2:00 pm Wednesday, November 22, 2017
Junior Topology: Random Simplicial Complexes in R^n by Gill Grindstaff in RLM 12.166
Classically, the Erdős-Rényi method provides a way to generate random graphs by choosing edges of a complete graph with some positive probability. This has a natural extension to random simplicial complexes, which has been covered extensively. Recently, the work of M. Kahle and some of our friends from probability have provided a different construction of a random simplicial complex, using persistence complexes over point processes in R^n. I will briefly review TDA, provide an overview of point processes, and give some results about the asymptotic Betti numbers of such a random complex, which vary according to the type of point process. This has implications for statistical testing in topological data analysis of manifolds, where the persistent homology of a finite point set is assumed to represent topological features of the underlying space and not the sampling method, and is of independent interest in stochastic geometry, where topological features of random objects (e.g. wireless networks or synthetic materials) often have a practical impact. For this talk, I will focus on more topological aspects of the calculations, specifically how the Betti numbers of a geometric simplicial complex can be bounded by subgraph counts in the 1-skeleton. Submitted by
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