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Applied Topology and Algorithmic Semi-Algebraic Geometry

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thesis
posted on 10.05.2022, 14:24 authored by Negin KarisaniNegin Karisani

Applied topology is a rapidly growing discipline aiming at using ideas coming from algebraic topology to solve problems in the real world, including analyzing point cloud data, shape analysis, etc. Semi-algebraic geometry deals with studying properties of semi-algebraic sets that are subsets of $\mathbb{R}^n$ and defined in terms of polynomial inequalities. Semi-algebraic sets are ubiquitous in applications in areas such as modeling, motion planning, etc. Developing efficient algorithms for computing topological invariants of semi-algebraic sets is a rich and well-developed field.

However, applied topology has thrown up new invariants---such as persistent homology and barcodes---which give us new ways of looking at the topology of semi-algebraic sets. In this thesis, we investigate the interplay between these two areas. We aim to develop new efficient algorithms for computing topological invariants of semi-algebraic sets, such as persistent homology, and to develop new mathematical tools to make such algorithms possible.

Funding

DMS-1620271

History

Degree Type

Doctor of Philosophy

Department

Computer Science

Campus location

West Lafayette

Advisor/Supervisor/Committee Chair

Saugata Basu

Additional Committee Member 2

Tamal K. Dey

Additional Committee Member 3

Petros Drineas

Additional Committee Member 4

Elena Grigorescu

Additional Committee Member 5

Hemanta K. Maji