Research

My research interests lie primarily in tropical geometry and toric varieties. At present, I am especially interested in an important class of tropical cycles known as regular cycles. Briefly, a tropical k-cycle [F] is called regular if there exists a k-tuple of global PL convex functions (T_1, ..., T_k) such that the tropical intersection product has degree 1.

Regular tropical cycles are beautiful from several perspectives:

1. Fundamental examples arise from Esterov–Gusev's Mixed Volume 1 Classification Theorem.

2. Further fundamental examples come from Alex Fink's Classification Theorem.

3. The interplay between the underlying algebraic variety and its tropical compactification (v.s. wonderful compactifiaction).

4. The ''Tropical Minimal Model Program'' — a conjectural framework for classifying such regular tuples.

5. The associated toric data, toric Chow ring and Minkowski weights.

6. Special examples: Bergman fans, Grassmannian, hyperplane arrangements.

7. The algorithmn for classifying 2-dim regular tropical fans on R^4.

8. ...

Publications & Preprints:

1. Tropical fans supporting a reduced 0-dimensional complete intersection (arXiv).

2. Toward a classification of Tropical complete intersection number one (arXiv).