I work in polyhedral geometry, at the crossroads of combinatorics, topology and algebra.

Tropical geometry

Mukai Grassmannians and Mukai’s model for curves, tropically. Tropicalization turns an algebraic variety into a polyhedral complex that retains much of its geometry. We study the tropicalization of Mukai Grassmannians.

Ongoing: with Michael Joswig, Kevin Kuehn, Bernd Sturmfels.

Toric topology

Toric manifolds of small Picard number. Toric manifolds correspond to complete nonsingular fans, and their Picard number is the number of rays minus the dimension. We use the wedge operation on simplicial complexes to classify PL spheres and toric manifolds of small Picard number.

[slides] — with Suyoung Choi, Hyeontae Jang

Ongoing: enumeration of PL spheres and toric manifolds of small Picard number, with Suyoung Choi.

Combinatorial optimization

Box-integrality and integer decomposition. A polyhedron is box-integral if it stays integral after intersecting it with any integer box. I study the integer decomposition properties of such polyhedra via triangulations, box-total dual integrality, and totally equimodular (TE) matrices, a generalization of totally unimodular matrices.

[slides, poster] — with Patrick Chervet, Roland Grappe

Ongoing: recognition and characterization of TE matrices, with Roland Grappe.

Operad theory

Set-theoretic operads. Operads encode the operations acting on a category of algebras. I study operads on sets, in particular on simplicial complexes, and their action on polyhedral products, which give topological models for toric manifolds.

[slides]

Ph.D. thesis