Archive for the ‘Matt’ Category

Parallel Parking and the Geometry of Differential Equations

October 3, 2007

I’ve lately been doing some research in the general area of geometric PDEs inspired by the intricate theory of minimal and constant mean curvature surfaces. This has given me the chance to apply differential-geometric techniques to problems which I used to believe could only be approached analytically. To introduce some of these ideas, I had started to write a post on the parallel parking problem — but I got scooped by that sneaky upstart blogger Charles over at Rigorous Trivialities!

Still, I think parallel parking is a great way to starting thinking about the geometry which governs differential equations. So let’s go down the rabbit hole…

Surfaces and Spinors

September 28, 2007

A huge problem that many people initially have with differential geometry is over-representation. Manifolds are conventionally defined via absurdly large, Zorn’s-lemma’d maximal atlases, giving an enormous sea of possible coordinates which could represent an object. Maps with some property from one surface to another are often defined relative to local charts, involving many arbitrary choices of coordinates, covers, and so forth. Tensor fields might be built up as collections of numbers with absurd transformation rules. In that mass of symbols, where is the geometry?

Conversely, who would expect upon first inspection that the trace and the determinant of a matrix are coordinate-invariant quantities? What about the Riemann curvature tensor? Would you have seen that the Gaussian curvature is intrinsic, yet the mean curvature is extrinsic? Can you identify the Willmore energy as invariant under Mobius transformations, despite being constructed from Euclidean invariants? What if it was written in local coordinates?

This overabundance of representations in differential geometry is a mixed blessing. They give us a thousand ways to look at a problem, but 995 of those obscure the geometric features we are trying to understand. Ideally, we would like to have a language for geometry which is rich enough that we can efficiently describe and compute with geometric objects, yet weak enough that we cannot state non-geometric facts. This might be a bit much to ask in general, but many big advances in differential geometry have come through finding clever representations for certain classes of geometric objects which allow us to easily read off or insert geometric information. At Greg’s request, I’d like to share a modern version of one of the classical representation theorems in the geometry of surfaces.

Antisymmetry in Geometry, I

September 13, 2007

Isabel’s post at God Plays Dice asking for a geometric entity realizing the dot product got me thinking again about a question that my friend Ron asked me several years ago: “Why do geometers always focus on antisymmetric tensors and ignore the symmetric ones?”

It certainly seems that antisymmetric objects have more grit to them — they correspond to things like fields of infinitesimal k-planes, or symplectic structures, or fluxes. Things you can imagine getting your hands on. Symmetric objects, on the other hand, are things like Riemannian metrics and the dot product, or the shape operator. They seem to measure things, which is far more subtle than being things. I’m always happier with a 2-form over a quadratic form: I can see the former.

There is a lot to say about this subject, and exploring it could take us to some fun places. As a first step in that direction, I’d like to share a proof that every manifold admits a metric which may be new to you.

Non-nonstandard Calculus, I

August 28, 2007

I alluded in one of my very first posts here to a calculus class that I was teaching using the ring $\mathbb{R}[dx]/(dx^2 = 0)$. The class was a six-week, 5-day-per-week intensive course covering the usual material of a college Calculus I course. I’ve been promising Greg and Jim that I’d write up some of my experiences with the course. I think I have had enough time to process the experience — let’s talk non-nonstandard calculus. (more…)

A Cute Problem

August 28, 2007

Jim told me this cute problem about two years ago. I was reminded of it while reading the recent posts over at Ars Mathematica. For your enjoyment:

Prove that in the poset of subsets of $\mathbb{N}$, there exists an uncountable chain(!)

Sum Divergent Series, III

August 2, 2007

One excellent reason to believe that these Cauchy-divergent sums can be assigned reasonable values comes from the fact that equations like $1 + 1 + 2 + 5 + 14 + 42 + \dots = e^{-\pi i/3}$

and $1 + 2 + 4 + 9 + 21 + 51 + \dots = 1 + 2 + 6 + 22 + 90 + \dots = -i$

have real, finite combinatorial consequences. These are the sums of the Catalan numbers, Motzkin numbers, and Schroder numbers, respectively. By taking these divergent sums seriously, we are led to new results. As a matter of fact, a new combinatorial theorem came out of the comments in the last post, thanks to Isabel (of God Plays Dice) noting that the sum of the Motzkin numbers should be $-i$: there is a bijective algorithm (explicitly constructed) which converts Motzkin trees to 5-tuples of Motzkin trees.

Today I would like to pose the question “How do we know that our divergent sums are meaningful in situations where we can’t immediately find a finite consequence?” And, of course, finally get to the promised puzzle.
(more…)

Sum Divergent Series, II

July 30, 2007

Last time we saw hints of how the geometric series was trying to tell us that there was some “truth” behind the equation $1 + 2 + 4 + 8 + 16 + \dots = -1$

So how do we get at the truth behind such an audacious claim? One fantastically powerful way to sum divergent series such as this is through regularization: finding an analytic function of $z$ through which we can relate the divergent sum to some limit in the complex plane. In particular, we can find reasonable values for surprisingly unreasonable sums like $1 + 1 + 1 + 1 + \dots$

Sum Divergent Series, I

July 28, 2007

Every few months, I get in a particular mood that inspires me to look for reasonable finite values to assign to superficially divergent sums. I’d like to share some of them with you and start a discussion of just what “reasonable” means in this context. Finally, I have an open (and open-ended) question on very divergent series for you all to have a crack at.

Harmonic Digression

July 12, 2007

I’ve been meaning to submit this to the Proofs Without Words column ever since I discovered it way back when I was learning calculus. At the time, I wasn’t very impressed by showing that the harmonic series diverged using integral approximations for some reason. I wish I could remember why — it would probably make me a better calculus teacher. This is what I came up with to show the divergence more directly(?):  I’ll leave the interpretation as a puzzle to the reader.

Calculus and Complexes

July 4, 2007

I am now 8 days into teaching a six-week, every-morning-at-8:30 summer intro calculus class. In order to (1) make all the material fit into such a short time, (2) leverage the student’s (theoretically) good grasp of algebraic manipulation and (3) because I visualize infinitesimals more often than limits, I decided to teach as much of the course as possible using a somewhat ill-defined version of the real numbers that includes infinitesimals. In particular, the students have been working over the ring $\mathbb{R}[dx]$ where $(dx)^2 = 0$. We additionally extend the ordering on $\mathbb{R}$ to $\mathbb{R}[dx]$ by defining $0 < dx < r$ for any positive real $r$ (this is vital if we want to extend piecewise-defined real functions to $\mathbb{R}[dx]$ since we need the order predicates to extend).