Toby Anderson

Thesis Advisos: Dr. Dagan Karp (Harvey Mudd College - Department of Mathematics), Dr. Siddarth Kannan (UCLA - Department of Mathematics)

Contact: tobanderson@hmc.edu

The Dual Boundary Complex of Moduli Spaces of Cyclic Compactifications

Moduli spaces provide a useful method for studying families of mathematical objects. We study certain moduli spaces of algebraic curves, which are generalizations of familiar lines and conics. This thesis focuses on the dual boundary complex of the moduli space of genus-zero cyclic curves. This complex is itself a moduli space of graphs and can be investigated with combinatorial methods. Remarkably, the combinatorics of this complex provides insight into the geometry and topology of the original moduli space. In this thesis, we investigate two topologically invariant properties of the dual boundary complex. We compute its Euler characteristic, and we provide a conjecture and multiple possible proof techniques for calculating its homotopy type. Finally, we briefly discuss additional questions that might provide interesting future investigation of this complex.