Random spaces and groups

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We will study topological and geometric invariants of various random objects.

On one hand, randomness is useful in modeling nature. So various kinds
of stochastic topology and geometry are relevant to physics,
topological data analysis, etc. On the other hand, the probabilistic
method allows us to prove "existence" theorems via measure-theoretic
arguments, in many cases when no constructions are known.

We will survey this area, including recent and ongoing work, and point
out many opportunities for future research directions along the way.

Topics will include:

––1-dimensional examples: Thresholds for connectivity and the
appearance of cycles in random graphs (Erdős–Rényi model, random
geometric graphs)
––higher-dimensional examples: Homology of random simplicial complexes
(abstract and geometric models)
––Hyperbolicity and property (T) of random groups (Gromov density
model, triangular model, random fundamental groups)
––Genus of random surfaces
––Random 3-manifolds (Dunfield–Thurston model)
––Other models of random metric spaces and applications
(Johnson–Lindenstrauss lemma, Bourgain's metric distortion theorem,
Vershik's random metric spaces)