Most recently my work is focused on applied algebraic topology. It is a new mathematical discipline studying mathematical problems arising in engineering, statistics and computer science. Applied algebraic topology also uses topological techniques to solve problems in various scientific and industrial applications (such as data analysis, computer vision etc).
In my recent work I studied discontinuities appearing in robot motion planning algorithms. I introduced a topological invariant of configuration spaces, TC(X), which measures their navigational complexity. With S. Tabachnikov and S. Yuzvinsky we discovered an unexpected link between the theory of robot motion planning algorithms and the classical immersion problem for real projective spaces; a popular article about this work was published in [D. Davis, "Algebraic Topology: There's an App for That", Math Horizons, September 2011]. These results were developed further by J. Gonzalez and P. Landweber who studied a symmetric version of TC(X) introduced earlier in my joint work with M. Grant.
Jointly with my collaborators we studied configuration spaces of mechanical linkages playing an important role in various applications (such as statistical shape theory, robotics, molecular biology). With D. Schuetz we found explicit formulae for the Betti numbers of planar polygon spaces. Jointly with J.-Cl. Hausmann and D. Schuetz, we solved a problem about varieties of linkages raised by K. Walker in 1985: we proved that these varieties are fully classified by their combinatorial invariants encoding the bar lengths (Walker's conjecture).
With T. Kappeler, we initiated a probabilistic approach to topological invariants of linkages and their configuration spaces. We proved that for linkages with large number of bars whose lengths are random, the topology of the configuration space is predictable to a large extent. More precisely, we found asymptotic behaviour of expectations of the Betti numbers of these random manifolds. In my subsequent work I established a universality phenomenon for the asymptotic values of the expectations of the Betti numbers. C. Dombri and C. Mazza developed these results further.
With A. Costa we studied topological invariants of graph groups and Eilenberg-MacLane spaces associated to random graphs. This research continued in my joint work with R. Charney: we studied automorphisms of right angled Artin groups associated to random graphs.
Some of results mentioned above are presented in my monograph "Invitation to Topological Robotics" published in 2008 by the EMS.
I am one of the organisers of the UK research network "Applied Algebraic Topology" which is supported by the LMS and the EMS, see http://homepages.abdn.ac.uk/mark.grant/pages/AATwebpage.