Michael Farber

This research will address problems of applied algebraic topology and will develop mathematical methods relevant to various applications in engineering, biology, statistics and network science. We shall also employ methods of applied algebraic topology to tackle some classical pure topological open questions (the Whitehead Conjecture).

The first part of this project studies various manifolds of linkages; our efforts will be focused on analysing new classes of linkages (linkages with telescopic legs, fixed-angle linkages, linkages in high-dimensional spaces, as well as topological invariants of manifolds of linkages with random length parameters), which require development of novel mathematical tools. The second part of this research is an attempt to exploit our recent results about probabilistic confirmation of the Whitehead conjecture to obtain a deterministic solution of this classical open problem.

The mathematical theory of random simplicial complexes which we plan to develop in this research, may find applications in the theory of large networks. Traditionally one models networks by graphs with nodes representing objects and edges representing connections between the objects. However if we are interested not only in pairwise relations between the objects but also in relations between multiple objects we may use the high dimensional simplicial complexes instead of graphs as mathematical models of networks.