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.
2009-2010: As part of the UROP program, I mentored MIT
undergraduates Giuliano Giacaglia and Linda Zayas-Palmer in a research problem about bijections between spanning trees and
recurrent configurations of the sandpile model. Jim Propp helped supervise
the project starting in fall 2009, and we proved some
conjectures of his about local-to-global properties of
rotor-router walk. A paper is in the works. In fall 2010 Damien Jiang
joined the group and we are working on characterizing the identity
elements of sandpile groups of circulant graphs.
18.177: Stochastic Processes. In Spring 2009 I
taught four guest lectures in Scott Sheffield's graduate topics class,
covering random walks and electrical networks, uniform spanning trees, Wilson's algorithm, loop-erased walk and
convergence to SLE.
(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