I study abelian networks, an invention of the statistical physicist Deepak Dhar, who
called them "abelian distributed processors."
In one point of view, abelian networks are discrete dynamical systems with a certain strong convergence property.
From another point of view, they are a means of associating algebraic invariants to graphs. In yet a third viewpoint, they are a model of asynchronous computation: nodes
of the network pass messages along edges to perform a computation requiring no central control over timing.
(with Y. Peres) Internal erosion and the
exponent 3/4 describes how an unusual exponent arises from a very simple erosion process in one dimension. The proof we give
is due to Kingman and Volkov (2003), who thought of this not as an erosion process but as a model of a
Orlik-Solomon Algebras of
Hyperplane Arrangements, an expository paper proving the basic theorem
of Orlik-Solomon and Brieskorn on the cohomology ring of the complement of
a complex arrangement, along with some remarks about the associated
combinatorics of the intersection lattice.
Confounding factors for
Hamilton's rule, the final paper for an anthropology class I took in
2002. I found that the rule is surprisingly
sensitive to changes in Hamilton's original hypotheses, which casts some
doubt on the evolutionary stability of kin selection.
A graph on n=24 vertices having no Hamiltonian cycle, in which every set of k<22 vertices
is adjacent to at least k+3 vertices:
Some basic results on Sturmian
words, written before I knew that's what they were called. These
results are all known in some form. Theorem 1 and the surprising
corollary to Theorem 2 go back to Morse and Hedlund (1940). I don't know
if Theorem 2 appears anywhere in exactly this form.
The beginning of the factor tree of the Sturmian word of slope sqrt(2)/2 and intercept zero:
Odds and Ends:
has more pixels than the population of the earth! (See here for the story of how it was