I made a movie recently for my advisor. The movie is so pretty, that I thought I’d share it here: may I present to you randomly drawn dots, where two dots are the same color when they touch!

I’ll be a bit more explicit: a dot is drawn at a random location; if it does not overlap any previous dots, it gets a new color. Otherwise, the dot takes the color of the component it touches. Sometimes a new dot connects many components, and in this case, the new component takes on the color of the largest among the old components.

There’s a lot of neat questions to be asked about such a process: for instance, after drawing n dots, how many components should we expect to see? As you can see in the movie, when you draw only a few dots, most of those dots are isolated and have their own color; but after drawing a ridiculously large number of dots, they are all connected and the same color. And inbetween, something more interesting happens.

Here’s an example of “something more interesting” taken from a larger picture than the above movie:

In this fun paper,

Kreck, Matthias. An inverse to the Poincaré conjecture. Arch. Math. (Basel) 2001. 98–106. MR.

it is pointed out that

• homology is a very basic invariant, and
• closed manifolds are very basic objects

and so a very basic question is: what sequences of abelian groups are the homology groups of a closed simply connected manifold?

It isn’t very hard to realize any sequence of abelian groups up to the middle dimension, but that middle dimension is tricky (e.g., classify $(n-1)$-connected $2n$-manifolds).

Anyway, I was wondering: is this realization question solvable for homology with coefficients in $\Z/2\Z$ or $\Q$?

For $X$ a metric space, and $S \subset X$, define the length spectrum of S to be $D_S := { d(x,y) : x, y \in S }$. It might be better to call this the “distance spectrum” or “distance set.”

Ian, during his Pizza seminar, gave the following definition: a set $S \subset \R^n$ is a $k$-distance set if $D_S$ has cardinality no greater than $k$. In words, the distances between points in a $k$-distance set take on no more than $k$ possible values.

The question that Ian answered is the following: how big can a $k$-distance set in $\R^n$ be? Clever linear algebra shows that the size grows polynomially in $n$ with degree $k$. A related exercise is the following: suppose $S \subset \R^n$ and $D_S$ is countable; prove that $S$ is countable.

Now here is my question: suppose $S \subset \R^n$ and $D_S$ is measurable with measure $m$. Can one then bound the measure of $S$? Ian asked this for the counting measure, but presumably one can get results for Lebesgue measure. Likewise, one can ask this for spaces other than $\R^n$.

All this talk of spectra has gotten me thinking very vaguely about a bunch of stuff–some random ideas! One context in which I have seen spectra is for lattices in Lie groups; I don’t know, but definitely ought to know how much control the spectrum exerts on the lattice. As a baby example, it is true that one can recover a lattice $\Lambda \subset \R^2$ from its length spectrum? Similarly, a Riemannian manifold has a length spectrum, and the “marked length rigidity conjecture” asks how much of the Riemannian structure is related to this. For information:

Furman, Alex. Coarse-geometric perspective on negatively curved manifolds and groups. 2002. 149–166. MR.

Finally, it is possible to define a “spectral distance” (I’m mis-using so many word here!) between two lattices in a Lie group. Namely, given $\Lambda_1, \Lambda_2 \subset G$, define $d(\Lambda_1, \Lambda_2) = d_{GH}(D_{\Lambda_1}, D_{\Lambda_2})$, i.e., the Hausdorff distance between their spectra. Though you’d probably want something slightly more refined (to count multiplicities). You could likewise say that two manifolds are “nearly isospectral” if their spectra are not so far apart in Gromov-Hausdorff distance. I have no idea whether this is a good idea; it probably isn’t. In any case,

Sunada, Toshikazu. Riemannian coverings and isospectral manifolds. Ann. of Math. (2) 1985. 169–186. MR.

constructs isospectral manifolds, and it would be interesting to know how easy it is to construct nearly isospectral manifolds. A braver person than I might conjecture that two manifolds are isospectral if they are $\epsilon$-nearly isospectral for small enough $\epsilon$.

At last, can one detect arithmeticity of a lattice from its spectrum? I suppose if I were really hip, I would ask: can a geometer hear arithmeticity? I think Sunada’s examples are all arithmetic?