2 research outputs found
Persistence for Circle Valued Maps
We study circle valued maps and consider the persistence of the homology of
their fibers. The outcome is a finite collection of computable invariants which
answer the basic questions on persistence and in addition encode the topology
of the source space and its relevant subspaces. Unlike persistence of real
valued maps, circle valued maps enjoy a different class of invariants called
Jordan cells in addition to bar codes. We establish a relation between the
homology of the source space and of its relevant subspaces with these
invariants and provide a new algorithm to compute these invariants from an
input matrix that encodes a circle valued map on an input simplicial complex.Comment: A complete algorithm to compute barcodes and Jordan cells is provided
in this version. The paper is accepted in in the journal Discrete &
Computational Geometry. arXiv admin note: text overlap with arXiv:1210.3092
by other author
Contributions to Persistence Theory
This paper provides a method to calculate the bar codes of a PCD (point cloud
data) with real coefficients in Section 3. With Dan Burghelea and Tamal Dey we
developed a persistence theory which involves level sets discussed in Section
4. This paper is the Ph.D thesis written under the direction of Dan Burghelea
at OSU.Comment: arXiv admin note: text overlap with arXiv:1104.5646 by other author