Jasper Bown

Thesis Advisor: Dr. Javier González Anaya (Harvey Mudd College - Department of Mathematics)

Second Reader: Dr. Patricio Gallardo, (University of California, Riverside - Department of Mathematics)

Contact: abown@hmc.edu

Connecting graphs, polytopes, and geometry: toric moduli from nestohedra

Projective toric geometry connects the study of geometric spaces and concrete combinatorial constructions through polytopes. 

On the geometry side, a moduli space is a geometric space where each point corresponds to a geometric object up to some equivalence. One well studied moduli space is M0,n, which describes n points in projective space up to Möbuis transformations. A related space is Td,n, the moduli space of n points in d dimensional affine space up to translation and scaling. Each of these spaces can be compactified according to a set of weights ⍵. For adequately small ⍵ these compactifications are toric and can be described by polytopes. 

On the combinatorics side, there are many constructions of polytopes such that the structure of the faces encodes combinatorial data. An important example is the associahedron which captures information about adding parentheses to a product of n elements. Devadoss and Carr introduced the graph associahedron which constructs a polytope encoding combinatorial data arising from a graph. Further generalizations to hypergraphs result in nestohedra. 

Ferreira da Rosa, Jensen, and Ranganathan studied the subset of toric compactifications of M0,n which are described by graph associahedra. My thesis uses nestohedra to examine two key generalizations:

• How can all toric compactifications of M0,n be described combinatorially?

• How can these combinatorial descriptions generalize to Td,n?