Challenges of Applied Algebraic Topology

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.

The results of this pure mathematical research are expected to be of significant interest to the mathematical community, they will be published in mathematical journals of highest standing and will be presented at important mathematical conferences worldwide.
Mathematicians of all countries will be able to use our progress and develop our results further in their subsequent work.

One of the main objectives of this research is an attempt to solve the Whitehead conjecture (which is open since 1941) using new probabilistic techniques and employing the results which were already obtained recently by my research group. Progress along these lines will be of great interest and will have significant scientific impact.

Progress with regards to the other objective of this research (topology of configuration spaces of linkages) will also have great impact on the community of researchers and may have important applications outside mathematics in the future (after being fully developed). Our research task of studying mathematically the fixed-angle linkages in 3-space is inspired by the protein folding problem and might be continued later in collaboration with biologists. However at present we impose no restriction on self-intersections of chains and thus our results should be viewed as a first approximation.

The task of studying linkages with telescopic legs requires new tools of Morse theory since the configuration spaces are manifolds with corners. A version of Morse theory for manifolds with corners is available, but applying it requires more difficult analysis which we plan to implement as part of this research. This technique will be of great interest for topologists.

Finally, the goal of studying the Betti numbers of high dimensional d>3 linkages (both in deterministic and probabilistic settings) requires a new mathematical approach since the corresponding configurations spaces generically have singularities. The results about linkages have been used in statistical shape theory and our more general results can be applied as well. Progress in this research goal will have an important scientific impact, both in mathematics and in other sciences.