Research

My research is in the broad area of Discrete Mathematics, including probabilistic combinatorics, combinatorical game theory, graph theory, and discrete geometry.

My recent focus has been on the Abelian sandpile, and problems related to the Local Lemma. My paper with Charles Smart on the convergence of the Abelian Sandpile gives a new paradigm—based on the set Γ—for understanding the sandpile's limiting process. Recent work with Lionel Levine and Charles Smart has used this paradigm to analyze the fractal geometry of the sandpile process, through a computationally established Apollonian property of the integer superharmonic functions on the integer lattice. In more recent work, we prove this Apollonian property underlying the Sandpile process. Future directions include the possibility of establishing much stronger notions of convergence,

My work with the Lefthanded Local Lemma was the first successful application of a Local Lemma to games, made possible by the fact that the Lemma allows one to ignore all problematic dependencies in a dependency graph in "one direction", eliminating those which normally undermine such an application to games. More recently, I have shown that the Lefthanded Local Lemma is best possible for the graphs to which it applies in a surprising way (which never holds for the normal Lovàsz Local Lemma).

Other topics of research have included:

• Optimum density contructions of k-chromatic-critical triangle-free graphs (k≥ 6), giving the first nontrivial families of graphs where we can determine the optimum density of critical members
• (with J. Beck and S. Vijay) Exponential lower bounds on the Hales-Jewett number and the `halving' Hales-Jewett number, which imply the existence of infinitely many Tic-Tac-Toe games which cannot be solved by Ramsey theory
• A surprising characterization of Euclidean shapes resilient to an erosion operation, including certain convex sets and certain `fractal' shapes
• A finite goal set in the plane, which a first player cannot build before a perfectly-playing second player in a Euclidean game, as a counterpoint to a suprising theorem of Beck which asserts that any finite goal set can be built by the first player (not necessarily before the second player)
• The expansion of vertex-transitive graphs of infinite degree, which, unlike transitive graphs of finite-degree, always have unimodal distance sequences.