Random Topology

Abstract

A recent area at the interface of probability/combinatorics and topology/geometry has been random topology. The idea is to understand the topological properties of random geometric and combinatorial structures. We will explore the topology and geometry of different random simplicial and cubical complex models. Many ideas developed for random graph models will be extended to higher dimensional notions, for example, connectedness can be thought of as Betti so one can ask about cycles in terms of Betti 1. We will explore two models of random subcomplexes of the regular cubical grid: percolation clusters, and the Eden Cell Growth model. We will also study a more combinatorial model, the fundamental group of random 2-dimensional subcomplexes of an n-dimensional cube. We will also examine the properties of the Linial-Meshulam model for simplicial complexes, an extension of the Erdos-Renyi random graph model. We will discuss the notion of a giant component for the Linial-Meshulam as well as the properties of minimum spanning cycles. On the road, we will understand some algorithms that were developed and used for analyzing and doing computational experiments on these models. The probabilistic method and ideas such as branching processes will be used extensively.

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Regular lectures: Winter semester 2022/2023

09.12.2022, 02:30