Challenges of Applied Algebraic Topology

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In this research we shall address mathematical challenges of understanding the topological properties of configuration spaces associated with linkages of new types, including linkages with telescopic legs and fixed-angle linkages, as well as multidimensional linkages which arise in algebraic geometry and mathematical physics and in some problems of statistics. The configuration spaces of linkages with telescopic legs (i.e. legs having variable lengths) are generically manifolds with corners and we plan using Morse theory techniques for finding relations between the metric parameters of linkages and the topological invariants of their configuration spaces. The fixed-angle linkages and their configuration spaces provide a good approximation to the variety of shapes of protein backbones and therefore information about topological properties of these spaces can be potentially used in the protein folding problem. Configuration spaces of multidimensional linkages are generically manifolds with singularities; their study represents significant mathematical challenges and requires new mathematical tools. We also plan to apply mixed probabilistic-topological techniques and study topological invariants of linkages (of various types) with random length parameters under the assumption that the number of bars of the linkage is large (tends to infinity). We hope to be able to generalise the previously obtained results of this type to new important classes of linkages. As part of this research we will also use the methods and results of applied algebraic topology to tackle a well-known classical topological problem known as the Whitehead conjecture. It was raised by J.H.C. Whitehead in 1941 and remains open despite multiple attempts of mathematicians. The Whitehead conjecture claims that a subcomplex of an aspherical 2-dimensional complex is also aspherical. Our recent results (2012) prove a probabilistic version of the conjecture. More precisely, we showed that aspherical 2-complexes produced randomly satisfy the Whitehead conjecture with probebility tending to one. In this research we shall try to exploit these probabilistic results hoping to obtain a full deterministic solution to the conjecture.