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Preliminary Exam Talks
Reeb Graphs: Applications and Generalisations
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Abstract
The main topic of this talk is Reeb graphs and their applications and generalisations. I start
by giving some background material. Then I will talk about the past research on algorithms
for Reeb graph computation. I then introduce the Reeb spaces and discuss efficient
algorithms for their applications. Several related problem on dynamic Reeb graph updates
are also discussed.
In the second part of the talk, as an application of the Reeb graphs we discuss vertical and horizontal homology classes. Vertical homology classes can be computed using the Reeb graph. However, we give relations between computing horizontal Betti numbers of simplicial complexes and rank computation. This shows that in general computing Betti numbers of simplicial complexes is as hard as computing Betti numbers of matrices. This will also be true for complexes embedding in 4-space, which is of special interest to us. In the end, we present the open questions and future research directions.
Advisor(s): Pankaj Agarwal
Herbert Edelsbrunner, Sayan Mukherjee, Paul Bendich