Research Interests

My research is in the area of abstract polytopes, which are combinatorial generalizations of convex polytopes, surface tilings, and space-filling tessellations. This vibrant field brings together combinatorics, group theory, geometry, and topology. The bulk of the research focuses on regular polytopes, which have the highest degree of symmetry. I work primarily with these and with chiral polytopes, which are "half-regular" in a particular way. My current project is the classification of "tight" polytopes, which are the smallest polytopes with a given set of combinatorial data. For example, what is the smallest abstract polyhedron whose faces are p-gons and where each vertex is incident on q edges?

Dissertation

Internal and external invariance of abstract polytopes

Publication List

Talks

  • The search for small polytopes, 2da Escuela en Simetrías de Estructuras Combinatorias, UNAM, Cuernavaca, Mexico, July 2017.
  • Tight chiral polyhedra, AMS Special Session on Geometry and Combinatorics of Polytopes, Rutgers University, New Brunswick, NJ, November 2015.
  • The search for tight polytopes, Discrete Geometry Seminar at Northeastern University, Boston, MA, March 2014.
  • Orbit graphs and transitivity of k-orbit polytopes, AMS Special Session on Discrete Geometry of Polytopes, Boston, MA, April 2013.
  • Minimal equivelar polytopes, MIT Combinatorics Seminar, Cambridge, MA, November 2012.
  • Minimal equivelar polytopes, Seminar in Geometry, Algebra, Singularities, and Combinatorics at Northeastern University, Boston, MA, November 2012.
  • Constructing abstract regular polytopes with external symmetries, UMass Boston Mathematics Colloquium, Boston, MA, May 2012.
  • Constructing abstract regular polytopes with external symmetries, WPI Mathematics Colloquium, Worcester, MA, January 2012.
  • Constructing self-dual and self-Petrie polyhedra, AMS Special Session on Combinatorial Geometry of Polytopes at the Joint Meetings of the AMS, Boston, MA, January 2012.
  • Constructing self-dual chiral polytopes, Workshop on Symmetry in Graphs, Maps and Polytopes at the Fields Institute, Toronto, Canada, October 2011.

CV

Research Statement

 

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