sphere doesn’t have a systolic triangulation (because G-B says that there exist a vertex adjacent to less than 6 triangles, so the link of v is a circle of lenght < 6 …). As a consequence, manifolds of dimension ≥3
*are not*systolic (because in manifolds links of codim 3 simplices are 2-spheres)

Well, unfortunately I didn’t get very far, but writing this took me ~2 hours. Tomorrow I’ll change somehow the mode of writing.

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Today we just came to Bedlewo Math Conference Center, all the real stuff is starting tomorrow (it’s after midnight here, so actually today), but we had some time to drink few beers and talk about mathematics. I learned about at least two *very* cool things.

As a bonus, read about my *very very* weird mathematical dream :-).

(*)

This one wins all the prizes for today; it was a question on pre-admission test for phd studies in Wroclaw (one of the biggest math centers in Poland). Hardcore version: prove or disprove whether it’s possible that an interesection of parabola and circle on a plane can consist of exactly two different point of which one is a point of tangency and the other is not a point of tangency.

The real version (softened by examining committee) was actually to prove that it’s possible :-).

For me it was a real surprise! Quite counterintuitive, *very* interesting. I won’t give you any hints, ’cause it’s best to ponder it alone for a moment.

(*)

The second one is about Mapping Class Group. MCG is fairly simple concept. One takes a fixed manifold X and defines MCG(X) as a quotient of all homeomorphisms X –> X modulo these homeomorphisms which are homotopic to identity. It is discrete group. The soft version of what we talked about is to find a surjective morphism from MCG(S³ x S³) to SL(2,Z). This is fairly easy and uses only a fact that S³ is a group.

The hardcore version, to which solution I don’t understand, is to find a kernel of this morhpism. It has something to do with Milnor’s exotic spheres (!).

(*)

And here’s my dream: Jean-Louis Loday is apparently giving some kind of a lecture. He writes some stupid philosophical question on a blackboard, which all the audience, including me, treats very seriously. He asks for some comments and I raise my hand. He chooses me to speak but just before I begin to speak he starts to laught at me, because I don’t like writing style of Jean Pierre Serre (He explicetely points to “Course in Arithmetics” by JPS). Then all the audience laguhs at me :-).

There were also some non-mathematical parts of this dream which I’ll mercifully omit. :-)

(*)

I hope to write some notes from tomorrow’s lectures.

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This year we have a serious problem with choosing a topic/book – we have three very interesting propositions and choosing only one is very difficult.

I’ll write here briefly about our previous camps (in case somebody was interested in proved ideas for such a camp) and about propositions for this year’s camp (in case somebody wanted to comment on that, which would be appreciated).

(*)

First camp was devoted to John Milnor‘s “Topology from differential point of view“. I was a bit frustrated before this camp (it was after 1st year of studying math) – during a whole year I had been learning very tedious things: calculus, general topology, set theory, basic algebra etc. This camp, with this specific book, was exactly what I needed. This book is superbly written, there are practically no tedious part of it (yet it is strictly mathematical text, not a “popular mathematics”) and it finishes with results that were very impressing to me, despite my lack of mathematical sophistication.

Some main points of this book are Sard’s theorem (with very easy to understand proof), properties of a degree of a smooth map, Euler characteristic through vector fields and computation of some homotopy groups of spheres through cobordism theory. We had two “adults” with us and they supplemented us with some additional topics (like, say, Hopf fibration and different points of view on Euler characteristic). Generally I think It’s very good to have some “adults” on a camp when one isn’t mathematically sophisticated enought – I remember that I felt much more surely when they were there to correct any misunderstandings.

There were plenty of drawings on a blackboard, reasonings were nontrivial and intuitive at the same time – I was relieved that there indeed is mathematics that I strived for for one long year. Milnor became my personal hero (we were also told at this camp what did Milnor prove about 7-dimensional spheres and it seemed just awesome).

(*)

However, at the time I suspected that maybe it’s not really Milnor who is the best, but rather all good mathematicians, in particular all Fields medallists, write very nice-to-read books. Accordingly, on the next camp (it took place in february, because first camp was such a huge fun and we wanted to repeat it as soon as possible) we decided to give a chance another Fields medallist, Jean Pierre Serre, and his book “A course in Arithmetics“.

Well, in a way, this camp was also a success, for example because I met there Olek Zablocki, by now my good friend, whose speed in mathematical reasoning was at that time astonishing to me. Meeting him then was a very cool experience. Also, we did some crazy swimming in the see when the temperature was far below the zero, there was heavy wind blowing and snow was everywhere around. That was really cool. We made a bet about who will go further into the see and one who won was a girl! You wouldn’t guess this if you saw her…

However, Serre is not a good hero in this story. Serre is a bad guy. Say whatever you want (“Serre is a beauty lover (easthetics lover)” , for polish readers: “Serre to pięknoduch”, quote is by A.S. Bialynicki-Birula, my favourite lecturer in Warsaw), but I was so dissapointed by Serre. I suspect that it’s very different when one listens to his lectures, but his book is very hard to read for beginners. At any given time, I just didn’t know what we’re doing and, even worse, where we’re actually heading. No comments of any kind on why what we’re doing is important or interesting. Only this year I really have learned why some of these things may be absorbing.

I plan to read his book once more in a near future to see my reaction. Maybe now I’ll say that it’s beautiful and easthetically pleasing BUT don’t take this book as a book for a camp for if you’re not sophisticated enough (maybe it’s a good book for an event where “adults” do most of a talking, but it’s not my kind of a camp.)

(*)

Accordingly, following september we quite well knew what we’re going to do: Milnor’s “Morse theory“. Actually we’ve done only first two chapters: the one on finite dimensional Morse theory and the “Rapid course on differential geometry” one. Again: Milnor is simply the best. The book is quite much more sophisticated than his “Topology from…” but basic truths about it remain the same: superbly written, provides a reader with geometrical intuition, non-tedious in every aspect. Sorry, but I just so feel like writing it once more: Milnor is the best.

My friend, Jarek Kedra (who was an “adult” on most of our camps) wrote once a paper with Milnor’s wife, Dusa McDuff. Accordingly, he once went to Stony Brook (that’s where Milnors reside) to talk with her. When, just before, he told me about his plans, I asked him to collect Milnor’s autograph on some of his book or a t-shirt. Unfortunately, Jarek just laughted at me. Nevertheless, Jarek is also sort of the best, as he actually shook hands with Milnor. Wow. :-)

(In case Jarek is reading: you’re the best nevertheless, without you and two other “adults” our camps wouldn’t be the same :-)

(Yeah, definetely. If not you and Gal, who would go and swim naked in a sea? :-)

(*)

After that september I started my two-years period of studying in Warsaw (which I’m finishing just now) and so it was hard for me to organize a winter camp. For the following september I quite knew what I’m willing to do – read two other parts of Milnor’s “Morse theory”). In Warsaw I made friends with some great young people and almost all of them agreed to come to Wiselka. Accordingly, I think this last camp was the coolest one. During the year before the camp, most of us attended advanced differential geometry course (which I especially enjoyed, as I’ve already knew some of it from last camp) and “Morse theory” allowed us to see very interesting applications of it. Highlights of a book are, among the others, investigations of a topology of Lie groups (including a version of Bott periodicity), results on homotopy groups of spheres (inluding Freudenthal Suspension Theorem and some theorems relating curvature to topology (“if curvature is everywhere < 0 then the space is homeomorphic to affine space”)

True anecdote: on this camp there were also some younger people which decided to have a camp on different topic (they said that what we’re doing is to hard for them – I think they were wrong). They weren’t as focused on mathematics as we were (on the camp we talk mainly about mathematics, even on the beach). On one occasion one girl from a younger group told us something like: “I can’t understand you. Normal young people should talk about sex and you talk about mathematics.” :-)

(Maybe it’s because there’s only one girl among us :-)

(*)

Sideremark: one so feels that book by Milnor and Stasheff “Characteristic classes” and Milnor’s “Lectures on an h-cobordism theorem” are not really wholy written by Milnor (the latter is a lecture notes written by two of his students). They lack clarity of Master’s works.

(*)

So I come to the subject of this year’s camp. One proposition, quite natural perhaps in the light of above, is to learn a bit about singularities in topology. More specifically, Jarek Kedra’s proposition is to learn about fundamental group of a completion of a (perhaps singular) curve in complex projective plane (from Ichiro Shimada’s paper) (one reason it’s interesting is a theorem by Denis Auroux which states that every symplectic 4-manifold can be covered by complex projective plane and this (branched) covering can be more-or-less understood in terms of a set of singular points of this covering (which is a curve) and monodromy around this curve (which has to do with fundamental group). Shimada’s notes are a bit to little for a camp and we would add some chapters from Milnor’s “Singular points of complex hypersurfaces“) I don’t know this book but it’s Milnor so it must be good (and Jarek Kedra says these two papers could nicely complete each other).

The second proposition is more or less mine: to learn basics of TQFT (Topological Quantum Field Theory). I motivate it in two ways: first, it’s connected to physics, and physics is interesting. We could perhaps learn how it’s actually connected to physics (maybe ask somebody to give us one or two lectures on these connections which nobody of us understands at all?). Second, it’s connected to mathematics, apparently quite strongly: I was on Ulrike Tillman’s lectures devoted to the proof of Mumford Conjecture, and speaking freely in a language of various field theories was apparently a prerequisite (and TQFT is perhaps the simplest of field theories, so learning it seriously would be perhaps a good starting points for further studies). Our text would be probably Frank Quinn’s “Lectures on Axiomatic Quantum Field Theory” (from this book). Jarek Kedra said that he really likes this idea and proposed some further texts to choose from: V. Turaev’s “Quantum invariants of knots and 3-manifolds” and M.Atiyah’s “The geometry and physics of knots”.

(Field theories in mathematics were coinvented by Sir Michael Atiyah. The only mathematical book which I wholy read by myself is Atiyah/McDonald “Commutative Algebra”. It is almost as good as Milnor’s books.)

(*)

And finally, the Introduction to Noncommutative Geometry camp. Also sort of my proposition, but all agreed that it could be a good idea (NGC is strongly represented among Warsaw mathamaticians and we’re all “aware” of the existence of NCG)

However, my proposition at least partially is provoked by this NCG blog, on which some mystical statements are being told. And I like this sort of mysticism. I like the idea of investigating properties of space of Penrose tilings – I can’t see any method that would allow it, and Alain Connes claims that NCG makes it possible.

The problem is: none of us can say what it’s really about and it’s hard for us to estimate whether we’ll like it or not, based only on texts we don’t really understand. We could ask one of aforementioned Warsaw mathematicians to give us an advice, but, well, at least I don’t know how to talk with them – they simply know to much (Anecdote: “You want to learn Mathematics? Go ask Tomek Maszczyk what is 2+2″). NCG is a very vast subject and it’s hard for us to ask specific question. And asking nonspecific question provokes answer which fails to be understandable :-).

So, I wrote one of the authors of above blog, Masoud Khalkhali (actually, it’s his idea to write about a math camp on a blog):

*Hello!
Every september me and my friends organize a “math camp”. The idea is as follows: around june (i.e. now) we choose what we want to learn, we
choose a specific text (less than 200 pages long), everyone (~10 people) chooses a part of it and becomes “an expert” on it. Finally we meet and talk what we’ve learned.*

*Partly because of your blog (I put there few comments as sirix), we consider making sort of “Introduction to NCG camp”. However, we don’t know whether we could learn anything meaningful (because of the vastness of topics of NCG). We know more or less some homological algebra (first few chapters of Weibel, not everybody knows derived categories), algebraic topology (~Hatcher), differential geometry/topology (one of previous camps was on Milnor’s “Morse theory”, we learned also Milnor’s “Characteristic classes”), functional analysis (all the standard topics “up to” spectral theory of bounded self-adjoint operators). Most of us don’t know any physics.*

*For example, our other idea for this year’s camp is Frank Quinn’s “Lectures on axiomatic TQFT” (table of contents).*

*Could you provide us with an idea of what we could do on our camp if we were willing to learn some NCG?*

Here’s what he answered me:

*Dear Lukasz Grabowski,
Having an NCG camp is an exciting idea! Your background sure is enough to start off. I can think of three, or 4 things right now: A chapter
from Alain Connes’ 1994 book (available online on his website at http://www.alainconnes.org). The survey paper (joint with Marcolli) A walk in the noncommutative garden
http://arxiv.org/PS_cache/math/pdf/0601/0601054v1.pdf*

*I have also put two survey articles on NCG on the archive.
http://front.math.ucdavis.edu/math.QA/0702140
http://front.math.ucdavis.edu/math.KT/0408416*

*These might give you an idea and hope it is of any help. If you need more consulting , surely I will be more than happy to discuss things with
you. Also discussing such things in the blog is a good idea since I am sure will be beneficial to other users as well.
Sincerely
Masoud (khalkhali)*

I looked on above papers, and I like especially one of them: “Very basic NCG”. I think it would be very good to teach us some language. But we have this rule that on every camp there must be some interesting and nontrivial theorem. So we’d perhaps have to take something from Alain Connes’ book. That’s sort of risky: for a layman like me Connes has one serious advantage: he’s, like Milnor, a Fields’ medallist; and he has one serious disadvantage: he’s, like Serre, French ;-). So the question is whether his book is more Milnorish or Serreish in flavour :-).

So, as you see, we have a serious problem here. All three are tempting but we can choose just one. Feel free to comment on that.

I’ll let you know about any progress in decision making process.

One more thing: If you’re by any chance a math student from, say, Germany or Czech Republic then we could perhaps make a “joint camp” this time or in future – Leave a comment or mail me if you like the idea (my mail is on “About” page).

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Accordingly, first few paragraphs aren’t very relevant, as they describe my admiration of Siegel’s proof.

I use Unicode for mathematical notation. Enjoy :-)

I got back to Warsaw, having spent the Easter in my hometown. The first lecture I attended after the break was “Modular Forms” and the lecturer served us a big chunk of juicy mathematics. Namely, he presented Carl Ludwig Siegel‘s proof of transformation formula for Dedekind eta function.

It’s not that this proof is extremely difficult. It isn’t; it’s understandable for anybody after basic course in complex analysis. Rather, it’s extremely tricky, and, because it’s relatively short (thus easy to follow), this trickiness is what makes it a cool thing.

I’d like to write what is the theorem and how Siegel proved it. (Or rather, I’ll use it as an excuse to write a summary of what I’ve learned so far :-)

First of all, the proof is really tricky. It is so tricky that after getting a general idea I decided that it’s impossible for normal human being to come up with something like that. If you come to the same conclusion after reading this post then it’s perhaps worth to have in mind that Siegel’s proof wasn’t a first proof of a result I’ll describe in a moment. AFAIK, previous proofs were (much) longer, but they were more straightforward. Still, because of such things I wonder whether First Class Mathematicians are actually normal human beings…

Let me review, for introduction and motivation, what I’ve learned. (If you want to focus on the meet, skip few paragraphs.) What we’re generally interested in are elliptic curves. For our purposes elliptic curve is a 2-dimensional torus with a chosen complex structure (not almost complex structure). As usually in mathematics, what we really care about are just isomorphism classes of objects, and in our case apropriate isomorphism is obviously a holomorphism (a diffeomorphism whose differential commutes with multiplication by i = √-1) (and “obviously” means here nothing but “by definition”). By some neat theorem every elliptic curve (“torus with a chosen complex structure”) is isomorphic to a torus of a form C/Λ, where Λ is some lattice in ℝ^{2}=C

(Of course, if we were only interested in topological properties of a torus it wouldn’t matter which lattice Λ we choose (say, whether we choose lattice generated by vectors (1,0) and (0,1) or (1,0) and (1,1)). C/Λ is topologically always the same torus. However, complex structure is different; you can visualize this by imagining a single torus with two complex structures coming from above two lattices and asking yourself what does the multiplication by i do on a tangent plane on this torus. Answer: for the first lattice multiplication by i is a rotation by π/2, for the second it’s some other linear transformation).

For this reason, the name “elliptic curve” is really reserved for tori of form ℂ/Λ (not tori with some weird complex structure). I adopt this convention from now on (and when I write “torus” it means “elliptic curve” as well, unless I explicitely state that I mean topological torus (torus without complex structure)) .

(You may argue that elliptic curve is a stupid name for a torus – torus is not a curve but something 2-dimensional. Well, the reason is that torus is 1-dimensional if regarded as a complex manifold. You may have heard that elliptic curve is a set of points in ℝ^{2} given by equation y^{2} = a∙x^{3} + b∙x + c, or similar – such set points indeed looks like an honest 1-dimensional curve. To obtain such a curve from 2-dimensional torus with complex structure one has to embed (holomorphically) this torus in ℂP^{2} (complex plane) and look at the intersection of torus with some affine part of ℂP^{2}.)

Now let’s wonder when two such tori, ℂ/Λ and ℂ/Λ’, are isomorphic. It’s easy to see that if Λ’ is a rotation of Λ then tori are isomorphic. Also (which is only very small ε less obvious), if Λ’ is a stretching of Λ then tori are isomorphic. These two properties are summarized by saying that if Λ’=αΛ, α∊ℂ, then ℂ/Λ ≃ ℂ/Λ’. The converse is also true and not hard to understand if one knows that isomorphism between two elliptic curves must come from an ℝ-linear isomorphism of ℂ (and this also follows from aforementioned neat theorem (I believe so :-).

It follows that every elliptic curve is isomorphic to the one of form ℂ/Λ for Λ generated by two complex numbers (previously I wrote “vectors” instead of “complex numbers” in this context) of which first is equal to 1 and second to some other complex number τ∊ℂ. Additionally, we can take τ to be from upper half-plane (because we don’t care about orientation on our tori, so that ℂ/ ≃ ℂ/ (β* is a complex conjugate of β, ≃ is induced by ℝ-linear transformation of ℂ, complex conjugation), even though lattices and are in general different. Let’s adopt convention that generators are always written in anticlockwise order).

Notice that τ is not something assigned to a torus ℂ/Λ, but rather to some chosen generators u,v of Λ. If we change generators of Λ then we’ll also change τ. It’s straightforward to check that if new generators are u’=a∙u+b∙v, v’=c∙u+d∙v then τ will change in the following manner: τ’=(aτ+b)/(cτ+d) (check it yourself). Notice also that, because u’ and v’ are also generators of Λ, matrix

a b

c d

is an element of SL_{2}(ℤ). So we have an action of a group SL_{2}(ℤ) on an upper half-plane.

We can now do something VERY cool, which I hear mathematicians often do and I always find a bit exciting. Instead of looking and investigating properties of single elliptic curve, we look at a set of all elliptic curves. We have a morphism from this set to H/SL_{2}(ℤ), where H is upper half-plane and the action of SL_{2}(Z) on it is the one just described. This morphism takes elliptic curve C/Λ, choses some pair of generators (u,v) of Λ and sends the elliptic curve to an orbit which contains τ associated to (u,v) (it’s easy to see that τ=v/u).

Now, we cooked up the action of SL_{2}(ℤ) on H precisely in such a way that this morphism is well-defined – that is, it doesn’t depend on choice of generators of Λ. Additionally, because two elliptic curves ℂ/Λ and ℂ/Λ’ are isomorphic precisely when Λ’=αΛ, we see that unisomorphic elliptic curves are mapped to different points of H/SL_{2}(ℤ). (And, of course, every point of H/SL_{2}(ℤ) is an image of some elliptic curve.)

Summarizing, isomorphisms classes of elliptic curves are precisely points of H/SL_{2}(ℤ) (more precisely: there is bijection between…). Cool! We have given a set of isoclasses of elliptic curves additional geometric structure in a very natural way! (natural is the key word; there is plenty of bijections between set of isoclasses of elliptic curves and your favorit geometrical object X but for most of these bijections there is no way to translate properties of X into useful information about isoclasses of elliptic curves)

Let me compare situation in which we are to the one often encountered in classical mechanics: we investigate there a motion of some set of objects in ℝ^{3}. So we cook up a space (called “phase space”) whos points are in correspondence with all possible configurations of our objects (for example, phase space of two particles is ℝ^{12} – point of this ℝ^{12} encodes positions and velocities (which are 3-vectors) of both particles). Now, real-valued functions on the phase space are called sometimes “observables” (however, more often this word is used for something in quantum mechanics which I unfortunately can’t understand), because they correspond to things we can observe (for example, in this ℝ^{12} function that sends a point to its first coordinate may correspond to observing first spatial coordinate of first particle; function that sends a point of ℝ^{12} to a sum of squares of last six coordinates (perhaps with some coefficients, if masses of both particles are not equal to 2) corresponds to observing kinetic energy of whole system).

I woudn’t hesitate to call functions on a phase space “properties of a system” or “modular functions” – modular comes from a word “moduli” which means (more or less :-) “property” (not “This is my property, so go away or I’ll shoot you.” but “Properties of this kind of plants are interesting.”).

Similarly, we call functions on H/SL_{2}(ℤ) “modular functions” or “elliptic modular functions” to emphasize that we’re interested in moduli of elliptic curves (“in properties of elliptic curves”). Before I give precise definitions, let me guess what some of you might think right now: “Ok, so functions on ℍ/SL_{2}(ℤ) are like observables, ℍ/SL_{2}(ℤ) itself is like a phase space and elliptic curves are like configurations of a mechanical system. So maybe we can go further with this analogy and ask what on the ellpitic curves part of a diagram is like a mechanical system?”.

Most curiosly, there is a group of physicist that pursue this point of view! They call themselves “String theorists”, they’re a little bit weird and AFAIU they claim something like this: basic things in universe are not particles or whatever but tiny “strings” – 1-dimensional compact manifolds (1-dimensional compact manifold = finite number of circles). If we draw a “movement” of such a string in a spacetime we get a two dimensional manifold. For example, if we are dealing with some peaceful string that doesn’t change its spatial position, we’ll get a pipe in a spacetime. However, 1-circle string can change into a 2-circle string at some point in time – if this happens we’ll get “pants”. Also, 1-circle string can change itself into 2-circle string and then back into 1-circle string – what we’ll get in this situation is topologically a torus without 2 disks. Also, it may happen that at a moment 0 there is no string, at a moment 1/4 there is a string consisting of one circle, later on it changes itself into two circles, then back to one circle, and at the moment 3/4 it again disappears – in this case the 2-dimensional manifold we get in a spacetime is a torus.

Although it’s not very important from string theory point of view, we now focus on this last example, so that we make a connection with elliptic curves. Physicists say that the string in any given moment has some internal tension which gives rise to a complex structure on the resulting torus. And yes, they also say that a movement of a string (that is, 2-dimensional torus with a complex structure) is pretty much the same from the physical point of view if tori are isomorphic. Phase space of all possible movements of a string (movements of this specific type: appear, change into two circles, back into one circle, disappear) is therefore a ℍ/SL_{2}(ℤ). We can roughly end our analogy with saying that “movement of a string” is like a mechanical system. In this case, elliptic modular functions could be interpreted as “properties of a motion of a string” (or just “observables” :-).

I hope that what I wrote in last 2 paragraphs is not a total nonsense :-).

Let’s back to reality. Theorem Segal proved roughly states that some explicetely given function on ℍ (upper half-plane) is actually almost a modular function (that is, that this function is almost invariant under action of SL_{2}(ℤ) on ℍ. Let’s start with precise definitions and statement of a thereom.

Suppose we have a meromorphic function f: ℍ –> ℂ (ℍ denotes upper half-plane). We’ll say that f is a “modular function” iff following two conditions hold:

a) f is invariant under action of SL_{2}(ℤ) on ℍ. That is, for every matrix A∊SL_{2}(ℤ) and for every τ∊ℍ we have f(A∙τ)=f(τ)

b) There exists m∊ℕ such that Fourier series of f is as follows: f(τ) = Σ_{k≥m}e^{2πiτ}.

Note that by a) f is a periodic function with period 1 (because one has a matrix

1 1

0 1

in SL_{2}(ℤ) which sends every τ∊ℍ to τ+1), so f has some Fourier series – be requires that this series doesn’t have infinitely many negative terms.

Clearly, modular function gives rise to a meromorphic function ℍ/SL_{2}(ℤ) –> ℂ. However, we’ll be interested also in functions on ℍ which don’t give rise to functions ℍ/SL_{2}(ℤ) –> ℂ but to sections of vector bundles over ℍ/SL_{2}(ℤ). That’s a reason for the following definition:

Suppose we have a meromorphic function f: ℍ –> ℂ and k∊ℕ (so k=0,1,…). We’ll say that f is a “modular function of weight 2k” iff following two conditions hold:

a) For every matrix A∊SL_{2}(ℤ) and for every τ∊ℍ we have f(A∙τ)=(cτ+d)^{2k}f(τ), where (c d) is a lower row of a matrix A.

b) There exists m∊ℤ such that Fourier series of f is as follows: f(τ) = Σ_{k≥m}a(k)∙e^{2kπiτ}.

Note that (cτ+d)^{2k} = 1 for (c d) = lower row of matrix

1 1

0 1

so modular functions of weight 2k are periodic with period 1 and again above remark applies. Also, we see that “modular functions of weight 0” are just “modular functions”.

Let’s consider ℂ-linear bundle Λ_{ℂ} of complex differential forms over ℍ (of course, it’s isomorphic to trivial bundle). Its sections are forms g(z)∙dz, where g is some complex valued function on ℍ. We can also consider k^{th} tensor power of this bundle: Λ_{ℂ}^{⊗k}. Now, it’s straightforward to check that modular function f of weight 2k gives rise to a section of Λ_{ℂ}^{⊗k}, namely f(z)∙dz^{k} (here dz^{k} =dz⊗dz⊗…⊗dz).

We now define a star of the evening: The Dedekind eta function, η. It is a function ℍ –> ℂ defined by the equation

η(τ) = e^{πiτ/12}∙Π_{n≥1}(1 – e^{2πinτ}).

(Let me remind that a group SL_{2}(ℤ) is generated by two matrices

1 1

0 1

and

0 -1

1 0

which act on ℍ by sending τ to τ+1 and sending τ to -1/τ, respectively. It’s straightforward to check that to check if given meromorphic function is modular of weight 2k it’s enough to check a) only for above 2 matrices (and check b), of course).)

One easilly checks that η is meromorphic and that it fulfills b). We’ll investigate to what extent η fulfills a).

Unfortunately, η is not a modular function. Indeed, one sees easilly that η(τ+1) = e^{πi/12}∙η(τ). However, this is still nice transformation law – 24th power of η still has chance to be a modular function of weight 12. Indeed, this is the case, as (among the oher things) the following theorem shows:

Theorem With A Tricky Proof By Siegel: η(-1/τ) = (-iτ)^{1/2}∙η(τ).

Before the tricky proof, let me once more remark that indeed from this follows that 24th power of η is a modular function (this is straightforward). However, this theorem has many other consequences (which one can find for example in Apostol’s book. The following proof also can be find there, in 3rd chapter).

Tricky Proof By C.L. Siegel: Check it out in Apostol’s book :-)

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In this article I’ll be putting links to polish notes from consecutive lectures as I complete writing them.

They are a little bit chaotic, but I’ll answer any questions with pleasure. Also, I’ll put here any comments that I’ll receive by mail from my friends.

W oznaczeniach Gamma jest zmienione na G.

Jestesmy zainteresowani przestrzenia moduli M_gn, ktorej punkty odpowiadaja powierzchnia Riemanna genusu g z wycietymi n dyskami wraz z wybrana struktura zespolona (tak sadze – conformal structure to chyba to samo co struktura zespolona).

Precyzyjniej: definiujemy (1) przestrzen Teichmullera T_gn wszystkich struktur zespolonych na powierzhni F_gn genusu g z wycietymi n dyskami, (2) grupe klas map (Mapping Class Group) G_gn jako pi_0(Diff(F_gn)), gdzie Diff(F_gn) to grupa dyfeomorpfizmow F_gn, ktore zachowuja orientacja i sa stale na brzegu. G_gn dziala na T_gn, zatem definiujemy (3) M_gn = T_gn / G_gn.

Hmmm. Cos pokrecilem, bo tak jak ja zdefiniowalem G_gn i T_gn to nie ma dzialania G na T. Caly Diff dziala na T. Ponizej zakladam, ze definicje T i G sa dobre :-)

(Pozniej dowiedzialem sie, ze nic generalnie nie pokrecilem – patrz ponizsze uwagi – Diff i G sa czesto homotopijnie rownowazne i daltego mozna skosntruowac przestrzen Teichmullera dla G, szczegolow nie znam, ale madry czlowiek tak powiedzial.)

Uwagi: (1) G_gn jest skonczenie prezentowalna. Generatory dajace skonczona prezentacje to np. twisty Dehna. Np. G_0,2 = Z, G_1,0 = SL_2(Z) ( “0,2” to jest rurka a “1,0” to torus)

(2) Jest odwzorowanie Diff(F_gn) do G_gn (skladowe spojnosci przechodza na punkty im odpowiadajace) i ono jest homotopijna rownowaznoscia (czyli skladowe spojnosci Diff(F_gn) sa homotop. rownowazne punktom) gdy 2-2g-n0. Argument: n>0 czyli wyjelismy dysk. Jezeli jakis element G_gn zachowuje strukture zespolona na F_gn (chyba G_gn to jest jednak Diff… – a moze chodzi o dzialanie z dokladnoscia do homotopii? Wtedy na mocy (2) bylby sens) to mozemy dolepic dysk (lub dyski) rozszerzyc dzialanie tego dyfeomorfizmu do dyfeomorfizmu powierznni bez brzegu, ktory jest staly na dysku (dyskach) i zachowuje strukture konforemna – a to jest niemozliwe (chyba dlatego, bo taki dyfeomorfizm (tak naprawde holomorfizm) podnosilby sie do holomorfizmu plaszczyzny hiperbolicznej stalego na jakims dysku; Gdy n=0 to stabilizatory sa skonczone.

(3) T_gn jest homeo z R^{6g-6+3n}. UT dawala heurystyczny argument gdy g=0 ale nie zrozumialem

Jest tez inny opis M_gn dla n>=1 i g>1. Mianowicie, dla n>=1 M_gn jest BG_gn (na mocy (2) i (3)). Dla g>1 mamy, jak wyzej napisalem, G_gn = Diff(F_gn) (rownosc homotopijna), wiec BG_gn = BDiff(F_gn), natomiast to jest rowne tak zwanemu M^top_gn.

M^top_gn robimy tak: Bierzemy przestrzen wszyskich mozliwych wlozen F_gn w R^niesk. To jest przestrzen homotopijnie rownowazna punktowi (tw. Whitneya) i dziala na niej Diff(F_gn) (poprzez “reparametryzacje zrodla”) wolno. Iloraz tego dzialania to M^top_gn. Zatem z definicji jest to BDiff(F_gn). Punkty M^top mozna utozsamiac z podrozmaitosciami R^niesk (oczywiscie). Zatem to jest “intuicyjna” przestrzen moduli.

Uwazka: twierdzenie Whitneya mowi tak naprawde, ze przestrzen wlozen dowolnego zrodla w R^niesk jest homotopijnie trywialna. (Nie jest to twierdzenie trywialne: nietrudno sie przekonac, ze dla zrodla rownego dwom punktom sprowadza sie do sciagalnosci S^niesk.)

Powyzsze uwagi pokazuja tez, ze (dla n>=1 i g>1) pierscien kohomologii M_gn to pierscien kohomologii BDiff(F_gn), a to jest pierscien klas charakterystycznych wiazek o wloknie F_gn. Filozoficznie to jest fajne: klasy charakterystyczne wiazek o wloknie F to kohomologie przestrzeni moduli “wszystkich F-ow”.

Homer-Ivanov udowodnili, ze H*BG_gn jest niezalezne od g,n gdy (mozliwe, ze zle spisalem) *niesk} G_g{g,1}. (Granica jest wzieta poprzez wiadome wlozenie F_{g,1} –> F_{g+1,1}.

Miller-Morita udowodnili, ze H*BG_niesk stensorowane z Q zawiera pierscien wielomianow Q[K_1, K_2,…] gdzie K_i sa pewnymi elementami stopnia 2i.

Hipoteza Mumforda: Tak naprawde jest rownosc: H*BG_niesk tensor Q = Q[K_1, K_2,…]. Zostala udowodniona przez Madsena i Weissa.

Dalej mozliwe, ze cos mocno pokrece.

Zmieniamy troche temat. C(X) to kategoria strun w X – jej obiekty to niesparametryzowane zbiory okregow w X a morfizmy to niesparametryzowane kobordyzmy miedzy dwoma danymi zbiorami okregow. To ma motywacje fizyczne (ktorych, poza najprostszym “te petle maja zastapic czastki i wtedy wszystko podobno ma szanse byc bardzie elegancko” nikt chyba nie jest w stanie wyjasnic :-( ). Wielcy tego swiata z idei fizycznych wydestylowali, ze mozemy badac C_d(X). Wydaje mi sie, ze UT powiedziala, ze ze wzgledow technicznych wygodnie patrzec na obiekty tej kategorii jak na d-wymiarowe niesparametryzowane podrozmaitosci w R^niesk wraz z odwzorowaniem do X, podobnie morfizmy to kobordyzmy (niesparametryzowane) w R^niesk wraz z odwzorowaniem do X. Chyba ze wzgledow technicznych Bedziemy rozwazac tez C_d'(X) – podkategorie C_d(X) w ktorej dopuszczalne morfizmy sa tylko takie, ktore maja niepusty cel. (W orginale C_d'(X) jest oznaczane C_{d, d krecone}(X)).

Wielkie twierdzenie mowi, ze BC_d(X) jest homotopijnie rownowazne Omega^{niesk-1} (G_{-d} dziobek X_+). UT powiedziala na razie tylko, ze ta Omega to dobrze znany i rozumiany obiekt. G w nawiasie jest grube i nie ma chyba nic wspolnego z G_gn lub G_niesk powyzej. Z tego twierdzenia podobno juz nietrudno wynika hipoteza Mumforda (w nawiasie w notatkach dopisalem sobie “bo Z x BG_niesk ma kohomologie jak Omega(C_2′)” – nic nie rozumiem). (kilka dni pozniej juz w W-wie przypominam sobie, ze nawias to jest nietrywialna czesc twierdzenia, by Ulrike Tillmann)

Mamy funktor “kwantowej teorii pola” z przestrzeni topologicznych do “Graded Abelian Groups”, ktory przestrzeni X przyporzadkowuje QF^d_*(X) := pi_{*+1} (BC_d(X)). Nierozumiem za bardzo co bada B od kategorii (tak naprawde to nie mam tez intuicji co do B od grupy) wiec tym bardziej nie rozumiem tego funktora. (dopiesek pozniejszy: jezeli tw. Whitneya dziala naprawde dla kazdego zrodla to chyba zaczne myslec o BG jako o przestrzeni wszystkich wlozen G w R^niesk). W kazdym razie, powyzsze wielkie twierdzenie podobno mowi m.in., ze ten funktor jest uogolniona teoria kohomologii. UT powiedziala, ze “it might be a little bit surprising” (bo w takim razie ten funktor jest wyliczalny)

Byla zajawka dowodu rownowaznosci BC_d(X) = Omega^{niesk-1}…, ale na ostatnim wykladzie bylo to przedstawone lepiej, wiec na razie nic nie napisze.

Jako rzeczy, z ktorych korzysta sie w pelnym dowodzie UT wymienila tw. Phillipsa o submersji: Niech M bedzie otwarta podrozmaitoscia (co to znaczy??? jest zbiorem otwartym w jakims R^M?). Jezeli mamy odwzorowanie f:M–>X i jakiekolwiek “nakrycie” tego odwzorowania F: TM –> TX ktore jest submersja to mozemy (f,F) zamienic na homotopijnie rownowazne (g,Dg) dla jakiegos g.

(Lubie to twierdzenie, bo umiem je zapamietac :-) Trudne jest ono?)

Inny ingredient to klasyczna teoria kobordyzmu. Taki napis sie pojawil: Omega_d = pi_d(Omega^niesk MSO), gdzie Omega_d to d-wymiarowe rozmaitosci z dokl do kobordyzmu, a MSO to spekturm Thoma.

Uwaga na koniec: my zajmujemy sie caly czas niezwartymi przestrzeniami moduli. Badanie uzwarcen jest trudniejsze (jest to mniej wiecej teoria niezmiennikow Gromova-Wittena), ale imc Galatius ma program, ktory w zamierzeniu przenosi czesc teorii niezwartych przestrzeni moduli na przypadek zwarty.

Chetnie zobaczylbym jakies odpowiedzi na takie pytania:

1) Chcialbym nauczyc sie rozumiec napisy postaci Omega^niesk, spektrum, itp. Jak to zrobic? (tego typu napisy to byla chyba najwazniejsza rzecz, ktorej nie znalem)

2) Co dokladnie z ta przestrzenia Teichmullera? Powiedzmy, ze grupa G dziala wolno na przestrzeni sciagalnej X i ze G jest homotopijnie rownowazna jakiemus swojemu ilorazowi – jak skonstruowac przestrzen sciagalna Y na ktorej ten iloraz wolno dziala?

3) Gdzie znajde tw. Phillipsa o submersji?

4) Zawsze chetnie uslysze jakiekolwiek komentarze fizyczne. W szczegolnosci: co kwantowa teoria pola mowi fizykom?

Przeglad Teorii Pola

(|_| oznaczac bedzie zawsze rozlaczna sume)

(na koncu sa pytania)

CFT – Conformal Field Theory (matematyczne sformulowanie – chyba G. Segal). Dzialamy w kategorii Segala S, ktorej obiekty to C_n, n=1,2,…, n-tki okregow: C_n := S^1 x {1,2,…,n}. Morfizmy: element Mor (C_n, C_m) to powierzchnia Riemanna wraz z wybrana struktura zespolona i mapowanie brzegu na C_n |_| -C_m , skladanie to sklejanie powierzchni (czyli trzeba tu pare rzeczy posprawdzac, np. ze struktury zespolone mozna jakos sensownie skleic). Do danych, ktore tworza kategorie Segala nalezy jeszcze “struktura monodalna” (“monodal structure”) – tzn. (jak zrozumialem) musimy wybrac produkt tensorowy, wybieramy sume rozlaczna.

Uwazka: Mor(C_0,C_0) = duza podwojna suma rozlaczna po M_{g_i}0 (suma przestrzeni moduli z poprzedniej czesci notatek).

CFT to funktor monoidalny (czyli ma zachowywac produkty tensorowe) (S, |_|) –> (Vect/C, tensor).

(Jest tu jeszcze jakis myk – UT przemknela sie nad tym ale ktos z publiki ja drazyl – zdaje sie, ze ten funktor ma byc topologiczny, tzn. on ma uwzgledniac topologie w zbiorach morfizmow. Nie jestem pewien, patrz nizej)

Podobno taki funktor jest trudny ale “well understood in terms of Kac-Moody algebras & loops groups & …”

20 lat temu Segal patrzyl na kategorie S(X), gdzie X jest symplektyczny ze str. prawie zespolona. Obiekty to elementy map(C_n, X) a morfizmy miedzy dwoma mapami u oraz v to pary (K, psi) gdzie K jest morfizmem w S miedzy zrodlem u i zrodlem v a psi jest prawie-holomorficznym odwzorowaniem K–> X, ktore obcina sie do u i v tam gdzie powinno.

Monoidalny funktor (S(X), |_|) –> (Vect/C, tensor) nazywamy “obiektem eliptycznym”. Podobno jest to “way to give geometric meaning to elliptic cohomology”.

(Tu myk jest wiekszy, bo obiekty kategorii maja naturalna strukture topologiczna, a w Vect takiego czegos nie ma – wobec tego ten funktor nie idzie tak naprawde do Vect tylko raczej do jakiegos rodzaju grassmanianu)

Klasyczna teoria pola: kategoria to P(X) – paths w X. Obiekty to punkty X, morfizmy to odwzorowania [0,t] –> X, ktore zaczynaja sie i koncza tam, gdzie trzeba. Klasyczna teoria pola to funktor E: P(X) –> Vect/C (z mykiem jak wyzej). Zdaje sie, ze slyszalem, ze przykladem jest wiazka nad X z koneksja. Nie rozumiem jaki to ma zwiazek z klasyczna teoria pola w ujeciu Lagrangianowym. E daje nam BE: BP(X) –> B(Vect/C). BP(X) jest homotijnie rownowazne X (nie wiem czemu, ale to chyba jest “klasyczne”), wiec dostajemy odwzorowanie z X do rozlacznej sumy BGL_n. UT cos zaczela mowic o zwiazkach z K-teoria, nie zrozumialem. O fizyce nie mowila.

UT kazala chyba na BP(X) patrzyc jak na motywacje do badanie BS(X) (i wielowymiarowego odpowiednika S(X) ). Okaze sie pozniejm, ze BS(X) jest homotopijnie rownowazne Omega^niesk(G_{-2} dziobek X_+), ktore konstruuje sie za pomoca grassmannianow.

Teraz cos prostego: TFT, topologiczna teoria pola. Fizycznie malo znaczaca, ale wymyslil ja Witten w nadzieji, ze w dobrym przyblizeniu bedzie opisywala pola “o malych energiach, widziane z duzej odleglosci”. Kategoria nazywa sie Cob_d. Obiekty to klasy dyfeomorfizmow zamknietych zorientowanych rzmaitosci wymiaru d-1, morfizmy to klasy dyfeomorfizmow kobordyzmow. W szczegolnosci Cob_2 = pi_0(S) w odpowiedni sensie (obiekty te same co S, morfizmy w S miedzy dwoma obiektami tworza jakas przestrzen topologiczna, dla utworzenia morfizmow miedzy tymi dwoma obiektami w Cob_2 bierzemy pi_0 tej przestrzeni. Skladanie morfizmow dostajemy stosujac pi_0 do skladania morfizmow (gdzie “skladanie morfizmow” to funkcja miedzy produktem dwoch przestrzeni topologicznych a trzecia przestrzenia topologiczna).

Topologiczna Teoria Pola to monoidalny funktor E: (Cob_d, |_|) –> (Vect/C, tensor). I tu akurat chyba nie ma myku (ale moze jest :-).

Twierdzenie Folha: W wymiarze d=2 Topologiczne Teorie Pola sa w bijekcji ze skonczonie-wymiarowymi algebrami Frobeniusa nad C (alg A jest alg. Frobeniusa jesli posiada “1” i funkcje Tr: A –> C, ktora ma te wlasnosc, ze parowanie <x,y> :=Tr(xy) jest niezdegenerowane).

Idea: E(S^1) jest zatem przestrzenia wektorowa. To bedzie nasza algebra A. Poniewaz funktor jest monoidalny to dwa okregi przerzuca na A tensor A. Zatem przerzuca morfizm “majtki” na morfizm A tensor A –> A, to jest mnozenie w naszej algebrze.

(Jest przemienne, bo interesuje nas tylko typ dyfezomorfizmu, A morfizm “dwie skrzyzowane rury” przechodzi na automorfizm A tensor A – wydaje mi sie, ze jeden gosc powiedzial, ze trzeba dodatkowo zalozyc, ze ten automorfizm to jest “switching”, UT przez chwile chyba mowila, ze to wynika z aksjomatow, nie wiem jak to sie skonczylo. Byc moze zatem trzeba zalozyc, ze TFT to monoidalny _symetryczny_ funktor.). (update: ten gosc do konca twierdzil, ze trzeba zalozyc, ze “ten automorfizm to jest “switching””. to chyba sensowne, bo inaczej nie widze przeszkod, zeby zalozyc, ze dwie skrzyzowane rury przechodza np. na “graded commutative switching”)

Poniewaz funktor jest monoidalny to przerzuca zbior pusty na C, zatem morfizm czapka i odwrocona czapka na morfizmy C –> A i A –>C. Pierwszy to jest nasz trace, drugi to jedynka. To, ze dostalismy algebre z jedynka i morfizmem Tr widac “od razu”. Pomyslu wymaga tylko sprawdzenie, ze Tr jest niezdegenerowany. Trzeba sprytnie wykorzystac morfizm genusu 0 ze zbioru pustego do dwoch okregow (i zauwazyc, ze < , > to obraz morfizmu genusu 0 z dwoch okregow do zbioru pustego i ze wszystkie morfizmy dyfeomorficzne z rurka sa identycznoscia, wiec przechodza na identycznosc). Nie jest to bardzo trudne, ale wymaga zrobienia rysunku.

Przyklad algebry Frobeniusa: gdy mamy zwarta rozmaitosc M to A = H*(M, C) (z kub-produktem), Tr = calka.

Ostatnia teoria pola to TCFT – Topological Conformal Field Theory. Kategoria,w ktorej pracujemy to C_{*}S. Obiekty to znowu skonczone zbiory okregow, morfizmy to C_{*}(morph S), gdzie S jest kategoria Segala a C_{*} to funktor brania lancuchow singularnych (morph S to przestrzen topologiczna wiec ma to sens – porownaj powyzszy paragraf “Teraz cos prostszego…”)

TCFT to monoidalny funktor E: (C_{*}S, |_|) –> (kompleksy lancuchowe, tensor). Taki funktor indukuje (H_{*}S, |_|) –> (zgradowane przestrzenie wektorowe, tensor).

W S jest podkategoria “drzew” P, obiekty takie same jak S, morfizmy to powierzchnie genusu 0 ktore maja dokladnie jeden wychodzacy okrag w kazdej skladowej spojnosci. Dla H_{*}P istnieje czesciowy analog twierdzenia Folha:

(Uwazka moja: jasne, ze kategoria H_{*}P jest duzo latwiejsza niz P, bo morfizmy w tej pierwszej maja szanse tworzyc skonczenie-wymiarowe przestrzenie liniowe, a w tej drugiej morfizmy sa punktami pewnych rozmaitosci)

Twierdzenie Getzbera: Funktor E: (H_{*}P, |_|) —> (Zgradowane prz. wekt., tensor) nadaje strukture BV-algebry na E(S^1).

BV-algebra to twor duzo bardziej skomplikowany niz algebra Frobeniusa (nazwa pochodzi od nazwisk: ?Batnahu?-?). Nie musi byc chyba laczna, maja byc operacje mnozenia, trojkatu i nawiasu, ktore spelniaja pewne zaleznosci. D-d jest trudniejszy niz twierdzenia Folha bo teraz zeby okreslic *, trojkat i nawias trzeba brac elementy homologii pewnych przestrzeni moduli. Dokladniej, * to generator 0wych homologii przestrzeni moduli pow. Riemanna genusu 0, z dwoma dziurami wejsciowymi i jadna dziura wyjsciowa; trojkat to generator pierwszych homologii przestrzeni moduli rurki. Nawias jest trudniejszy: jest to pewien element pierwszych homomologii tej samej przestrzeni co dla *, okreslony przy uzyciu twistow Dehna tej przestrzeni (ma to sens, bo twisty Dehna sa elementami odpowiedniej grupy klas map G, zatem okreslaja petle w BG, a te daja elementy homologii BG (a BG to odpowiednia przestrzen moduli).

Rozszerzenie tego twierdzenie (to chyba tw. Tillmann?): Jezeli funktor E: H_{*}P –> Zgradowane prz. wekt. podnosi sie do H_{*}S to t^3 [x,y]=[t^3x,y]=0 oraz Trojkat(t^3 *x) = t^3*Trojkat(x), gdzie

t (tak zwany propagator) to generator zerowych homologii przestrzeni moduli powierzchni genusu 1 z jedna dziura wejsciowa i jedna dziura wyjsciowa. Idea dowodu pojawila sie na nastepnym wykladzie, bardzo chcialbym ja zrozumiec (bo pracujemy w homologiach morfizmow! Wydaje mi sie w tej chwili, ze jest to zupelnie inny poziom trudnosci niz w przypadku tw. Folha), wiec moze jeszcze napisze.

Z tego twierdzenia plynie wniosek, ze gdy t jest odwracalne to struktura BV-algebry jest trywialna (tzn. trojkat =0, nawias=0). Ciekawe skad pochodzi nazwa propagator (wymyslili ja chyba fizycy)?

Pytania:

1. Jaki jest zwiazek kategorii drog na rozmaitosci P(X) i funktora klasycznej teorii pola z klasyczna teoria pola w sensie minimalizacji lagrangianu?

2. Co fizycznie oznacza propagator (element t z przedostatniego paragrafu)?

(H oznacza funktor brania homologii, H* – kohomologii)

(na koncu sa pytania)

Poczatek wykladu sluzyl wprowadzeniu pojecia operadu. To pojecie bylo dla mnie nowe i musialem sie douczyc po wykladzie. Polecam artykul w wikipedii oraz artykul Johna Baeza: http://math.ucr.edu/home/baez/week220.html

Lubie krotkie definicje (gdy juz zrozumiem dlugie), najkrotsza jest taka: operad to multikategoria z jednym obiektem. Jest to jednak male oszustwo: John Baez na taki operad mowi “planar operad” a “prawdziwy operad” to dla niego (i dla UT tez) planar operad z dzialaniem grupy permutacji (jezeli nie wiesz o czym mowa to koniecznie odwiedz strone Baeza – sa tam odpowiednie rysunki).

Troche dluzsza definicja jest taka: operad to kategoria z produktem, ktorej obiekty to liczby naturalne ( z zerem), produkt na obiektach jest dodawaniem, a morfizmy sa postaci n –> 1 i produkty takich (morfizm id_n to produkt n morfizmow id_1).

Najdluzsza (i najjasniejsza) definicja jest taka: operad E to nastepujace dane: zbiory E_n (nalezy o nich myslec jak o zbiorach operacji, ktore przyjmuja n argumentow i zwracaja jeden) na ktorych dzialaja n-te grupy permutacji (“zmiana kolejnosci argumentow”), wyrozniony element 1 in E_1 oraz funkcje E_n x E_{k_1} x … x E_{k_n} –> E_s (gdzie s to suma wszystkich k_i) – skladanie operacji – ktore spelniaja aksjomat lacznosci i dobrego zachowywania sie z wyroznionym elementem.

Acha: E_n nie musza byc zbiorami – wystarczy, ze sa elementami jakiejs kategorii z produktem (monoidal category). My bedziemy glownie zainteresowani operadami w Top z produktem kartezjanskim. Do operadu mozna przylozyc funktor zachowujacy produkt i otrzymac inny operad – my bedziemy zainteresowani przykladaniem funktora kompleksow singularnych (dostajemy operad w kompleksach lancuchowych z produktem tensorowym) i funktora homologii (dostajemy operad w zgradowanych przestrzeniach wektorowych z produktem tensorowym)

Przyklad operadu: Na stronie Baeza jest dosc dokladnie opisany operad Lie odpowiadajacy operacjom w algebrze Lie. Operad Ass odpowiadajacy lacznej operacji mozna (chyba) opisac tak: zbior Ass_k to {planarne drzewa z korzeniem i k liscmi o 3walentnych wewnetrznych wierzcholkach wraz z numeracja lisci}/~, gdzie ~ to relacja utozsamiajaca wszystkie drzewa o k lisciach z ta sama numeracja. Skladanie operacji to sklejanie drzew. (gdy zapomniec o relacji to mamy operad Bin odpowiadajacy ogolnej operacji binarnej. Istnieje odwzorowanie operadow Bin –> Ass. Odwzorowanie operadow najprosciej zdefiniowac jako funktor miedzy odpowiednimi multikategoriami lub kategoriami).

Teraz, gdy mamy konkretny zbior X z laczna operacja to mozemy utworzyc kategorie, ktorej obiekty to X x X, X x X x X, itd. a morfizmy sa generowane przez laczna oepracje. Mamy oczywisty funktor z Ass do X.

Jezeli mamy operad C w jakiejs ustalonej kategorii z produktem (tzn. C_n sa obiektami tej kategorii) to mozemy mowic o C-algebrze: C-algebra to obiekt X wraz z morfizmami T: C_n “x” X^n —> X, gdzie “x” oznacza produkt wydzielony przez dzialanie grupy permutacji (na pierwszym skladniku z definicji operadu na drugim “componentwisely”). Byc moze cos trzeba zalozyc o kategorii dodatkowego, zeby taki “x” zawsze istnial. Morfizmy te maja byc kompatybilne ze skladaniem operacji w operadzie. UT nie sprecyzowala tego, ale najprawdopodobniej chodzi o to, ze mozna najpierw zlozyc dane operacje i wykonac T lub wykonac dwa razy T.

Pierwszy operad jaki UT zdefiniowala to “operad moduli”, oznaczany S. S_n to zbior (suma rozlaczna po wszystkich g>=0 przestrzeni moduli M_g{n+1}), gdzie n+1 w indeksie oznacza n wchodzacych dziur i jedna wychodzaca. Znow jest problem jak poprzednio: jak skladac operacje (trzeba zadac jakos strukture zespolona): wydaje mi sie, ze UT powiedziala, ze nalezy uzywac kolnierzykow. (W tym przykladzie lepiej powiedziec, ze S_n to nie zbiory tylko obiekty kategorii Top.) Nie jest dla mnie jasne, jak zdefiniowac dzialanie grupy permutacji na tym operadzie. Widze dwie mozliwosci: albo wejsciowe dziury sa jakos numerowane (bardziej sie przychylam) albo dolepiane sa jakies permutujace rurki, ale tu trzebaby jakos wybrac strukture zespolona na tych rurkach i raczej w ten sposob dastalibysmy grupe warkoczy.

Nastepny przyklad to (wedlug Baeza jest to pierwszy wymyslony przez ludzkosc operad, by Peter May) “operad malego d-dysku” (little d-disk operad), oznaczany D^d. Ustalmy dysk jednostkowy B. D^d_n to zbior takich wlozen Phi skonczonej ilosci ponumerowanych rozlacznych kopii B w B, ktore obciete do konkretnego komponentu sa okreslone przez srodek obrazu i jego promien. Skladanie jest okreslone w jedyny mozliwy sposob (wymagajacy narysowania obrazka, patrz artykul Baeza, choc on akurat ma wersje “little d-cube operad”)

Potem byly przyklady algebr nad operadami. Pierwszego nie zrozumialem: mamy dowolny operad C w jakiejs kategorii, ktorej obiekty to zbiory. Jezeli mamy jakis obiekt Z tej kategorii z wyroznionym punktem *, to mozemy utworzyc wolna C-algebre generowana przez Z: C(Z) = |_| C_n “x” Z^n/~. Smar rozlaczna jest po wszystkich n, “x” oznacza produkt wydzielony przez grupe permutacji. Relacji nie zrozumialem, nie wiem po co jest wyrozniony punkt. W dodatku potem napisalem, ze dla operada modul wyrozniony punkt to czapka in M_0,1 pomiedzy zbiorem pustym a jednym okregiem. Nic nie rozumiem.

Konkretniejszy przyklad jest taki: przestrzen topologiczna, ktora jest druga przestrzenia petli nad czyms z wyroznionym punktem:: X =Omega^2(Y,*). W takiej sytuacji X jest algebra nad operadem malego 2-dysku. Rzeczywiscie, X to przestrzen Map((D^2,S^1), (Y,*)). Trzeba powiedziec jak ustalony element operadu – czyli wlozenie kilku “malych” dyszczkow w “duzy” dysk – dziala na kilku takich mapach: dziala mianowicie tak, ze zwraca mape z “duzego” dysku do Y, ktora wszystko co nie jest w “malych” dyszczkach pakuje w wyrozniony punkt a na malych dyszczkach zachowuje sie tak jak mapy na ktorych dzialamy. Rysunek czyni cuda – patrz strona Baeza.

Twierdzenie Getzbera z poprzedniego wykladu mozna wyslowic w jezyku operadow mniej wiecej tak: Nadanie struktury H_{*}P-algebry ustalonej zgradowanej przestrzeni wektorowej A_{*} jest rownowazne nadaniu tej przestrzeni struktury BV-algebry.

Podobnie rozszerzenie Ulriki Tillmann: Jezeli dana struktura HP algebry podnosi sie do struktury HS algebry to odpowiedni element t (patrz poprzedni wyklad) spelnia t^3[x,y]=0 dla wszystkich x,y (itd. – patrz poprzedni wyklad) (czyli “t^3 anihiluje nawias”)

(UWAGA: Piszac poprzednie notatki sadzilem, ze t to element algebry (nie napisalem tego, ale tak sadzilem: napisalem w pewnym momencie t^3 *x) – tak nie jest, t to oczywiscie morfizm tej algebry.)

Pojawil sie dowod powyzszego faktu Ulriki T.: dla przypomnienia, t to wartosc funktora na elemencie (generatorze) zerowych homologii przestrzeni moduli X powierzchni genusu 1 z jedna dziura wejsciowa i jedna dziura wyjsciowa; nawias to wartosc funktora na elemencie pierwszych homologii przestrzeni moduli Y powierzchni genusu 0 z dwoma dziurami wejsciowymi i jedna dziura wyjsciowa, generowany przez twist Dehna wokol dziury wyjsciowej. Mamy odwzorowanie H(X)xH(Y)xH(Y)xH(Y) = H(X x Y^3) –> H(Z), gdzie Z to przestrzen moduli powierzchni genusu trzy z dwoma dziurami wejsciowymi i jedna dziura wyjsciowa. Na mocy okreslenia odwzorowania X x Y^3 –> Z, interesujacy nas element H(Z) to element H_1(Z) generowany przez twist Dehna “pomiedzy” dwoma dziurami wejsciowymi a pierwszym otworem (otwor to jest to, czego torus ma jeden a sfera zero :-). Ale z drugiej strony mamy wlozenie Z –> Z’, gdzie Z’ to przestrzen moduli pow. takich jak dla Z ale z zalepionymi wejsciowymi dziurami. Jasne, ze interesujacy nas twist Dehna przechodzi przy tym odwzorowaniu na zero, a z drugiej strony, na mocy twierdzenia o stabilnosci Homera-Ivanova, na H_1 to odwzorowanie jest izomorfizmem.

Wracajac na chwile do operada malego dysku, mamy nastepujacy, klasyczny wg UT, rezultat (odwrocenie przykladu powyzej): Jezeli X jest D^d algebra a pi_0(X) jest grupa to X jest Omega^d(Y) dla pewnego Y. Nie rozumiem tego: co to znaczy, ze pi_0 jest grupa (w naturalny sposob)?

Potem pojawily sie jakies tabelki operadow, nic wtedy nie rozumialem, bardzo zaluje, bo padaly ciekawe slowa (grupa warkoczy i cos w stylu “grupa warkoczy z dziurami” – tak zapisalem, ale nei wiem o co chodzilo). W pierwszym rzedzie tabelki byl zbior C_n danego operadu (czyli w kolejnosci: D^2_n, P_n, S_n i D_n^niesk (nie wiem co to jest)), w drugim rzedzie bylo napisane pi_1(C_n) i tu byly powyzsze dziwne grupy, w trzecim wierszu byl iloraz homotopijny C_n przez dzialanie grupy permutacji (niestety nie wiem, czym jest iloraz homotopijny – bardzo chetnie bym sie dowiedzial)

Uwaga: nietrudno wyobrazic sobie, ze pi_1(D^2_n) to faktycznie jest n-ta grupa warkoczy, ktore “zaczynaja sie i koncza tak samo”.

Z tego wszystkiego mialo wynikac – a moze byla to niezalezna uwaga? – ze “istnieja modele operadow z odwzaraniami D^2 –> P –> S –> D^niesk odpowiadajacymi mapom na pi_1”. (Po wykladzie padlo pytanie jak sie konstruuje odwzorowanie S –> D^niesk wiec byc moze nie wszystko bylo oczywiste (odpowiedzi nie zrozumialem))

Mam jeszcze jakies pol strony notatek,, ale nic sensownego z nich juz nie wynika.

Pytania:

1) dzialanie grupy permutacji na operadzie S?

2) Wolna algebra nad operadem

3) Homotopy quotient

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* !Clarification update!* I didn’t mean that if you’re physicist then I don’t want to see your comments – I do even more! It’s just that I have generally bad experience with talking about physics with physicists

So here goes my post:

Yo,

I’m reading Sheldon Katz’s book “Enumerative geometry and string theory”, chapter 11. I know nothing about QFT or physics so please use polite language (for example, D-brane, string, SUSY are very impolite (unless [explained or given with reference] for dummies) while connection, Sobolev space, Poincare duality etc are honey for my ears).

SK says that we’ll be doing this QFT on (n+1) or (n+0) dimensional X, called spacetime; and that we’ll be interested in fields on X of mainly two types:

1) sections of vector bundles

2) maps from X to another manifold Y

First question: what physical meaning have fields of second type? I hope that fields of first type are old-school fields, like 4-form coming from Maxwell equations, etc. – am I right?

Second “question”: well, actually I know that unfortunately I’m not right, because SK gives following weird example: (0+1) dimensional X and fields are real-valued functions. He introduced Action Functional S(x(t)) = “here stands precisely the standard integral used for deriving Newton’s laws of motions)”. Why it’s weird for me? Because X is called “space-time” and here it isn’t space-time. Rather, on X there are fields which can be interpreted as functions from time to “real” spacetime. So it’s weird – is it normal? Should I abandon this nice idea that space-time in QFT roughly corresponds to real space-time?

Please, be polite. Thanks!

—

[ sirix ]————sirix-at-univ-szczecin-pl————-

https://sirix.wordpress.com (my maths&physics-oriented blog)

Just noticed that I again used this stupid comma before “that”. Fixed it, as well as some other typos.

If I learn something nontrivial, I’ll report here.

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It’s not that there was nothing that interested me in itself – for example, I really wanted to know why when one passes from category of chains over abelian category to category localized in quasiisomorphisms (see below) then one gets all homotopic morphisms identified. In aforementioned book there are few definitions and diagrams -“nothing is happening”, but I don’t get illuminated even after longer moment of looking at them.

Correspondingly, my knowledge of derived/triangulated categories, simplicial objects and derived functors of nonadditive functors (these were the main topics of a course) is a little bit like a knowledge of history – I know essential facts, I even see some causal connections between them, but I miss something – a spirit of mathematics?.

I’d love to see some comments on how you are/were learning homological algebra. Especially, I’d *love* to know how (for example) Terrence Tao learned HA (as everybody probably knows, there is non-zero chance of meeting him around wordpress.com, so who knows :-). My General Theory of Geniuses(;-) predicts that he (or maybe He?:-) spent as much time as I did (or less, of course) on the given topic, but his unconsciousness checked all the details and communicated them to his consciousness, one day or another (actually, my GTG says that my unconsiousness also checked all the details, but it saved them for itself, damn bastard.)

Below follows a survey of what I’ve learned on this HA course, part I: derived categories, with some additional thoughts on the subject.* I believe you may find it worth reading if you already know some basics of HA (nothing more than classical derived **functors**) and you’d like to read some informal introduction, to have some view on a matter before studying it for real. *Comments are appreciated.

(*)

Derived categories. Is there any motivation for them? Well, for *learning them* there definitely is some – gossip I heard says that physicists are using them in non-trivial way. Actually,Homological Mirror Symmmetry is formulated using derived categories. I don’t know any details. I’d love to know.

(And since this thing is “Fightin’ the resistance of matter” (*and* I am listening to some old-style-grungy depressing music) I’d like to use this opportunity to express my bitter sorrow that I have to learn mathematics this way. I envy everybody “on the front of research” that they have so much fun out of doing mathematics, and I learn things that I know they’re using but don’t know actually why, so they seem artificial and not funny……)

Yes… So let’s pretend I know the motivation :-) Yay! I suppose you know few things about classical derived functors. You take an additive functor F: A –> B (A and B are abelian categories) which is, say, left exact, which means that for every exact sequence 0 -> X -> Y -> Z -> 0 you get exact sequence 0 -> F(X)-> F(Y)-> F(Z). In general the last arrow isn’t onto, so you want to measure how strongly it’s not onto. And you invent derived functors R^{i}F which have a property that they fit into long exact sequence:

0 -> F(X)-> F(Y)-> F(Z)->

-> R^{1}F(X)-> R ^{1} F(Y)-> R ^{1} F(X)->

-> R^{2}F(X)-> R^{2}F(X)-> R^{2}F(X)->…

When we’ll have derived categories everything’ll be more elegant: All the information about derived functors will be encoded in only one functor DF: DA -> DB (DK denotes derived category of K). Greatest change in conciseness will be seen perhaps in derived functor of composition. You might remember that to compute R^{*}(F∘G) one needs to write down a spectral sequence(AAA!!! NOOT A SPEECTRAAL SEEEEQUEEENCE!!!!). After introducing derived categories we’ll have very elegant D(F∘G)= DF ∘ DG. This is something I still need to understand – I don’t really know where the spectral sequence has gone (but I’m happy he went away, he may stay there if he wishes to). However, this statement’s elegance is quite convincing that derived category is right thing to do.

From more philosophical point of view – what’s the most important derived functor? No, it’s not some ?%*& ext or tor. It’s *cohomology*. It might be that you didn’t know that, so i’ll write explicetely: Cohomology of a topological space X can be realised as a derived functor of a Global Section functor Γ:*Sheaves*(X)->*Ab *which takes a sheaf on X and gives back an abelian group of all global section of this sheaf. For example, if you want to get de Rham cohomology of a manifold you simply take sheaf which on every connected open subset takes value **R **(real numbers)- it’s obvious from definitions that derived functors of Γ evaluated on this sheaf are de Rham cohomology groups, because de Rham complex is (by Poincare lemma) a resolution of sheaf **R**(and this resolution is suitable for computing derived functor, though it’s not injective – this isn’t very hard).

Taking other sheaves and evaluating derived functors of Γ on them leads to some different cohomology theories, which can measure some other things than ordinary cohomology theory does. So, taking derived category of *Sheaves*(X) and studying DΓ could be seen as “studying all cohomology theories in one time” which sounds like a cool thing to do.

Motivation for a following definition comes also from the desire to identify any object of A with its resolution. This in turn might be motivated by a following observation: if you want to compute Tor^{i}(X,X), value of classical derived functors of functor _⊗X on object X, you take injective *resolution* of X, tensor it with X and compute homologies.(BTW, at least when resolution of X is finite you could instead take resolution of X and tensor it with *resolution of X*, form a Tot complex and compute homologies – this also suggests that resolution and an object are very similar from HA point of view). So, if you somehow identified object with its resolution you could say simply “to compute value of derived functor on an object just compute original functor on an object”:-). It’s a little bit tricky, but basically this is what we’ll end with.

There are also some other motivations listed in Gelfand&Manin.

Ok, let’s fix an abelian category A. Let ChA be a category of (cohomological) chain complexes over A (perhaps bounded or bounded from right or bounded from left – constructions will not change at all, outcome will be called D^{b} od D^{–} or D^{+}). Motivated by all the above we want to identify two objects in ChA if there is an quasiisomorphism between them (quasiisomorphism is a morphism which induces isomorphism on homology). Accordingly, we make a following definition.

Derived category of A is a pair (DA, Θ), where DA is an additive category and Θ: ChA –> DA is a functor that carries all quasiismorphisms to isomorphisms, such that this pair is universal for this property (if there is other such pair (B,Ψ) than there exists *unique* functor DA–>B such that suitable diagram commutes)

There exists a very simple proof of existence. Let DA be a category whose objects are the same objects as these of ChA (that is, Ob(DA)= OB(ChA)). Set of morphisms between two objects X and Y in DA, Mor_{DA}(X,Y), is a set of paths between X and Y in a directed graph whose vertices are objects of A, and whose edges are

i) morphisms in A

ii) edges x_{s}, for every quasiisomorphism s: K->L, that go from vertex K to vertex L.

with a following relation: two paths in this graph are equivalent if you can obtain one from the other by a series of basic operations:

i) exchanging any edge f∘g with path consisting of edges g and f

ii) exchanging paths of form s∘x_{s} with id.

Composition of morhism is defined by conjunction of paths in the above graph.

Above construction is called localisation of ChA with respect to quasiisomorphisms.

This is it. There is obvious functor Θ from Ch(A) to DA. Pair (DA, Θ) has a suitable universal property (no, I *didn**‘t* check all the details but it stands to reason ;-). But, DA just constructed is poor to work with – it’d very hard to decide even whether given path represents a trivial morphism. So we should work some more.

Definition: Suppose we’re given category B and class of morphisms C ⊆ MorB. We say that this class C is localizing in B iff

**i)** id_{X}∊C for ever X∊ObB

**ii)** if α, β ∊ C then α∘β ∊ C, if it makes sense

**iii)** for every α∊C and f ∊ MorB, α: X->Z, f: Y->Z there exist β∊C, g∊MorB, β: W->Y, g: W->X, such that suitable diagram commutes (draw it)

**iv)** for every α∊C and f ∊ MorB, α: X<-Z, f: Y<-Z there exist β∊C, g∊ MorB , β: W<-Y, g: W<-X, such that suitable diagram commutes (draw it)

**v)**if α∘f=α∘g for some α∊C, f,g∊MorB then there exists β∊C such that f∘β=g∘β

Proposition: If C is localizing in B then localisation of B with respect to C has a simple description. Every morphism between to objects X and Y in this localisation can be represented by a graph (denoted later (1))

X <-α- W -f-> Y

for some α∊C, f∊MorB. Moreover, two such diagrams

X <-α- W -f-> Y

X <-α’- W’-f’-> Y

represent the same morphism iff there exist β:Z->W, g:Z->W’, β∊C, g∊MorC, such that suitable diagram commutes.

Word about a proof: from a property **iv)**it follows that indeed every morphism in a localisation has suitable representation. Proof of the latter part is carried out in Gelfand&Manin in a following way: they define category B’ to be a category with objects same as in B and morphisms given by diagrams of form (1). They define suitably composition of such diagrams ((1) can be regarded as a fraction of form f/α and composition is realized by adding such fractions) and use all the properties **i)**–**v)**to prove that everything is well defined *and *that B’ has a suitable universal property. So indeed this B’ must be itself a localisation.

However, class of quasiismorphisms is not localizing in ChA. Cha!:-) We have to build an intermediate category out of ChA – so called homotopy category of A, denoted KA. Its objects are same as these of ChA, and

Mor_{KA}(X,Y):= Mor_{ChA}(X,Y)/{morphisms which are chain homotopic to 0}

One painfully checks that quasiisomorphisms are localizing in KA, so KA localized in quasiisomorphisms has nice description guaranteed by above proposition.

*Now follows actually something interesting.* We don’t know whether localized KA is equivalent do DA. We have a functor from DA which is identity on objects and which is obviously surjective on morphisms (it’s really obvious). We don’t know whether it’s injective on morphisms. That is, we don’t know if the following is true:

*Fact: If f,g: X->Y are homotopic in ChA then their images are equal in DA.*

It turns out that it is true (G&M, III.4.3), but for me it’s kind of a miracle – DA is a localisation in quasiisomorphisms. The construction doesn’t say a word about homotopic morphisms. Proof occupies one page in G&M (so it’s about 0.9 pages to long for me to really understand :-).**If anybody has any philosophical remarks on this I’ll greatly appreciate if he’ll share them!**

Now, when we have our shiny derived category, we move slowly to derived functors. Before, let’s observe that we have a functor from our original category A to DA (realize object of A as a chain complex concentrated in dimension 0, send it to DA). I claim that this functor is an equivalence of categories A and full subcategory of DA consisting of chain complexes with nonzero homology only in dimension 0 (called “category of H^{0} complexes”).(This is one of very few things I actually checked so here you are :-). We have to check two things, that our functor gives a bijection Mor_{A}(X,Y)–> Mor_{DA}(X,Y) for every pair of objects X,Y ∊ A, and that every H^{0 } complex of DA is isomorphic to one concentrated in dimension 0.

Both are easy: for the latter one we need to take H^{0} complex Z and build a diagram

Z <-α- V -β-> H^{0}(Z)

with both α and β quasiisomorphisms. For V^{i} one can take Z^{i}^{} for i<0, ker(d^{0}) for i=0 and 0 for i>0. For the first one let’s check surjectivity. So we have a diagram

X <-α- Z -f-> Y

and we need to check that it’s equivalent to a diagram

X <-id- X -g-> Y

(these diagrams are images of morphisms from A). For g one takes what f∘α^{-1} generates on H^{0}, and for suitable β:V->Z, h:V->X one takes V as above, with obvious morphisms. Injectivity is pretty obvious,’cause no matter what representation for morphism you take, you always get the same thing on homologies.

For chain complex X^{*}(or rather cocomplex), or its image in KA or DA, we define shifted complex X[i]^{n}:= X[i+n]. For ~2 years I couldn’t remember this definition – there is always a problem “in which direction is it moving?”. But 2 days ago I finally managed to invent a suitable visualisation: I see integer points on a line which *never move.*Under this line I see an original complex and then I imagine that this complex moves violently in a direction opposite to the order of integer numbers by i places.

We turn to derived functors, for some simple case for now: suppose we’d like to compute classical Ext^{i}_{A}(X,Y) for some X,Y∊A. You can do it in a standard way, but there is also following:

Fact: Ext^{i}_{A}(X,Y) ≅ Hom_{DA}(X[k], Y[k+i])

(As I said in a motivation part, we have a functor Hom(_, Y): A -> A and we’d like to define DHom(_,Y): DA -> DA that would encode all the information about classical Ext^{i}(X,Y). Additionally it should be defined by pointwise use of normal Hom. The fact shows that we should have a chance to do something in this spirit)

Notice that left side doesn’t depend on k. However, it’s easy to see that right side doesn’t as well (up to natural isomorphism induced by “shifting functor”). This is very elegant: for example, one can now define product in Ext in a very simple manner:

Ext^{i}_{A}(X,Y) x Ext^{j}_{A}(X,Y)-> Ext^{i}^{+}^{j}_{A}(X,Y)

is realized by a composition of morphisms in DA. Things like long exact sequences associated to short exact sequence 0 -> X -> Y -> Z ->0 also follow from general abstract nonsense.(Namely, in more general setting of triangulated category T we always have that so called distinguished triangle – and above short exact sequence is an example of distinguished triangle in derived category – gives rise to apropriate long exact Hom_{T}_{}sequence. I hope to write few words on that in a next article.)

Word about a proof of a Fact: It’s easy to see that every element of classic Ext^{i}_{A}(X,Y) gives rise to an element of Hom_{DA}(X[k],Y[k+i]), for clarity let’s put k=0. Indeed, element of Ext group can be represented as an extension

0-> Y=V^{-i }-> V^{-i+1 }->…-> V^{0}-> V^{1}=X -> 0

This extension clearly defines element of Hom_{DA}(X,Y[i]), that is a diagram

X <-d^{-1}– V -> Y

where V is a 0 -> V^{-i}->…-> V^{0}-> 0. G&M prove that this assignment is an isomorphism.

Let’s focus for a moment on derived functor of a general additive left-exact functor F: A -> B. We’d like to obtain a DF: D^{+}A -> D^{+}B. Naively, we could take object of DA, represent it as a chain complex, and act on this chain complex with F. This is bad idea, because F in general doesn’t preserve quasiismorphisms – if it does, then I believe it’s exact (I’m sure there is an implication exactness => preserves quasiisomorphisms)(although, notice that F *defines* a functor on homotopy category KA, because it is additive, so it preserves chain homotopy). So we woudn’t end up with a functor, as functor must preserve isomorphisms (and quasiisomorphisms become isomorphisms in derived category).

What we’ll do instead is motivated by classical derived functors.

We say that class R⊆ObA is adapted to the functor F if following conditions hold:

i) R is closed with respect to taking finite sums

ii) Every object of A might be embedded in some object of R

iii) F takes bounded from the left, exact chain complexes consisting of objects of R into exact sequences (for right exact F one would have to take here complexes bounded from the right).

Now, to make long story short, It’s pretty simple to see that F defines a functor G from K^{+}(R) localized in quasiisomorphisms to DB, where K^{+}(C) is a full subcategory of K^{+}A consisting of these chain complexes whose all objects are from C. One proves that actually K^{+}(C) localized in quasiisomorphisms is equivalent to DA, so we can choose an equivalence Ω: D^{+}A ->”K^{+}(C) localized in quasiisomorphisms” and say that DF is a composition G∘Ω.

This definitions depends obviously on a choice of adapted class C and of equivalence Ω. That’s why one for a second forgets what we’ve done and adopts following definition for a derived functor:

* Definition: Given F: A->B as usually, derived functor is a pair consisting of*

*1) a functor DF: D ^{+}A -> D^{+}B which takes exact triples into exact triples*

* 2) a natural transformation ε _{F} of obvious functors*

* K ^{+}A -> K^{+}B -> DB and K^{+}A -> DA -> DB*

* that is universal with these properties (it there is some other pair (G, ε) then there is a unique transformation of functors*

* K ^{+}A —-> DA —DF—> DB and K^{+}A —-> DA —G—> DB*

* such that suitable diagram commutes.)*

Remark about 1): DA, and D^{+}A, are not abelian, but we call a triple exact if it comes from an exact triple in ChA (or Ch^{+}A).

It takes some trouble to prove that what we’ve constructed above is a derived functor, after equipping it with suitable ε_{F}.(5 pages in G&M. I don’t even guess what’s trivial and what’s not.)

But, modulo some checking, we’ve done what we wanted to :-)

(There’s a famous exercise in Lang’s Algebra, in the chapter concerning homological algebra:”Take any Homological Algebra notebook and prove all the theorems by yourself”. There is also a famous footnote in polish translation:”We advise a reader to omit this exercise at a first reading”)

I hope to read rest of G&M’s chapter on derived categories, using the “homological momentum” that the exam gave me. Maybe then I’ll write some (shorter) notes here.

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