Campus Only Senior Thesis
Bachelor of Science
Francis Edward Su
2021 Max K Chao-Haft
This thesis explores possible sink-source-cycle theorems for discrete vector fields on 3-dimensional polytopal complexes. We begin with an intuitive introduction to the Poincaré-Hopf index theorem and a detailed review of L. Glass’s graph-theoretic analogue of the Poincaré-Hopf index. This transitions to a discussion of the “filled board theorem” – a sink-source-cycle theorem in two dimensions – first discovered by R. Alvarado, M. Averett, B. Gaines, C. Jackson, M.L. Karker, M.A. Marciniak, F. Su, and S. Walker and later reproved, via a discrete index theorem, by S. Ammons. We attempt to extend this result to three dimensions and analyze the obstructions we encounter.
Chao-Haft, Max, "Discrete Vector Fields on Polytopal Complexes" (2021). HMC Senior Theses. 257.
This thesis is restricted to the Claremont Colleges current faculty, students, and staff.