Michael Farber

Topological robotics: motion planning in projective spaces

In this paper, we study one of the most elementary problems of the topological robotics: rotation of a line, which is fixed by a revolving joint at a base point. One wants to bring the line from its initial position A to a final position B by a continuous motion in space. The ultimate goal is to construct a motion planning algorithm which will perform this task once the initial position A and the final position B are presented. This problem becomes hard when the dimension of the space is large. Any such motion planning algorithm must have instabilities, that is, the motion of the system will be discontinuous as a function of A and B. These instabilities are caused by topological reasons. A general approach to study instabilities of robot motion was suggested recently in [6, 7]. With any path-connected topological space X, one associates in [6, 7] a number TC(X), called the topological complexity of X. This number is of fundamental importance for the motion planning problem: TC(X) determines character of instabilities which have all motion planning algorithms in X. The motion planning problem of moving a line in R n+1 reduces to a topological problem of calculating the topological complexity of the real projective space TC(RP n ), which we tackle in this paper. We compute the number TC(RP n ) for all n ≤ 23 (see

Homological algebra of Novikov-Shubin invariants and morse inequalities

It is shown in this paper that the topological phenomenon“zero in the continuous spectrum”, discovered by S.P. Novikov and M.A. Shubin, can be explained in terms of a homology theory on the category of finite polyhedra with values in a certain abelian category. This approach implies homotopy invariance of the Novikov-Shubin invariants. Its main advantage is that it allows the use of the standard homological techniques, such as spectral sequences, derived functors, universal coefficients etc., while studying the Novikov-Shubin invariants. It also leads to some new quantitative invariants, measuring the Novikov-Shubin phenomenon in a different way, which are used in the present paper in order to strengthen the Morse type inequalities of Novikov and Shubin [NSh2].

Topology of cyclic configuration spaces and periodic trajectories of multi-dimensional billiards

We give lower bounds on the number of periodic trajectories in strictly convex smooth billiards in R m+1 for m ? 3. For plane billiards (when m = 1) such bounds were obtained by Birkho5 in the 1920s. Our proof is based on topological methods of calculus of variations — equivariant Morse and Lusternik– Schnirelman theories. We compute the equivariant cohomology ring of the cyclic con guration space of

Determinant lines, von Neumann algebras and

In this paper, we suggest a construction of determinant lines of finitely generated Hilbertian modules over finite von Neumann algebras. Nonzero elements of the determinant lines can be viewed as volume forms on the Hilbertian modules. Using this, we study both

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torsion

combinatorial and

Novikov type inequalities for differential forms with non-isolated zeros

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Robot motion planning, weights of cohomology classes, and cohomology operations

analytic torsion invariants associated to flat Hilbertian bundles over compact polyhedra and manifolds; we view them as volume forms on the reduced

Topology of closed 1-forms and their critical points

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Topology of configuration space of two particles on a graph, II

homology and cohomology. These torsion invariants specialize to the the classical Reidemeister-Franz torsion and the Ray-Singer torsion in the finite dimensional case. Under the assumption that the