My three main subjects are toric topology, combinatorial optimization, and operad theory. I work with the following objects:

Toric topology

Toric manifolds with small Picard number. This research focuses on toric manifolds, which correspond to complete nonsingular fans in the n-real space. In particular, we use the wedge operation on simplicial complexes to study toric manifolds with small Picard number, that is, whose associated fans have only slightly more rays than their dimension.

[slides] — with Suyoung Choi, Hyeontae Jang

Combinatorial optimization

Box-integrality and integer decomposition properties. Box-integrality is a property of some integer polyhedron to remain integer after a truncation with any integer box. Integer decomposition properties is the study of integer points in convex polyhedra and how to find “minimal” generators of these integer points. I study integer decomposition properties of polyhedra satisfying box-integrality, using discrete geometry, triangulation of polyhedra, box-total dual integrality, and a class of matrices that generalizes totally unimodular matrices: totally equimodular matrices.

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

Operad theory

Set theoretic operads. The purpose of operads is to encode universally all the operations acting on any category of algebras. I study operads on sets and how they behave with respect to different set theoretic operations, with a specific focus on operads on simplicial complexes. They appear in the theory of polyhedral products, which helps building topological models for toric manifolds.

[slides]

Ph.D. thesis