Topology of random geometric complexes: a survey <s> Open problems / future directions <s> This expository article is based on a lecture from the Stanford Symposium on Algebraic Topology: Application and New Directions, held in honor of Gunnar Carlsson, Ralph Cohen, and Ib Madsen. <s> BIB001 </s> Topology of random geometric complexes: a survey <s> Open problems / future directions <s> We consider the topology of simplicial complexes with vertices the points of a random point process and faces determined by distance relationships between the vertices. In particular, we study the Betti numbers of these complexes as the number of vertices becomes large, obtaining limit theorems for means, strong laws, concentration inequalities and central limit theorems. As opposed to most prior papers treating random complexes, the limit with which we work is in the so-called ‘thermodynamic’ regime (which includes the percolation threshold) in which the complexes become very large and complicated, with complex homology characterised by diverging Betti numbers. The proofs combine probabilistic arguments from the theory of stabilizing functionals of point processes and topological arguments exploiting the properties of Mayer–Vietoris exact sequences. The Mayer–Vietoris arguments are crucial, since homology in general, and Betti numbers in particular, are global rather than local phenomena, and most standard probabilistic arguments are based on the additivity of functionals arising as a consequence of locality. <s> BIB002
We close by mentioning several possible directions for future research. • Sharper results in the thermodynamic limit. Proving strong results for expectation of Betti numbers in the critical regime remains a challenging problem. The best result so far is that if where C > 0 is some constant which depends on the underlying distribution on R d and the degree k BIB002 . It would be a breakthrough to write an explicit formula for C and we expect that the results would find applications in TDA. • Connections between the various models. Is there a model for random geometric complex which approximates the sub-level sets of the Gaussian random field? See  and BIB001 for introduction and overview of Gaussian random fields and their topological properties. • Torsion. It may be surprising that all of the results in this survey for homology do not depend on the choice of coefficients. In dimensions d ≥ 4 and higher, random geometric complexes will have torsion in integer homology. • Higher-dimensional percolation theory. All of the random geometric complexes discussed here are analogues of random geometric graphs where the number of vertices n is finite, but where n → ∞. Percolation theory is of a very different flavorone considers an infinite random graph, by taking a random subgraph of a lattice, and then analyzes large-scale structure such as whether or not an infinite component appears. Analogous lattice models