Archive for the ‘Greg’ Category

The Cohomology of Quotients

February 2, 2008

    We’ve organized a mostly informal Topics in Noncommutative Algebra seminar this semester, and I’m talking first in it.  I’m eventually going to be talking about a paper of Ginzburg’s connecting Hochschild and cyclic cohomology to the equivariant cohomology of representation schemes.  Unfortunately, the trouble about talking about fun results like that is that you need to cover alot of background material; as such, I’m doing what is turning out to be a two lecture series on equivariant cohomology and its deRham version.  I figured I’d mirror these talks with a couple of posts, and maybe even talking about Ginzburg’s paper if I get enough prereqs covered.

    Today I’m just going to be talking about topological equivariant cohomology.  Let’s start with a nice space (say, a CW complexM and a Lie group G which acts on M.  Unless this action is free and proper, the quotient space M/G might be a poorly behaved space.  Take, for example, \mathbb{Z} acting on S^1 by some irrational rotation; the quotient isn’t even Hausdorff.

    The motivating question of equivariant cohomology is: “Is there a good cohomology theory for the pair (M,G), which is H^\bullet_{CW}(M/G) if G acts freely and properly?” The hope is that this will shine some more light on the hidden internal structure of the bad quotients.

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An Almost-Proof of the Four Color Theorem

January 26, 2008

I recently gave an Olivetti (our graduate colloquium) on chord diagrams, effectively covering my first two posts on the subject. In preparation for the talk, I read a little bit more about some cool things one can do with them, and I finally got around to reading a paper of Bar-Natan on connections with the Four Color Theorem. I figured I should write a post on it before everyone completely forgot what I’ve said already; that said, this post should be readable even if you didn’t read my other posts on chord diagrams.

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A Resolution of a Tensor Algebra

January 24, 2008

    This post is about projective resolutions of algebras, thought of as a bimodules over themselves.  As long as B is an associative, unital algebra (which it always will be in this post), there is a canonical projective resolution of B, called the bar resolution, which is sufficient for most purposes.  However, this resolution is of infinite length, and so it isn’t useful in bounding projective dimensions of modules.  For those purposes, it is natural to look for finite projective resolutions of B.

    I came across such a problem in my research, and came up with limited and ultimately unhelpful results.  My interest was in the case that B is a tensor algebra T_AM of an algebra A over a bimodule M.  Under what conditions on A and M would a nice, finite resolution of T_AM exist?  My result is as follows:

Let A be an algebra, M be a bimodule over A, and let \Omega^1A denote the kernel of the multiplication map m: A\otimes A \rightarrow A.  If \Omega^1A is projective as an A bimodule, and Tor_1^A(M_A,_AM)=0, then there is a projective resolution of T_AM of length 3.

This is kinda neat, but its not super useful unless it can be used to produce projective resolutions of T_AM modules.  Hence, the second result:

Let A, M, and \Omega^1A be defined as above.  If \Omega^1A is projective and M is flat as a right A module, then any left T_AM module has a projective resolution of length 3.

Sadly, I wanted the assumptions of the former to prove the latter, which my techniques don’t.

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My Favorite Math Party Trick

January 12, 2008

    I have a minor personal triumph to relate today.  So, I try to have several cute, quick math puzzles/facts on hand to bust out at parties, because I am cool like that.  The most popular of these is a participation-based trick that goes as follows (feel free to play along at home):

    Take a pen and paper and draw a quadrilateral.  There are no restrictions (it can be concave or self-intersecting), but don’t make it too close to the sides of the paper.  Now, for each edge, draw the square containing that edge that is outside the quadrilateral.  Put a dot in the center of each of the four squares, and draw a line connecting opposite dots, ie, those that came from opposite edges.

The Punchline: The lines you just drew are the same length, and perpendicular.

(If you lack pen and paper, theres an applet here)

    It works pretty well because it isn’t very sensitive to sloppy geometry on the part of the artist; so you will pretty consistantly get ‘perpendicular-looking’ lines.  Also, I often bend the truth a bit and attribute it to Napolean Bonaparte, even though he proved something different but closely related.  The usual attribution of this result is to van Aubel, who to my knowledge conquered very little of Europe.

    I’ve been using this for a couple of years now, and periodically I attempt to find a nice geometric proof without passing to coordinates.  Such a proof eluded me up to last night, when I came up with a reasonably nice vector-based proof.  Its a little cheap since using vectors isn’t totally different from passing to coordinates, but in my mind a geometric proof is one which can be done only with pictures (though I will use words for laziness’ sake).  For the record, I was unaware of van Aubel’s proof until this morning, which is a more traditional geometric proof, but a little bit more indirect.

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The Efficiency of Random Parking

January 8, 2008

   At dinner recently, a friend mentioned the following problem:

    There is a street of length x (not necessarily an integer) on which cars of length 1 wish to park.  However, instead of parking in a nice organized way, they park at random, picking uniformly from the availible positions to park (they are apparently jerks).  Assuming no cars leave, and continue to arrive until no more can fit, what is the expected number of cars that will fit?

     For instance, if the street is length 2, then the first car will almost always park so that no other can fit,  and so the expected number of cars is 1.  If the street is length 3, the first car can never prevent a second from fitting; but a third car almost never will fit, and so the expected value is 2.  The trend of simple answers is broken at street length 4, where the expected number of cars is \frac{11}{3}-\frac{4}{3}\ln(2).  This came up as a fun, back-of-the-envelope type problem, but the answer is actually an iterated integral that becomes prohibitively difficult after a few steps.  Perhaps a intrepid reader can solve more steps than I was able to.

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Chord Diagrams and Lie Algebras

December 25, 2007

    Merry Christmas!  In this post, we will build on some of the previous posts about chord diagrams.  In a bit of a tangent from previous thoughts, we will explore the relationship between chord diagrams and Lie algebras.  Explicitly, last time we came up with a relation, which we will henceforth call the IHX-relation:

  STU relation

Remember that this was really another aspect of the 4T relation for regular chord diagrams.  We will see how this relationship is a pictorial representation of the Jacobi identity, which allows us to interpret generalized chord diagrams modulo the IHX relation as instructions on how to combine a large number of Lie brackets.

    The first step is to introduce the Penrose’s tensor notation, which is a very natural tool for writing down instructions on manipulating tensor powers of vector spaces.  If we have a Lie algebra with an invariant inner product, we can turn a large class of graphs with extra data into such instructions.  This class of graphs includes chord diagrams, and we will see that in this framework, the IHX relation and the Jacobi relation are the same thing.

    I should warn readers that there’s not much of a punch-line to this post.  Instead of being about building to some nifty conclusion, this is about a different perspective on chord diagrams and a series of nifty things you can do with them.  Also, this is a very long post that is a bit rambly.  I wouldn’t mind some constructive criticism on how some of this stuff could have been explained better.

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Chord Diagrams: Understanding the 4T Relation

December 18, 2007

   Last time, we used knot theory as a way of motivating these funny things called chord diagrams, which were circles with a collection of chords.  They came up as ways of writing singular isotopy classes of singular knots, but there are many other ways of thinking of them.  They can also be thought of as fixed-point free involutions on 2*degree elements (modulo cyclic permutations) or a trivalent graph with a distinguished Hamiltonian cycle.  I list these other incarnations more as a way of appealing to a broader base, though the latter perspective will be relevant today.

    The topic of the day is the 4T relation, which I mentioned at the end of the previous post.  It, together with the 1T relation, determined which functions on the set of all chord diagrams of a given degree came from knot invariants.  The 1T relation came from a fairly straight-forward observation, while I didn’t even attempt to defend the 4T relation.

    One way of thinking of this relation is as a symptom of a larger structure.  We will generalize chord diagrams to a slightly broader context, and impose a relation there that is more natural.  From there it is obvious to see that these generalized chord diagrams can always be reduced in terms of the usual chord diagrams, and that the only evidence of this broader structure is the 4T relation.

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Chord Diagrams

December 13, 2007

    Well, its been quite awhile since the last post.  The real problem is that this blog can’t be a higher priority for me than, you know, important stuff like learning, teaching and research.  So, when time and energy get tight, its one of the first things to go.  It also doesn’t help that the more I bend my mind towards research, the less time I spend thinking about things that would actually make good posts (I have started and abandoned several posts on uninteresting research-type things in the last couple months).

     I’ll try to get back into the swing of things by talking about some stuff I really enjoy, but is far from my research: knots.  I’ve always had a soft spot for knot theory, since its like a poor man’s number theory; a source of simple problems which require techniques from advanced math to solve.  A good example of this are Tait’s conjectures, three basic conjectures from the 19th century that resisted proof until the discovery of the Jones invariant using techniques from analysis and representation theory (and secretly physics).

    Today, though, I’d like to talk about ‘chord diagrams’, a type of object subtly related to that of knots, and whose study can yield some interesting new knot invariants.  They also come up in a number of different areas (solving Feynman diagrams, the representation theory of lie algebras) that I know very little about.  If anyone reading is more familiar with some of the places these come up, please send me a reference.

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Symplectic Mechanics and Symmetry

October 18, 2007

   In my last post, I outlined how solving Newton’s F=ma for a single object moving in a static force field could be restated as flowing along a vector field in the tangent bundle (=cotangent bundle).  However, if I had written out the details of what flowing along the vector field meant, it would have been clear that it was just a fancy way of saying the particle accelerates in the direction of the force.  Is this just elaborate computational legerdemain, where I have used big words and new concepts to misdirect you from the fact that I am just moving the difficulty around?

    The aim of this post is to demonstrate that the symplectic perspective has its own merits, by describing the technique of ‘symplectic reduction’.  The principle here is a simple and beautiful one: if the mechanical system under consideration has a (good) symmetry, then there is some quantity one can assign to the possible states of the system which is conserved for all time.  This meta-theorem is called Noether’s Theorem, though we will not be using it in full generality.  Once a ‘conserved quantity’ is found (think of something like angular momentum), together with the symmetry it came from, the mechanical system can be reduced to one on a lower dimensional space.

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Classical Mechanics, The Symplectic Way!

October 12, 2007

    The goal for this post is to give a general outline of how to do some very basic Newtonian physics, using the language of symplectic topology (I prefer to say symplectic topology instead of symplectic geometry, since a symplectic manifold has no local invariants).  First, we’ll outline the basic kind of problem we care about, that of a particle sitting in a manifold with some forces acting on it.  Then, we’ll go over the basic symplectic techniques needed, and finally we’ll state how the law of physics looks like in this setting.

    This is a post I have meant to write for a while, since several times now I have thought about writing a post which would assume it as background.  I’m also hoping that writing about a topic so dear to my heart helps jog me out of my non-posting funk.

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