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Organizers |
Affine linking and winding numbers and the study of front propagation
by
Vladimir Chernov
Mathematics Department, Dartmouth College
Coauthors: Yuli B. Rudyak, Mathematics Department, University of Florida, Gainesville
Let M be an oriented n-dimensional manifold. We study the causal relations between the wave fronts W1 and W2 that originated at some points of M. We introduce a numerical topological invariant CR(W1, W2) (the so-called causality relation invariant) that, in particular, gives the algebraic number of times the wave front W1 passed through the point that was the source of W2 before the front W2 originated. This invariant can be easily calculated from the current picture of wave fronts on M without the knowledge of the propagation law for the wave fronts. Moreover, in fact we even do not need to know the topology of M outside of a part P of M such that W1 and W2 are null-homotopic in P.
We also construct the Affine winding number invariant win which is the generalization of the winding number to the case of nonzero-homologous shapes and manifolds other than R2. The win invariant gives the algebraic number of times the wave front has passed through a given point between two different time moments without the knowledge of the wave front propagation law.
The invariants described above are particular cases of the general affine linking invariant al of nonzero homologous submanifolds N1 and N2 in M introduced by us. To construct al we introduce a new pairing on the bordism groups of space of mappings of N1 and N2 into M. For the case N1=N2=S1 this pairing can be regarded as an analog of the string-homology pairing constructed by Chas and Sullivan, and it is a generalization of the Goldman Lie bracket.
Date received: May 21, 2003
Copyright © 2003 by the author(s). The author(s) of this document and the organizers of the conference have granted their consent to include this abstract in Atlas Conferences Inc. Document # cakb-29.