Quantum Bears

Hello all! It has been a very long time since I last wrote; I have been going to and from CERN as we’re preparing for the LHC. I’m in Geneva right now, and I just came back from watching En Pleine Nature (Into the Wild). Without giving away too much plot, this movie contains a bear that did not eat anyone. It made me think of Grizzly Man from a few years ago, in which another bear did. What impressed me most about Grizzly Man is that the probability of being eaten by a bear, should we find ourselves face-to-face, is not a simple 20%. It depends a great deal on who the bear is, what he thinks about humans, how hungry he is, what I smell like, the weather, his mood, etc. There’s a whole space of parameters, and some regions of this space are filled with nearly 100%, others with nearly 0%.

Let’s say we’re doing a scientific study of bears eating people. In our first experiment, we put 100 people in the woods and just count how many get eaten. Then we’d like to get more grant money, so we do a more in-depth analysis by controlling for several variables: some of our volunteers are smeared in honey barbeque sauce, others aren’t. Our sequence of studies slowly sharpen the focus on the bear-eating parameter space, identifying the high-probability regions and the low-probability regions. Where does this process end? If we could do infinitely many studies, would we find that each point is either 100% or 0% (deterministic bears)? Or not (uncertain bears)?

Framing a discussion of quantum mechanics in this way illustrates a feature that is often missed: the connection between quantization and uncertainty. Usually these two topics just fall out of the postulates with little indication that they are related. As it turns out, quantization makes fundamental uncertainty possible.

Odd Sums of Consecutive Odds

Oscar Wilde’s character Algernon said in The Importance of Being Earnest, “One must be serious about something, if one is to have any amusement in life.” Of course in Wilde’s typical ironic fashion, Algernon was only referring to his own dedication to frivolous diversions. In that spirit, allow me a few moments to tell a story about one of the odder sums of odd integers I discovered as a kid.

I remember that sometimes when I was bored — most especially during long, bi-weekly car trips with my parents — I would play various games with integers. I have no idea why, but at one point I memorized some huge list of powers of 2 (I can still remember the list from 1 to 65,536). I also computed the squares, cubes, and so forth of most of the smaller integers. As a result, I discovered on my own quite a number of interesting patterns in the integers. I don’t remember most of them, but there is one in particular that has stuck with me through the years.

Singular Integral Operators and Convergence of Fourier Series

I’m Peter “Viking” Luthy, a journeyman graduate student at Cornell. I’m an analyst, and my current research goals are in harmonic analysis with applications to and from ergodic theory. To avoid being called a hypocrite, Greg asked me to post on occasion and spread my analytic gospel — this isn’t the Everything-but-Analysis Seminar, after all.

My goal in this post is to go through the initial setup of a deep theorem of Carleson dealing with the convergence of Fourier series on . This theorem is almost universally interesting in and of itself. Additionally, it will give ample reason as to why people — myself included — care about objects called singular integral operators. This will also provide some impetus for some future posts as well, particularly one which will outline a famous construction of Fefferman and give some reasons why harmonic analysis in higher dimensions is distinctly harder than in dimension 1.

My Favorite Prime Number with Four Divisors

    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…

Equivariant DeRham Cohomology

    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 in terms of the cohomology of some big infinite-dimensional space .  This is good on a conceptual level, but unless 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 .

    The idea is to start by pining for the existance of a nice de Rham complex on .  We’ll say “Oh, if only it existed, it would look like this, and this…”.  Since 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 .  Such DGAs will be called ‘locally-free, acyclic -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 -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).

The Cohomology of Quotients

    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 complex)  and a Lie group which acts on .  Unless this action is free and proper, the quotient space might be a poorly behaved space.  Take, for example, acting on 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 , which is  if acts freely and properly?” The hope is that this will shine some more light on the hidden internal structure of the bad quotients.

Puzzles, Groups, and Groupoids

Over at Good Math, Bad Math, MarkCC has a nice post introducing groupoids which uses the fifteen puzzle as an example. I like this example a lot, and I thought it would be interesting to expand on it a bit. So I’m going to tell you:

1. Why the Rubik’s Cube is a finite group,
2. Why the fifteen puzzle is a finite groupoid, and
3. How to solve the fifteen puzzle.

I’m not going to assume any knowledge of groups or groupoids, but if you don’t know much group theory, you’ll have to skip over certain parts of the second half.

An Almost-Proof of the Four Color Theorem

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.

Convergence of Infinite Products

There is a simple convergence test for infinite products that I think deserves to be better known.

Theorem. Let be a sequence of positive numbers. Then the infinite product

converges if and only if the series

converges.

A Resolution of a Tensor Algebra

    This post is about projective resolutions of algebras, thought of as a bimodules over themselves.  As long as is an associative, unital algebra (which it always will be in this post), there is a canonical projective resolution of , 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 .

    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 is a tensor algebra of an algebra over a bimodule .  Under what conditions on and would a nice, finite resolution of exist?  My result is as follows:

Let be an algebra, be a bimodule over , and let denote the kernel of the multiplication map .  If is projective as an bimodule, and , then there is a projective resolution of of length 3.

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

Let , , and be defined as above.  If is projective and is flat as a right module, then any left  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.