438 Malott Hall
Ph.D. (2007) University of California at Berkeley
Probability and combinatorics
The broad goal of my research is to understand how and why large-scale forms and complex patterns emerge from simple local rules. My approach is to analyze mathematical models that isolate just one or a few features of pattern formation. A good model is one that captures some aspect of “scaling up” from local to global, yet is simple enough to prove theorems about! Recently I have been thinking about abelian networks, a generalization of the Bak-Tang-Wiesenfeld abelian sandpile model.
Apollonian structure in the abelian sandpile (with Wesley Pegden and Charles K. Smart), Geometric and Functional Analysis 26 (2016), no. 1, 306-336.
Threshold state and a conjecture of Poghosyan, Poghosyan, Priezzhev and Ruelle, Communications in Mathematical Physics 335 no. 2 (2015), 1003–1017.
Internal DLA and the Gaussian free field (with D. Jerison and S. Sheffield), Duke Mathematical Journal 163 no. 2 (2014), 267–308.
Equations solvable by radicals in a uniquely divisible group (with C.J. Hillar and D. Rhea), Bulletin of the London Mathematical Society 45 (2013), 61–79.
Logarithmic fluctuations for internal DLA (with D. Jerison and S. Sheffield), Journal of the American Mathematical Society 25 (2012), 271–301.
Parallel chip-firing on the complete graph: devil's staircase and Poincaré rotation number, Ergodic Theory and Dynamical Systems 31 (2011), 891–910.
Driving sandpiles to criticality and beyond (with A. Fey and D. B. Wilson), Physical Review Letters 104 (2010), 145703.