Font Size:
Characteristic classes that cut equal areas and equal perimeters and prevent highly regular embeddings
Last modified: 2014-03-09
Abstract
The properties of the regular representation bundles over the configuration space of k distinct points in the Euclidean space has classically been studied extensively by F. Cohen, Chisholm, V. Vassiliev, and many others.
Motivated by geometric problems we present new computations of twisted Euler classes, Stiefel-Whitney classes and their monomials as well as corresponding Chern classes of these bundles.
Thus, we not only extend and complete previous work, supplying for example a proof for a conjecture by Vassiliev, but also make progress in solving and extending variety of problems from Discrete Geometry, among them
(i) the conjecture by Nandakumar and Ramana Rao that every convex polygon can be partitioned into n convex parts of equal area and perimeter;
(ii) Borsuk's problem on the existence of "k-regular maps" between Euclidean spaces, which are required to map any k distinct points to k linearly independent vectors;
(iii) Ghomi and Tabachnikov problem about the existence of "l-skew smooth embeddings" from a smooth manifold M to a Euclidean space E, which are required to map tangent spaces at l distinct points of M into l skew subspaces of E .
(This lecture is based on joint work with Frederick Cohen, Wolfgang Lueck and Gunter M. Ziegler)
Motivated by geometric problems we present new computations of twisted Euler classes, Stiefel-Whitney classes and their monomials as well as corresponding Chern classes of these bundles.
Thus, we not only extend and complete previous work, supplying for example a proof for a conjecture by Vassiliev, but also make progress in solving and extending variety of problems from Discrete Geometry, among them
(i) the conjecture by Nandakumar and Ramana Rao that every convex polygon can be partitioned into n convex parts of equal area and perimeter;
(ii) Borsuk's problem on the existence of "k-regular maps" between Euclidean spaces, which are required to map any k distinct points to k linearly independent vectors;
(iii) Ghomi and Tabachnikov problem about the existence of "l-skew smooth embeddings" from a smooth manifold M to a Euclidean space E, which are required to map tangent spaces at l distinct points of M into l skew subspaces of E .
(This lecture is based on joint work with Frederick Cohen, Wolfgang Lueck and Gunter M. Ziegler)