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Computational Topology for Configuration Spaces of Disks in a Torus

Ritchey, Katherine
http://rave.ohiolink.edu/etdc/view?acc_num=osu1562945889197152

Abstract Details

2019, Doctor of Philosophy, Ohio State University, Mathematics.
In this thesis, we will look at how computational methods can be used to better understand the topology of configuration spaces of non-overlapping hard disks of equal radius. We would like to discern how the topology of these configuration spaces changes as the radius varies. This is a computationally tractable model for liquid-solid phase transitions. We will focus specifically on configuration spaces in the equilateral torus. A result by Baryshnikov, Bubenik, and Kahle shows that the topology of these configuration spaces can only change at "critical," or mechanically-balanced, configurations. We would therefore like to have a better understanding of the critical configurations that can arise. In the first few chapters, we discuss what is currently known about these configuration spaces and describe computational experiments performed to find thousands of critical configurations in the torus for small numbers of disks. In the second half of this thesis, we study our list of known critical configurations and their properties in more detail. We describe notions of index and non-degeneracy in terms of critical configurations. We also discuss computational methods used to classify the symmetries of each critical configuration. In addition, we mention some ways to find new critical configurations from known critical configurations. Finally, we describe computational methods used to reconstruct Morse graphs and to understand the path-connectedness of these configuration spaces as the radius changes for fixed numbers of disks.
Matthew Kahle (Advisor)
Jean-Francois Lafont (Committee Member)
Maria Angelica Cueto (Committee Member)
132 p.
Mathematics

Recommended Citations

Citations

  • Ritchey, K. (2019). Computational Topology for Configuration Spaces of Disks in a Torus [Doctoral dissertation, Ohio State University]. OhioLINK Electronic Theses and Dissertations Center. http://rave.ohiolink.edu/etdc/view?acc_num=osu1562945889197152

    APA Style (7th edition)

  • Ritchey, Katherine. Computational Topology for Configuration Spaces of Disks in a Torus. 2019. Ohio State University, Doctoral dissertation. OhioLINK Electronic Theses and Dissertations Center, http://rave.ohiolink.edu/etdc/view?acc_num=osu1562945889197152.

    MLA Style (8th edition)

  • Ritchey, Katherine. "Computational Topology for Configuration Spaces of Disks in a Torus." Doctoral dissertation, Ohio State University, 2019. http://rave.ohiolink.edu/etdc/view?acc_num=osu1562945889197152

    Chicago Manual of Style (17th edition)

osu1562945889197152
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This open access ETD is published by The Ohio State University and OhioLINK.