### 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

*Non-flat regular polytopes and restrictions on chiral polytopes*, preprint.*Internal and external duality in abstract polytopes*(with Mark Mixer), Contributions to Discrete Mathematics, to appear.*Open problems on k-orbit polytopes*(with Daniel Pellicer), preprint.*Tight chiral polyhedra*, Combinatorica, to appear.*Classification of tight regular polyhedra*(with Daniel Pellicer), Journal of Algebraic Combinatorics,**43**(2016) no. 3, 665-691.*Tight orientably-regular polytopes*(with Marston Conder), Ars Mathematica Contemporanea,**8**(2015), 68-81.*Symmetry type graphs of polytopes and maniplexes*(with María Del Río Francos, Isabel Hubard, and Micael Toledo), Annals of Combinatorics,**19**(2015), 243-268.*Chiral extensions of chiral polytopes*(with Daniel Pellicer), Discrete Mathematics,**330**(2014), no. 1, 51-60.*Minimal equivelar polytopes*, Ars Mathematica Contemporanea,**7**(2014), no. 2, 299-315.*Variance groups and the structure of mixed polytopes*, Rigidity and Symmetry, Fields Inst. Comm.,**70**, Springer, New York (2014), 97-116.*Constructing self-dual chiral polytopes*, European Journal of Combinatorics,**33**(2012), no. 6, 1313-1323.*Self-Dual, Self-Petrie Covers of Regular Polyhedra*, Symmetry**4**(2012), no. 1, 208–218.*Mixing chiral polytopes*, Journal of Algebraic Combinatorics,**36**(2012), no. 2, 263-277.*Mixing regular convex polytopes*, Discrete Mathematics,**312**(2012), no. 6, 763–771.

### 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.