A basic question of algebraic dynamics is whether an algebraic variety admits an endomorphism (self-map) of degree >1. For example, every abelian variety admits a multiplication by m endomorphism, which is an endomorphism of degree m2g. Similarly, projective space (and more generally, any toric variety) admits an endomorphism of degree >1. However, admitting an endomorphism is a rare property and there are not many examples beyond these! For technical reasons, we work with the notion of an int-amplified endomorphism.
Restricting to Fano varieties, we have the following conjecture: a smooth Fano variety admits an int-amplified endomorphism if and only if it is a toric variety. This conjecture is known to be true for smooth Fano varieties in dimensions 2 and 3. My main project currently is studying a generalization of this conjecture for Fano varieties with klt singularities. I have proved the conjecture for several new cases. As part of my approach I use Macaulay2 to compute local invariants of toric singularities.
Macaulay2
Macaulay2 is a computer algebra system developed by Michael Stillman and Daniel Grayson that is widely used in algebraic geometry and commutative algebra. I use Macaulay2 to do computations with toric varieties, using the NormalToricVarieties package. I am working on developing libraries in Macaulay2 for working with toric singularities.
I helped develop the ToricExtras package at the recent Utah Macaulay2 workshop.
Constructible derived categories
Derived categories are an interesting category-level invariant in algebraic geometry, with subtle connections to birational geometry. It is known, for example, that the standard flop induces an equivalence of bounded derived categories of coherent sheaves.
I show in the following note that the analogous functor in the setting of constructible derived categories (that is, bounded derived category of constructible sheaves) is never an equivalence.
[will add soon]
Computations of derived intersections
At the Cornell SPUR program, I proved some theorems on the commutative algebra of derived intersections, characterizing when the dg-algebra of functions on certain derived intersections is formal.