<p>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.</p>
<p>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.</p>
| Identifer | oai:union.ndltd.org:purdue.edu/oai:figshare.com:article/19586011 |
| Date | 20 April 2022 |
| Creators | Negin Karisani (12407755) |
| Source Sets | Purdue University |
| Detected Language | English |
| Type | Text, Thesis |
| Rights | CC BY 4.0 |
| Relation | https://figshare.com/articles/thesis/Applied_Topology_and_Algorithmic_Semi-Algebraic_Geometry/19586011 |
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