mm-1043 === Subject: Re: A Beautiful Differential Equation > HereÕs a differential equation that seems very beautiful to me: > fÕ(f*(Exp(kt)f(x))) = s(t)fÕ(x) I didnÕt expect a quick answer to so difficult a problem but Dr. Michael Ulm has solved the problem and posted an exquisite reply! See: Eugene Shubert http://www.everythingimportant.org === Subject: CohenÕs work in AC constructed a collection for which AC is required for any of its choice set. ---Nam === Subject: Re: CohenÕs work in AC >Hi all, >constructed a collection for which AC is required for any of its > choice set. It is fairly well described on pages 205--207 of JechÕs Set Theory (Math Reviews *80a:03062* ) KP -- E-MAIL: K.P.Hart@EWI.TUDelft.NL PAPER: Faculty EWI PHONE: +31-15-2784572 TU Delft FAX: +31-15-2786178 Postbus 5031 URL: http://aw.twi.tudelft.nl/~hart 2600 GA Delft the Netherlands === Subject: Re: Distances in a dense subset of R^2 Originator: israel@math.ubc.ca (Robert Israel) > Is there a dense subset S of R^2 such that for every pair of points > p,q in S the Euclideean distance d(p,q) is a rational number ? This problem, attributed to Ulam, is Problem 10 in _Old and New Unsolved Problems in Plane Geometry and Number Theory_ by Victor Klee and Stan Wagon, ISBN 0-88385-315-9. I donÕt know the current status of this problem. You could try searching for recent publications that cite the works cited by Klee & Wagon. === Subject: Re: Fourier Integral Operators > I may be applying Fourier Integral Operators to seismic > ray theory (and probably the transport equation portion) > for my Ph.D. thesis and would like to know of any > good online or printed recent or standard references > in the subject and I guess also in its subset > pseudodifferential operators. (warning: IÕm really no expert but nobody seems to answer, so...) - for fourier integral operators, a classic is: L.Hormander, The analysis of linear partial differential operators, vol. IV, Springer, 1985. - for pseudodiff operators, there are some lecture note by R.Melrose at: Anyway itÕs a very active area and a lot has been done since Hormander I think... Hope this helps. -- thomas. === Subject: mackey-functors does anybody know a paper where I can find a proof that K- and L-theory are mackey-functors? -- Marcus Meyer === Subject: Re: mackey-functors > does anybody know a paper where I can find a proof that K- and L-theory are > mackey-functors? Pray tell, what is a mackey-functor? Is it named after George W. Mackey? -- Chris Henrich To succeed it is not enough to be stupid: you must also be well-mannered.Õ -Voltaire === Subject: Motivic vs etale cohomology? Hi. Can anyone explain the relationship between motivic and etale cohomology? Does motivic cohomology generalize etale cohomology? Is there an advantage of motivic cohomology - is it a more complete theory in some sense? Perhaps a better connection with other generalized cohomology theories? -mb === Subject: Re: Motivic vs etale cohomology? > Hi. > Can anyone explain the relationship between motivic and etale cohomology? > Does motivic cohomology generalize etale cohomology? > Is there an advantage of motivic cohomology - is it a more complete theory > in some sense? Perhaps a better connection with other generalized cohomology > theories? The idea of both etale cohomology and motivic cohomology is to have a tool for algebraic varieties over an arbitrary field that works the way singular cohomology works for a nice topological space. For coefficients in Z/nZ, etale cohomology is the way to go. For coefficients in Z, itÕs motivic cohomology. The relationship between the two is provided by the universal coefficient theorem, as in topology. Or another way to say it is that motivic cohomology can accept any coefficient group, and for Z/nZ, it agrees with etale cohomology. === Subject: Re: Motivic vs etale cohomology? Originator: israel@math.ubc.ca (Robert Israel) >> Can anyone explain the relationship between motivic and etale cohomology? >> Does motivic cohomology generalize etale cohomology? ... >The idea of both etale cohomology and motivic cohomology is to have a >tool for algebraic varieties over an arbitrary field that works the >way singular cohomology works for a nice topological space. >For coefficients in Z/nZ, etale cohomology is the way to go. For >coefficients in Z, itÕs motivic cohomology. The relationship between >the two is provided by the universal coefficient theorem, as in >topology. Or another way to say it is that motivic cohomology can >accept any coefficient group, and for Z/nZ, it agrees with etale >cohomology. I have a vague memory (and perhaps the experts can help clearing it up) that the etale cohomology became popular because that was what Deligne used in his proof of the Weil conjectures. Grothendieck had hoped (?) for some other cohomologies, like crystalline cohomology and such, which proved to be both unworkable and unnecessary for that proof in question one was opting for. Then the idea is that one should have a cohomology over Z that could be determined if the p-adic localizations. So motivic cohomology was put in there (by Deligne?) for that purposes, in order to at least make that idea expressible. But when I was interested in these things, it was still a conjecture to go from the p-adic picture to the motivic one over Z. -- My guess it still is, because to prove it in full one would have proved things like the Generalized Riemann Hypothesis and stuff I do not recall. :-) * Email: Hans Aberg * Home Page: === Subject: Moufang iff conservative? Can anyone prove my conjecture that loops are Moufang loops if and only if they have the Frobenius conservation property? Moufang loops are Cayley multiplication tables for unsigned sets, with a 1 , left and right division, and the Moufang properties (zx.yz=z.xy.z and two equivalents). Conservation is Det[A] Det[B] = +- Det[AB] (Det has to be interpreted as a symmetric difference +-, because subtraction is undefined for Moufang loops). Frobenius showed that all groups are conservative; Octonions are Moufang and conservative. The significance of this is that the reduction of a Moufang Loop to an algebra, by defining some form of negation and subtraction, always creates a division algebra. I call such an algebra a hoop. Hoops are developed and demonstrated in http://library.wolfram.com/infocenter/Mathsource/4894/ Roger Beresford. Facts do not cease to exist because they are ignored (A. Huxley.) === Subject: Re: Number of minima of a function described by a feed-forward neural network > Let N_in be the number of inputs, N_hid, the number of neurons in the > hidden layer with sigmoidal activation functions, and a linear output > neuron. The inputs are all finite, real numbers. > > How many proper minima can this function have? If the minimum has > all second derivatives positive, I refer to it as a proper minimum. > > I think I have some results with strong and weak conditions, which > are somewhat surprising to me. Not being a mathematician, I have not > bothered to prove them. I would first like to know if someone has a > good or a complete answer to my question. > > > A. Bulsari > Have you seen this paper: > @inproceedings{auer96, > author = {P. Auer and M. Herbster and M. K. Warmuth}, > address = {Cambridge, MA}, > booktitle = {Advances in Neural Information Processing Systems}, > editor = {D. Touretzky and M. Mozer and M. Hasselmo}, > pages = {316--322}, > publisher = {The {MIT} Press}, > title = {Exponentially many local minima for single neurons}, > volume = {8}, > year = {1996} > } > Marcus. No, I have not. A single neuron does not produce a single local minimum in the function. The authors are most likely referring to the number of local minima in an error cost function, which is a very different issue. A. Bulsari === Subject: This WeekÕs Finds in Mathematical Physics (Week 197) Originator: baez@math-cl-n01.math.ucr.edu (John Baez) Also available at http://math.ucr.edu/home/baez/week197 This WeekÕs Finds in Mathematical Physics - Week 197 John Baez IÕve been away from This WeekÕs Finds for a long time, so I have a lot to talk about... so much that I scarcely know where to begin! In June I went to a big general relativity conference at Penn State, and I have a lot to say about that, but at the end of July I went to two conferences in Lisbon, and I want to talk about those a bit now. One was a workshop on categorification and higher-order geometry. This was run by Roger Picken and Marco Mackaay, and it brought together a bunch of people interested in how n-categories are affecting our notions of geometry. If youÕre interested in this, you might enjoy looking at the talk titles here: 1) Workshop on categorification and higher-order geometry, The other was the Young ResearcherÕs Symposium, a section of the International Congress of Mathematical Physics. This symposium allows old geezers to pass on their accumulated wisdom to young researchers before they go senile and forget it all. The youngsters also give talks, but I was invited as one of the old geezers. ItÕs a bit scary! Anyway, at these conferences I learned some cool stuff about elliptic cohomology from Stephan Stolz, and also some cool stuff about Monstrous Moonshine from Terry Gannon. It turns out theyÕre more related than I realized - and the relation involves string theory! I always love it when two things IÕm studying turn out to be related. So, IÕd like to tell you about this stuff... before I forget it. I gave a very sketchy introduction to elliptic cohomology in week149 and week150. One reason IÕm interested in this subject is that it seems to be a categorified version of something topologists are already fond of: K-theory. In K-theory, you study a space by looking at all the vector bundles over this space. By trying to categorify the concept of vector space, Kapranov and Voevodsky were led to the concept of 2-vector space, which is a category that acts sort of like a vector space. You can think of elliptic cohomology as a souped-up version of K-theory where you study a space by looking at all the 2-vector bundles on it! IÕll warn you right away, this isnÕt how *most* people think about elliptic cohomology - this is a fairly new approach due to Nils Baas, Bjorn Dundas and John Rognes. Most people think of elliptic cohomology as being related to string theory. But the two viewpoints seem to be compatible. HereÕs why: if you have a connection on a vector bundle, it gives a way to parallel transport a vector along a curve. People use this around in a gauge field - which is just physics talk for a connection. So now letÕs imagine you categorified this whole story. If you had a connection on a 2-vector bundle, and you believe that categorification increases the dimensions of things by one - which it often does - you might hope that this connection would tell you how to do parallel transport over a *2d* surface! And this in turn might tell you how *strings* change state when you move them around. Well, nobody has worked out all the details yet, but something like this seems to be going on... and I want to know what it is! IÕd like to explain what Stephan Stolz told me about this. I have to warn you, though: this stuff applies to a *new improved version* of elliptic cohomology, which became popular after the one I was talking about in previous Weeks. Some of the old stuff I said no longer applies to this new version. To minimize confusion, people call this new version the theory of topological modular forms. So, what is this thing? First of all, itÕs a generalized cohomology theory. Hmm. To make sure you understand that sentence, I need to give the worldÕs quickest course on generalized cohomology theories. For a more leisurely introduction see week149. Here goes: A spectrum is an infinite list of spaces E(n) where n ranges over all integers, such that each space in the list is the space of loops in the next space on the list. Given any space X, we can define the generalized cohomology groups of X to be h^n(X) = [X,E(n)] where [X,E(n)] is the set of all homotopy classes of maps from abelian groups. If you know about the good old familar ordinary cohomology groups H^n(X) of a space X, youÕll be pleased to know that these are an example of a generalized cohomology theory. YouÕll also be happy to know that lots of the basic theorems about ordinary cohomology theory hold for these generalized ones. The main one that *doesnÕt* hold is the one that says: H^n(point) = Z if n = 0 0 otherwise For a generalized cohomology theory, the cohomology of a point can be more interesting! In particular, if E(n) is something called a ring spectrum, the groups h^n(point) will form a graded ring. This happens in a lot of interesting examples. Okay, now youÕre an expert on generalized cohomology theories. As I said, the theory of topological modular forms is one of these things. So, to completely describe it, I just need to give you an infinite list of spaces tmf(n) forming a spectrum. Then for any space X we can define a list of abelian groups tmf^n(X) = [X,tmf(n)] and weÕre off and running. By the way, donÕt be freaked out that now IÕm using the same name for the spectrum and the generalized cohomology theory it gives - people do this a lot. Unfortunately, at present itÕs a lot of work to define these spaces tmf(n). People donÕt understand them very well, so they construct them by hand in a complicated way that poor folks like me have no chance of understanding. Fortunately, Stephan Stolz told me what people secretly think these spaces must be! Nobody has proved this yet or even made it into a precise conjecture, but itÕs so audacious - and it would explain so much - that I canÕt resist saying it: tmf(n) is the space of supersymmetric conformal field theories of central charge -n. ThereÕs a lot of fine print here that IÕm leaving out, and some that nobody even knows... but a supersymmetric conformal field theory is sort of roughly like a superstring vacuum: a world in which superstrings can romp and play. This is oversimplified and it will piss off string theorists, but never mind, right now IÕm just trying to make a very crude point: the theory of topological modular forms is sort of like studying a space by mapping it into the space of all possible superstring vacua! Zounds! Before we blow our minds contemplating the space of all superstring vacua, let me back off a bit and try to explain what any of this has to do with modular forms. Modular forms are a famous old concept from complex analysis. These days people do complex analysis not just on the complex plane but on more general Riemann surfaces, and this turns out to be crucial for understanding modular forms. We also use these surfaces to describe the worldsheets traced out in spacetime by the motion of a strings. So, it should not come as a shock that modular forms should show up in a generalized cohomology theory involving strings! But IÕd like to make this connection considerably more precise. To do this, IÕll reveal that the spectrum for topological modular form theory is a ring spectrum, and the abelian groups tmf^n(point) fit together in a very famous graded ring: itÕs the ring of MODULAR FORMS! Well, at least after we tensor it with the complex numbers, it is... but before we worry about that, I should say what modular forms are. IÕll start with a quick but unenlightening definition. First, a modular function of weight n is an analytic function on the upper half of the complex plane, say f: H -> C where H is the upper half-plane, which transforms as follows: f((az+b)/(cz+d)) = (cz+d)^n f(z) for all matrices of integers (a b) (c d) having determinant 1. Then, we say a modular function is a modular form if it doesnÕt blow up as you march up the upper half-plane to the point at infinity. There are only nonzero modular forms when the weight is a natural number. ItÕs easy to see that these form a graded ring: if you add two modular forms of weight n you get another one of weight n, and if you multiply two modular forms of weights n and nÕ, you get one of weight n+nÕ. This graded ring is the same as what you get by tensoring the graded ring tmf^n(point) by the complex numbers! In case youÕre wondering what this tensoring with the complex numbers business is all about: itÕs mainly just a way of killing off elements of a group that become zero when you multiply them by some integer. If youÕre a topologist these so-called torsion elements are really interesting. They make topological modular forms a lot more subtle than traditional modular forms as defined above. Topologists really go into raptures over torsion! But if youÕre a lowly mathematical physicist such as myself, struggling to understand even a little of whatÕs going on, you go ahead and kill the torsion by tensoring with C. And, IÕm pretty sure the new topological modular form theory is the same as the old version of elliptic cohomology except for stuff involving torsion. So, ignoring these subtleties, letÕs just say that tmf is a generalization of cohomology theory in which the integers get replaced by the modular forms when we calculate the cohomology of a point... where modular forms are some weird functions that show up in complex analysis! But what does this have to do with the idea that tmf is related to the space of all string theories? To understand this, we need a better understanding of modular forms: we need to see how theyÕre related to elliptic curves, and we need to see how these are related to conformal field theory. Then things will start to make sense. To do this, letÕs start with the worldÕs quickest course on know elliptic curves. For a more leisurely introduction, see week13, week125, and week126. An elliptic curve is what you get when you take the complex plane and mod out by a lattice, like this: * * * * * 0* * * * * * Topologically you get a torus, of course. But it also has the structure of an abelian group, coming from addition in the complex plane. It also has the structure of a compact Riemann surface - that is, a compact 1-dimensional complex manifold. So, a more precise definition of an elliptic curve is that itÕs an abelian group in the category of compact Riemann surfaces. With this definition, it turns out that we can rotate or dilate our lattice without changing the elliptic curve we get from it. More precisely, we get an *isomorphic* elliptic curve. So, any elliptic curve is isomorphic to one coming from a lattice like this: z z + 1 * * * 0 1 * * * * * * where z is in the upper half-plane. But, lots of different choices of z give the same elliptic curve! For example, we can replace z by z + 1 and still get the same lattice, hence the same elliptic curve. We can also replace z by -1/z. This turns the short squat right-leaning parallelogram in the above picture into a tall skinny left-leaning one - but after rotating and dilating this, we get back the parallelogram we started with, so we get the same elliptic curve. In fact, though itÕs not obvious from *this* way of thinking about the problem, itÕs easy to show that all the different choices of z that give the same elliptic curve are related by these two transformations. Now, the group of transformations of the upper half-plane generated by z |-> z + 1 and z |-> -1/z is precisely the group of all transformations z |-> (az+b)/(cz+d) where the matrix (a b) (c d) has determinant 1. This group of such transformations is called PSL(2,Z). So, the space of all isomorphism classes of elliptic curves is H/PSL(2,Z) where again H is the upper half-plane. Folks call this space the moduli space of elliptic curves. ItÕs a Riemann surface, and I drew a picture of it in week125. Okay, now youÕre an expert on elliptic curves. A while back, I defined a modular function of weight n to be an analytic function on the upper half-plane f: H -> C such that f((az+b)/(cz+d)) = (cz+d)^n f(z) for all transformations in PSL(2,Z). Now we can see what this equation really means. When n = 0, it just says f is *invariant* under PSL(2,Z), so it becomes a function on H/PSL(2,Z). Thus, modular functions of weight 0 are just analytic functions on the moduli space of elliptic curves! So, if youÕre trying to explain modular functions to your friends, just tell them theyÕre functions that depend on the shape of a doughnut - what could be simpler than that? Of course shape needs to be interpreted in a subtle way to make this true. Similarly, a modular function is a modular form if it doesnÕt blow up when Im(z) -> +infinity, which means that it doesnÕt blow up when your doughnut gets really long and skinny, more like a circle than an honest doughnut. The circle is like the ultimate low-calorie doughnut. In the language of string theory, where the surface of your doughnut is the worldsheet of a string, the Of course, when n is nonzero, modular forms of weight n arenÕt really invariant under PSL(2,Z): theyÕre only invariant up to a phase. I put this physics jargon in quotes because the fudge factor (cz+d)^n isnÕt really a unit complex number. But the moral principle is the same - and in string theory, this fudge factor really *does* come from a quantum mechanical phase ambiguity, called the conformal anomaly. (To make this up to a phase idea precise, we can think of modular forms of weight n as sections of some *line bundle* on the *moduli stack* of elliptic curves... but I explained this already in week125, and I donÕt want to say more about it now.) Now that we understand modular forms a bit better, we can begin to vaguely see why tmf^n(point) tensor C is the space of modular forms of weight n. HereÕs how. If you know a little about the path-integral approach to quantum field theory, youÕll know that one of the basic things you compute in any quantum field theory is a number called the partition function. YouÕll also know that this number is often infinite, or defined only up to some ambiguities... thatÕs why quantum field theory is tough. So, given that a conformal field theory is something like a string theory, and given that the worldsheet of a string is a Riemann surface, you shouldnÕt be surprised that given any compact Riemann surface and any conformal field theory we can try to compute a number called the partition function. Nor should you be surprised that this number is sometimes afßicted with ambiguities! So, restricting attention to the case where our Riemann surface is an elliptic curve, you should not be surprised that the partition function of any conformal field theory is a MODULAR FORM! If this modular form has weight 0, the partition function is an honest-to-goodness function on the moduli space of elliptic curves: for any elliptic curve the partition function is an actual number. But if the modular form has nonzero weight, the partition function is afßicted with phase ambiguities - where phase is in quotes for the same reason as before. In particular, if the partition function is a modular form of weight n, we say our conformal field theory has central charge -n. The central charge just tells us how the phase ambiguity works... though some jerk put in a minus sign to confuse us. Now think what this implies! Remember that tmf(n) is space of conformal field theories with central charge -n. Since the partition function of any such thing is a modular form of weight n, we get a map Z: tmf(n) -> {modular forms of weight n} This is a step towards seeing that tmf^n(point) tensor C = {modular forms of weight n} since at least thereÕs a relation between the two sides! To go further, use the definition of generalized cohomology: tmf^n(point) = [point,tmf(n)] and note that [point,tmf(n)] is the set of *connected components* of the space of supersymmetric conformal field theories of central charge -n. So, weÕd like to see why this is an abelian group, and why tensoring it with the complex numbers gives the space of modular forms of weight n. To see this, weÕd just need to show four amazing things: A) The partition function doesnÕt change as we trace out a continuous path in the space of conformal field theories of central charge -n. Thus, the partition function defines a map Z: [point,tmf(n)] -> {modular forms of weight n} B) The set of connected components of the space of conformal field theories of central charge -n forms an abelian group, and the above map is a group homomorphism. C) The kernel of the above homomorphism consists precisely of the torsion elements, so we get a 1-1 homomorphism Z: [point,tmf(n)] tensor C -> {modular forms of weight n} D) Any modular form of weight n is a linear combination of partition functions of conformal field theories of central charge -n, so the homomorphism Z: [point,tmf(n)] tensor C -> {modular forms of weight n} is also onto. Sorry, IÕm getting a little carried away... itÕs not good to put in so much detail when youÕre explaining stuff, but I just realized that we need these four amazing things to be true, and I couldnÕt resist writing them down. Learning by teaching is great for the teacher; sometimes less so for the student. Anyway: The first amazing thing must come from index theory and how the index of a Fredholm operator doesnÕt change when we deform it continuously. It must also use the fact that the partition function weÕre talking about can be written as such an index. This only happens because weÕre considering supersymmetric theories! Stephan Stolz emphasized to me that we really need to be using N = 1/2 supersymmetric conformal field theories; I havenÕt gotten around to understanding the N = 1/2 part. The second amazing thing is not really amazing. In fact, itÕs easy to see the whole graded ring structure of modular forms coming from operations on conformal field theories. IÕll explain that in a minute. The third amazing thing is a total mystery to me. ItÕs obvious that all torsion elements must lie in the kernel of a homomorphism from a group to a vector space, but itÕs utterly mysterious why the kernel consists *precisely* of the torsion elements. The fourth amazing thing is presumably some sort of calculation: you just need to find enough conformal field theories to make sure their partition functions generate the ring of modular forms. In fact, the ring of modular forms is generated by one of weight 4 and one of weight 6: these are both Eisenstein series, which are well-understood, so we just need someone to cook up conformal field theories having these as partition functions. Does anyone reading this know how to do it? (Irrelevant digression: The previous paragraph implies that all nonzero modular forms have *even weight*. To correct for this, some people stick in a factor of 1/2 when defining the weight of a modular form. I mention this only so youÕre forewarned when you read the literature.) Okay, let me round off this story by saying a little about how you add and multiply conformal field theories... and why. A conformal field theory assigns a Hilbert space to any compact oriented 1-manifold, and a linear operator going between Hilbert spaces to any Riemann surface with boundary going between such 1-manifolds. There are a bunch of axioms it needs to satisfy, invented by Graeme Segal. I wonÕt list these here, but the category theorists among you will quiver with delight upon learning that the most important of these axioms say a conformal field theory is a symmetric monoidal functor. Anyway, itÕs easy to take direct sums and tensor products of Hilbert spaces and also operators. This gives a way of defining the direct sum and tensor product of conformal field theories. When we take the direct sum of conformal field theories their partition functions add. When we take their tensor product the partition functions multiply. So, these operations on conformal field theories correspond precisely to the graded ring structure on modular forms! To see why this graded ring structure is interesting in string theory, I should be more precise about the relation between string theory and conformal field theory. Perturbatively, string theory in a given background is described by a conformal field theory. We can use this to calculate an operator for any Riemann surface with boundary: we think of this operator as saying how the string changes state given the conformal structure on its worldsheet. When a conformal field theory plays this role we call it a string vacuum. But, not any old conformal field theory will serve as a string vacuum! It has to be one with central charge 0, in order to have a partition function without any ambiguities. If the central charge is nonzero we say thereÕs a conformal anomaly and turn up our noses in disgust. However, people often build conformal field theories with central charge 0 out of ones with nonzero central charge. The simplest ways to build new conformal field theories from old are direct sums and tensor products. So, the graded ring structure on modular forms is sort of lurking around in string theory! To learn more about elliptic cohomology and its relation to conformal field theory, you should read this paper that Stephan Stolz is in the process of writing with Peter Teichner: 2) Stephan Stolz and Peter Teichner, What is an elliptic object? http://math.ucsd.edu/~teichner/Preprints/Oxford.pdf This paper is almost 80 pages long and they arenÕt even done yet! But donÕt be scared - it has an introduction thatÕs worth looking at even if the rest is too intense for you. If even the introduction is too intense, well, back off and try again later. For how elliptic cohomology is related to 2-vector spaces, read this: 3) Nils A. Baas, Bjorn Ian Dundas and John Rognes, Two-vector bundles and forms of elliptic cohomology, available as math.AT/0306027. IÕll quote the abstract because it will be enlightening to a few of you: In this paper we define 2-vector bundles as suitable bundles of 2-vector spaces over a base space, and compare the resulting 2-K-theory with the algebraic K-theory spectrum K(V) of the 2-category of 2-vector spaces, as well as the algebraic K-theory spectrum K(ku) of the connective topological K-theory spectrum ku. We explain how K(ku) detects v_2-periodic phenomena in stable homotopy theory, and as such is a form of elliptic cohomology. One thing this means is that these folks have not gotten the theory of elliptic cohomology by studying 2-vector bundles. TheyÕve gotten a theory which detects v_2-periodic phenomena, and is thus a form of elliptic cohomology. The point is, thereÕs an infinite tower of generalized cohomology theories, called the chromatic filtration. This has ordinary cohomology tensored with the complex numbers on the 0th level, complex K-theory on the 1st level, elliptic cohomology on the 2nd level, and so on up to infinity, where something called complex cobordism theory sits grinning down at us. Theories on the nth level detect v_n-periodic phenomena. Despite the best efforts of several homotopy theorists, I still donÕt understand what this means. But, Bott periodicity for complex K-theory is the paradigm of a v_1-periodic phenomenon, so weÕre talking about some heavy- duty generalization of that! Note that Baas, Dundas and Rognes donÕt talk about connections on their 2-vector bundles. The closest thing to this that people have used in elliptic cohomology is the notion of elliptic object, invented by Graeme Segal and improved by Stolz and Teichner. An elliptic object on a manifold M is like a way of moving strings around in M, so you can think of it as a recipe for 2d parallel transport. The funny part is, you need a conformal structure on your surface before you can do parallel transport over it! Stolz and Teichner do a great job of working out the following analogy: complex K-theory elliptic cohomology connections on complex vector bundles elliptic objects supersymmetric 1d field theories supersymmetric conformal field theories In particular, they show how the spectrum for complex K-theory can be built from the space of supersymmetric 1d field theories, just as the spectrum tmf is (conjecturally) built from some space of supersymmetric conformal field theories. Being an optimist, I canÕt help but hope this pattern goes on something like this: some cohomology theory that detects v_n-periodic phenomena connections on complex n-vector bundles some supersymmetric field theories on n-dimensional spacetime Who knows? Next I should say a word about the new versus old versions of elliptic cohomology. At this point things are going to get... ahem... a bit technical. Then IÕll talk about the connection to Monstrous Moonshine, and things will get really vague, and downright bizarre. The old version of elliptic cohomology was a specially nice sort of generalized cohomology theory called a complex oriented cobordism theory. I explained what these were in week149, and in week150 I explained how each of these things gives a formal group law. If you want an easily understood example of a formal group law, just take a group, pick coordinates near the identity of this group, and write out the group operation in terms of these coordinates as a power series. This works whenever your group is an analytic manifold and the group operations are analytic functions. The result is a formal group law. The word formal comes from the fact that weÕd actually be satisfies if the group operations were described by *formal* power series. Anyway, now consider the torus. A torus is a group in an obvious way - just a product of two copies of the group U(1) - but there are different ways to make it into a *complex* manifold where the group operations are *complex* analytic functions. A way of doing this is nothing other than an elliptic curve! In fact, each elliptic curve corresponds to a complex oriented cobordism theory, and we could call any one of these an elliptic cohomology theory, if we wanted. But itÕs better, actually, to glom all these different theories into one big universal theory. The most obvious way to attempt this is to take the moduli space of elliptic curves and cook up a formal group law over the algebra of functions on this space by stitching together all the formal group laws for each specific elliptic curve. This formal group law corresponds to a from a complex oriented cobordism theory called Ell. This is what I was calling the old version of elliptic cohomology. The new version, namely tmf, is a bit sneakier. I think itÕs the limit - in the sense of category theory - of the elliptic cohomology theories for all specific elliptic curves. The reason this is different than Ell is that some elliptic curves have nontrivial symmetries! Unlike Ell, tmf is *not* a complex oriented cobordism theory. But the difference is very subtle, and only involves 2-torsion and 3-torsion, that is, elements that vanish when you multiply them by some power of 2 times some power of 3. The reason the numbers 2 and 3 show up is apparently because the elliptic curves with nontrivial symmetries come from the square lattice: * * * * * * * * * * * * and the hexagonal lattice: * * * * * * * * * * * which have 4-fold and 6-fold symmetry, respectively. I already expounded on these symmetries in week124 and week125, and showed that theyÕre responsible for the mysterious role of the number 24 in string theory. So, itÕs nice to see them showing up here! In fact, they also show up in other devious ways, which I would love to understand better. For starters, they give a certain period-12 pattern in the theory of modular forms, which becomes a period-24 pattern if you define weights using the convention that IÕm using here. Lots of people know about this - see any introduction to modular forms, like this one: 4) Neal Koblitz, Introduction to Elliptic Curves and Modular Forms, 2nd edition, Springer-Verlag, 1993. I already vaguely explained this in week125. But, more deviously, these symmetries are also related to a certain period-576 pattern in topological modular form theory! The number 576 is 24 x 24. According to my vague memories of what Stephan Stolz said, the first 24 is the usual one in bosonic string theory. In particular, if we ignored subtleties involving torsion, elliptic cohomology would have period 24, with the periodicity generated by a conformal field theory of central charge 24 having an enormous group called the Monster as its symmetries! This is where Monstrous Moonshine comes in, and especially the work of Borcherds. (This canÕt be exactly right, because the most famous conformal field theory with the Monster as symmetries is not supersymmetric, and its partition function is the j-function, which is a modular function of weight 0, not a modular form of weight 24. So, my brain must have been a bit fried by the time we got to this really far-out stuff.) Where does the extra 24 come from? I donÕt know, but Stephan Stolz said it has something to do with the fact that while PSL(2,Z) doesnÕt act freely on the upper half-plane - hence these elliptic curves with extra symmetries - the subgroup Gamma(3) does. This subgroup consists of integer matrices (a b) (c d) with determinant 1 such that every entry is divisible by 3. So, if we form H/Gamma(3) we get a nice space without any points of greater symmetry. To get the moduli space of elliptic curves from this, we just need to mod out by the group SL(2,Z)/Gamma(3) = SL(2,Z/3) But this group has 24 elements! In fact, I think this is just another way of explaining the period-24 pattern in the theory of modular forms, but I like it. I especially like it because SL(2,Z/3) is also known as the binary tetrahedral group. To get your hands on this group, take the group of rotational symmetries of the tetrahedron, also known as A_4. This is a 12-element subgroup of SO(3). Using the fact that SO(3) has SU(2) as a double cover, take all the points in SU(2) that map to A_4. You get a 24-element subgroup of SU(2) which is the binary tetrahedral group. In fact, if you think of SU(2) as the unit sphere in the quaternions, the binary tetrahedral group becomes the vertices of a 4-dimensional regular polytope called the 24-cell! IÕm very fond of this polytope, and have already extolled its charms in week91 and week155. So, what pleases me now is that IÕve found a trail directly from the 24-cell to the appearance of the number 24 in string theory... and even the fact that topological modular form theory has periodicity 24 x 24. Of course I can barely follow this trail myself, and I probably got some stuff wrong - I hope the experts correct me! But the trail seems to be real, not just a will othe wisp, so I can now try to widen it and make it less twisty. ThereÕs more to say but IÕll stop here. I have given other references to monstrous moonshine in week66, but hereÕs a very pretty website about it: 5) Helena A. Verrill, Monstrous moonshine and mirror symmetry, http://hverrill.net/pages~helena/seminar/seminar1.html and here is a nice easy paper by Terry Gannon about it: 6) Terry Gannon, Postcards from the edge, or Snapshots of the theory of generalised Moonshine, available as math.QA/0109067. ------------------------------------------------------------- ---------- mathematics and physics, as well as some of my research papers, can be obtained at http://math.ucr.edu/home/baez/ For a table of contents of all the issues of This WeekÕs Finds, try http://math.ucr.edu/home/baez/twf.html A simple jumping-off point to the old issues is available at http://math.ucr.edu/home/baez/twfshort.html If you just want the latest issue, go to http://math.ucr.edu/home/baez/this.week.html === Subject: To study moonshine and its implications I have an opportunity for a graduate student (Masters or doctorate) or a post-doctoral student - to study automorphic functions and finite group characters as exemplified by Monstrous Moonshine. URL: http://cicma.concordia.ca/faculty/cummins/moonshine.html John McKay (Professor of Mathematics and of Computer Science) -- But leave the wise to wrangle, and with me the quarrel of the universe let be; and, in some corner of the hubbub couched, make game of that which makes as much of thee.