This research aims to shed light on three fundamental questions: (1) How do small-scale interactions combine to produce complex large-scale structure? (2) What kinds of computational tasks can be performed in a distributed system with no central control over timing? (3) What kinds of algebraic invariants can help distinguish between nonisomorphic graphs? Abelian networks tie these three questions together: From a dynamical point of view, an abelian network is a cellular automaton that produces complex patterns using local rules. From a computational point of view, an abelian network yields the same output regardless of the timing of events at its individual nodes. And from an algebraic point of view, an abelian network associates an invariant, in the form of an abelian group or commutative algebra, to its underlying graph.
What can a network of computers reliably accomplish if each computer in the network runs at a different and unpredictable speed? Answering this question would bring us closer to understanding complex networks like the brain, the economy and the internet. A related question asks what kinds of patterns can arise from simple interacting components in a situation where the order of interactions does not matter. A better understanding of pattern formation would aid in engineering desired large-scale outcomes by modifying only small-scale features of a system. This kind of problem arises, for example, in designing a road network to minimize traffic delays and in using nanotechnology to engineer stronger materials.
What can a network of computers reliably accomplish if each computer in the network runs at a different and unpredictable speed? This question can be formulated mathematically using the notion of an abelian network. In collaboration with Benjamin Bond, the PI developed the foundations of abelian networks, including describing the class of optimization problems such networks can solve arXiv:1309.3445, characterizing the finite abelian networks that halt on all inputs arXiv:1409.0169, and describing the states obtainable from arbitrarily large inputs arXiv:1409.0170. A particular example is called the abelian sandpile, an idealized model cascading sand. One reason this model is interesting is that it captures some of the mathematical features of disaster events such as avalanches, earthquakes and wildfires. The most salient of these features are heavy tails (indicating that large disasters are relatively likely) and long-range dependence (indicating that correlations develop between distant parts of the system). Among other advances, the PI arrived at a precise mathematical description of the threshold state of the abelian sandpile, which is a picture of what the system looks like when driven to the brink of instability. arXiv:1402.3283 In joint work with Wesley Pegden and Charles Smart, the PI uncovered fundamental properties of the scaling limit of the abelian sandpile on the two-dimensional square grid. This is the limit obtained by zooming out on larger and larger sandpiles. In this limit of zooming out, the mesh size of the grid becomes infinitessimal, so that one might expect the sandpile to acquire full rotational symmetry; but instead, it "remembers" that it used to live on a square grid. We now have a precise way to measure that lattice dependency, using a fractal called an Apollonian circle packing. arXiv:1208.4839, arXiv:1309.3267
