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# Two Problems in Applied Topology

In this thesis, we present two main results in applied topology.

In our first result, we describe an algorithm for computing a semi-algebraic description of the quotient map of a proper semi-algebraic equivalence relation given as input. The complexity of the algorithm is doubly exponential in terms of the size of the polynomials describing the semi-algebraic set and equivalence relation.

In our second result, we use the fact that homology groups of a simplicial complex are isomorphic to the space of harmonic chains of that complex to obtain a representative cycle for each homology class. We then establish stability results on the harmonic chain groups.

## History

## Degree Type

- Doctor of Philosophy

## Department

- Mathematics

## Campus location

- West Lafayette