Randomness and Geometric Structures

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For the first out of four topics, the main goal is to understand the typical sphere, where "sphere" here means a two-dimensional Riemannian manifold that is topologically a sphere, and "typical" means chosen according to a canonical "uniform" probability measure. The approach is to consider a limit of measures on discrete structures, where the concept of uniformity is meaningful. The co-investigator intends to construct a limiting measure for the graph metric induced by uniformly chosen planar graphs on n vertices. Experimental and heuristic evidence suggests that the limiting measure will have the structure of a continuum tree. For the second topic, the co-investigator intends to study the implications of previous work on the relationship between random walks and geometry on graphs in the context of Brownian motion on Riemannian manifolds. The third topic concerns the stochastic Loewner evolution, introduced by Schramm as a conjectured scaling limit for percolation and loop-erased walk. These and several other conjectures concerning this process have been proved and yet others remain open. The fourth topic is probability in groups. The co-investigator intends to continue to work on automorphism groups of regular trees, which are fundamental objects of study in the theory of p-groups. All four parts of the project involve more than one area of mathematics, increasing understanding between fields. The common motivation is to understand better fundamental mathematical objects that are often of utilitarian use. In particular, the second part concerns random walks on graphs, a topic which has been applied in the design of efficient computer algorithms. The fourth part concerns randomness in groups, a topic which has been used successfully in the telecommunications industry for encryption and error-correction.