On the zero-in-the-spectrum conjecture

Farber, M.; Weinberger, S.

doi:10.2307/3062113

All Outputs (7)

Homology of planar polygon spaces. (2007)
Journal Article
Farber, M., & Schuetz, D. (2007). Homology of planar polygon spaces. Geometriae Dedicata, 25, 75-92. https://doi.org/10.1007/s10711-007-9139-7

Moving homology classes to infinity. (2007)
Journal Article
Farber, M., & Schuetz, D. (2007). Moving homology classes to infinity. Forum Mathematicum, 19, 281-296. https://doi.org/10.1515/forum.2007.010

Closed 1-forms with at most one zero (2006)
Journal Article
Farber, M., & Schutz, D. (2006). Closed 1-forms with at most one zero. Topology (Oxford), 45(3), 465-473. https://doi.org/10.1016/j.top.2005.06.006

We prove that in any nonzero cohomology class ξH1(M;R) there always exists a closed 1-form having at most one zero.

Topology of closed one-forms (2004)
Book
Farber, M. (2004). Topology of closed one-forms. American Mathematical Society

This is a research monograph studying topological problems related to closed 1-forms. A closed one-form is a natural generalization of a smooth functions and the book describes generalizations of the classical Morse and Lusternik - Schirelmann theori... Read More about Topology of closed one-forms.

Topological complexity of motion planning (2003)
Journal Article
Farber, M. (2003). Topological complexity of motion planning. Discrete & Computational Geometry, 29(2), 211-221. https://doi.org/10.1007/s00454-002-0760-9

In this paper we study a notion of topological complexity TC(X) for the motion planning problem. TC(X) is a number which measures discontinuity of the process of motion planning in the configuration space X . More precisely, TC(X) is the minimal numb... Read More about Topological complexity of motion planning.

Topology of billiard problems, II (2002)
Journal Article
Farber, M. (2002). Topology of billiard problems, II. Duke Mathematical Journal, 115(3), 559-621. https://doi.org/10.1215/s0012-7094-02-11535-x%2C+10.1215/s0012-7094-02-11536-1

Part I. Let T⊂Rm+1T⊂Rm+1 be a strictly convex domain bounded by a smooth hypersurface X=\partialTX=\partialT. In this paper we find lower bounds on the number of billiard trajectories in TT which have a prescribed initial point A∈XA∈X, a prescribed f... Read More about Topology of billiard problems, II.

On the zero-in-the-spectrum conjecture (2001)
Journal Article
Farber, M., & Weinberger, S. (2001). On the zero-in-the-spectrum conjecture. Annals of Mathematics, 154(1), 139 - 154. https://doi.org/10.2307/3062113

We prove that the answer to the "zero-in-the-spectrum" conjecture, in the form suggested by J. Lott, is negative. Namely, we show that for any $n\ge 6$ there exists a closed $n$-dimensional smooth manifold $M^n$, so that zero does not belong to the s... Read More about On the zero-in-the-spectrum conjecture.