Archive for the ‘Greg’ Category

Rational Homotopy Theory

April 27, 2008

    I tend to think of homotopy theory a little bit like ‘The One That Got Away’ from mathematics as a whole.  Its full of wistful fantasies about how awesome it would have been if things could only have worked out.  Imagine if homotopy groups of spaces and homotopy classes of maps were as easy to compute as homology groups… we’d be teaching undergrads about Postnikov towers and topology might very well end up a subset of group theory.

   Instead, homotopy theory is a hopeless, incalculable mess in all but the trivial cases… that bitch.  The canonical example here is the Toda’s Table of the Homotopy Groups of Spheres.  That’s right, even the simplest imaginable case - homotopy classes of maps between spheres – is a wildly unpredictable mess with only a handful of a structure theorems.

   So homology and cohomology theories rule the day; not as powerful as homotopy groups, but infinitely more tractable.  However, recently I’ve become somewhat enamored of a weaker form of homotopy which is just weak enough where you can actually say things: Rational Homotopy Theory.  The general idea is to simply ignore any information coming from torsion homotopy groups.  After all, all the hideousness in Toda’s table is finite groups; we know the infinite homotopy groups, and they represent reasonably interesting phenomena.

    The main upshot of this is that all the information of a space (up to rational homotopy) can be packaged in a differential graded algebra.  Rational homotopy equivalence becomes quasi-isomorphisms, and so the question of whether two spaces are rational homotopic is very reasonable.  With some mild restrictions, it can be shown that spaces up to rational homotopy are the same as DGAs up to quasi-isomorphism.  This opens the door for recasting much of topology as purely algebraic constructions.

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Abelian Categories and Module Categories

April 10, 2008

    The blog’s been rather quiet lately, due in a large part to me being in research mode right now.  That also explains why my posts, when they occur, are mostly advanced.  I just don’t have much general-consumption math on the brain at the moment.

    Today, I’d like to talk about some of the more basic things in the subjects I tend to work (as a compromise).  As you might recall, an abelian category is a category where the set of morphisms between any two objects, Hom(A,B), is an abelian group; with some additional properties that make it nice enough to do things like homological algebra.  The classic example of an abelian category is Mod(R), the category of finitely generated left modules of some ring R.

    One can ask the question: how far is an arbitrary abelian category from being a module category?   One result in this direction worth knowing is the Freyd-Mitchell Embedding Theorem, which says that any abelian category has a fully faithful, exact embedding into some module category.  The principle use of this theorem is to make homological algebra proofs which assume the existance of ‘elements’ work.

    But, how can we tell if an abelian category is equivalent to a module category?  As we will see, finding such an equivalence is the same as finding a sufficiently nice element in the category, called a compact progenerator.  It can also be interesting to find multiple progenerators, giving us non-trivial equivalences Mod(R)\sim Mod(S), where we call R and S ‘Morita equivalent’.

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Morse Theory Indomitable

March 30, 2008

   I recently came across an excellent survey article, “Morse Theory Indomitable“, by Raoul Bott.  It starts with the basic history of Morse functions, and covers the additions of Smale and Witten, and the connections to symplectic reduction.  Though, even moreso than being a clear and concise overview of some beautiful mathematics, it is all liberally dosed with personal anecdotes from the life of someone who lived throughout virtually the entire story.  It was a throughly enjoyable read, even though I had seen almost all the contained math before.

Koszul Duality and Lie Algebroids

March 25, 2008

    Something has been bothering me about Koszul duality lately.  Well, technically many things have been bothering me about it, but here’s a particular thing that has been bothering me.   Usual (homological) Koszul duality assigns to an augmented R-dga A the algebra A^!:=RHom_A(R,R).  Since R is central, A^! is again an augmented R-dga, and so A^{!!} makes sense.  In many (all?) cases, A^{!!}\sim A (hence, duality).

    I am interested in the case when R is no longer central in A.  Then an action of A on R no longer is the same as an augmentation map, so we only assume we have a left A-module structure on R (where the contained copy of R acts by mutliplication).  We can still form A^! in the same way; however, A^! no longer contains R, so it seems impossible to form A^{!!}.

    However, I have reason to suspect that one should be able to form A^{!!}, and that it should often be quasi-isomorphic to A.  My evidence in this direction is a paper of Positsel’skii’s, “Nonhomogeneous Quadratic Duality and Curvature”, wherein he shows that for a very narrow class of algebras A which both contain and act on R, that A can be recovered from A^!.

     An important case of this duality is for the ring of differential operators \mathcal{D} on a smooth curve.  This has a canonical left action on \mathcal{O}, and so \mathcal{D}^!:=RHom_\mathcal{D}(\mathcal{O},\mathcal{O}) exists, and is in fact the deRham complex.  So, a specific case of my question is: is there a sense in which the deRham complex \Omega of a space acts on \mathcal{O}, such that something like RHom_\Omega(\mathcal{O},\mathcal{O})\sim \mathcal{D}?

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IAS Conference Question Dump

March 19, 2008

    Its now been several days since the conference at IAS, and I might as well do a quick wrap up.  Overall, the conference was great.  Almost all the talks were understandable, and contained new mathematics of a truly humbling quality.  There was a surprising collection of internet mathies: Ben Webster, Joel Kamnitzer, Peter Woit, Charles Siegel, Aaron Bergman and David Ben-Zvi (and maybe even some I’m forgetting or didn’t notice).  Also, I didn’t actually play Bad Talk Bingo; I had more than enough of my own math to do during downtime and bad talks.  It was standing room only for virtually every talk (though that was at least in part because we were in a small room when a big auditorium was being unused).  I even sat in the aisle next to Deligne for the first talk.

    One downside is that it was a pretty intimidating atmosphere for asking questions.  Usually, I’m pretty good about asking potentially stupid questions, but I wasn’t confident in my knowledge of the ‘basics’.  I asked some of the speakers my questions after the talks, and I tried to write them all down for further contemplation.  I figured I’d put ‘em here, both for my own reference and in case anyone out there knows the answer.  I’ve included the name of the talk the question is from; but in some cases, these are unrelated questions I was thinking about.

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Television Will Rot Your Brain

March 12, 2008

    A fun little math problem today.  It came about when I misheard a problem asked by a friend, and solved a slightly different problem.  It goes as follows:

    A child watches television everyday.  He always watches at least one hour a day, and he only watches tv in whole integer amounts of hours.  Concerned for his well-being, his parents impose a restriction: he can never watch more than 11 hours of tv in any 7 day period.  Show that there is some consecutive string of days in which the child watches exactly 20 hours of tv.

The number 20 is a relic of the problem that I misheard (show that this happens in any 11 week period).  Of course, why stop there?  Show that for any positive integer n, there is some string of consecutive days in which he watches exact n hours of tv.

    This begs an interest question: for what other ‘restrictions’ is this property true?  That is, what other numbers d and h have the property that if the child can only watch h hours of tv in a d day span, then he achieves every positive integer as a consectutive total?

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Bad Talk Bingo: Brainstorming

March 4, 2008

    Next week I am off to IAS for the conference on Algebro-Geometric Derived Categories and their Applications.  I am starting to get worried about it, since the list of talk titles is up and only a few of them appear to be welcoming to non-experts in their respective fields.  Also, when registering, there was no option under ‘Occupation’ for ‘Grad Student’, which struck me as a subtle hint that I wasn’t welcome.  Perhaps just being at IAS will compel speakers to turn the difficulty up to eleven.

    In any event, I am preparing for there to be a fair share of bad talks.  This, and other recent experiences with bad talks has given me the following idea of something to do: Bad Talk Bingo.  The idea is to create a 5×5 grid of events and signs that a talk is going badly, so that I can check off which ones happen and try to get 5 in a row.  My plan is that if I ever succeed in getting 5 in a row, I am allowed to pretend to get a phone call and rush hurriedly from the talk.

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Hyperbolic Discounting

February 25, 2008

    I have always been fascinated with the study of how people make decisions.  Its a complicated process that involves as much ‘gut instinct’ as rational evaluation, and is rife with systematic errors in judgement.  For a long list of common mistakes, check out Wikipedia’s list of cognitive biases, particularly the decision-making section.  The majority of them are things that most people are probably aware of, like picking a option just because other people picked it.  However, a couple of them are a bit surprising, especially the one I want to talk about today: the phenomenon of hyperbolic discounting.

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My Favorite Prime Number with Four Divisors

February 11, 2008

    A prank I recommend to readers is to use the number 91 when a group situation calls for a random prime number.  If done subtly enough, a decent portion of mathematicians will believe you.  Granted, its not a particularly funny prank…

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Equivariant DeRham Cohomology

February 10, 2008

    My lectures on equivariant cohomology are spinning a bit out of control.  The questions and lively discussion, while always welcome, have stretched what was meant to be a hand-waving tour through the basics into a three week mini-course (at least, I hope its only three weeks).  I’m starting to feel a bit sheepish, since the result I’m trying to get to might not really merit a full month-long preamble.

    Last time, I talked about how to define the equivariant cohomology of a space M in terms of the cohomology of some big infinite-dimensional space M\times E/G.  This is good on a conceptual level, but unless E is particularly nice, we will have a bitch of a time computing the cohomology of anything.  What we need is a more effective model for the cohomology of M\times E/G.

    The idea is to start by pining for the existance of a nice de Rham complex \Omega(E) on E.  We’ll say “Oh, if only it \Omega(E) existed, it would look like this, and this…”.  Since E was only defined by its properties (contractibilty and a free action), this amounts to listing what properties a differential graded algebra should have to correspond to those of E.  Such DGAs will be called ‘locally-free, acyclic G^*-algebras’.

    From there, its a three step process.  First, show that every such DGA computes the same cohomology.  Second, show that there is an (almost) universal locally-free, acyclic G^*-algebra called the ‘Weil algebra’, which is simple enough in structure to make computations effective.  Third, show that there exists any such DGA which correctly computes the equivariant cohomology (this last step should probably be first, but it isn’t very exciting).

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