Research

My research is primarily in commutative algebra, involving questions inspired by algebraic geometry. In particular, most of my work involves studying Rees algebras/blowup algebras.

Papers/Preprints

Eddie stands in front of a projector screen at a talk. He points out two specific points on a curve on a slide discussing how to study interesting points.

What are Rees algebras?

The Rees algebra (along with the associated graded ring and the special fiber ring) are known as the “blowup” algebras. They are associated with a process in algebraic geometry called “blowing up.”

Blowing up can help resolve singularities. Consider the image below. It has a singularity at the origin - where the curve crosses over itself.

There is a curve which crosses over itself at the origin on a 2-dimensional plane.

To “blow up” and remove the singularity, imagine the curve is like a stiff wire that you can “pull apart” in 3-dimensions, giving the following curve which projects onto our original curve.

In a 3-d system, the original curve which intersects itself is displayed on the xy-plane, but there is also a 3-d curve which does not intersect itself, but which would project to the 2-d curve.

The blowup algebras help us figure out the properties of this new curve which doesn’t have singularities.

But the Rees algebra also has applications in pure commutative algebra, geometric modeling, and the study of chemical reaction networks.

It’s definitely worth studying!

There is a blue surface and an orange surface which intersects to give a black curve.

Often, we want to find nonsingular surfaces whose intersection is the blowup. The equations of such surfaces are called the “defining equations” of the blowup, and knowing the defining equations is often key for understanding the blowup in general!

A screenshot from Zoom containing three people in separate display boxes. Each person is holding up a piece of wood with the same geometric figure present.

Eddie Price (top left), Alessandra Costantini (top right), and Matt Weaver (bottom) holding the blow up of the plane (burned into wood, made by Matt for all of us).

Collaborators

Please check out my collaborators!

This list includes collaborators even for preliminary projects.