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RESEARCH STATEMENT: DYNAMICS AND COMPUTATION IN ABELIAN NETWORKS

By Lionel Levine

Abstract

The broad goal of my research is to understand how large-scale forms and complex patterns emerge from simple local rules. My approach is to analyze specific mathematical models which 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 tractable enough to prove theorems about! Some of the models I’m working on include • Abelian sandpiles [BTW87], a model of self-organization and pattern formation. • Parallel chip-firing [BG92], a model of mode-locking and synchronization. • Internal DLA [LBG92], a model of fluid flow and random interfaces. • Rotor-router aggregation [LP09], a model based on derandomizing random walks. Abelian networks, invented by Deepak Dhar [Dha06], tie these models and many others together in a common mathematical framework. The mathematics involved is a mixture of probability and combinatorics, draws on techniques that originated in the study of partial differential equations, and has close connections with statistical physics and computer science. Figure 1. Pattern formation in a stable sandpile (left) and an exploding sandpile (right) in Z 2. Each site is colored according to the number of sand grains present

Year: 2011
OAI identifier: oai:CiteSeerX.psu:10.1.1.185.3304
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