Regularity in algebra, combinatorics, and geometry (26w5583)

The Banff International Research Station will host the "Regularity in algebra, combinatorics, and geometry" workshop in Banff from April 19 - 23, 2026.

Core objects of study in algebraic geometry and in commutative algebra are called varieties and rings, respectively. Varieties are geometric objects that consist of sets of solutions to polynomial equations. Each variety has a corresponding ring consisting of the polynomial functions that are defined on the variety.

There is a classical setting, called \emph{standard graded}, in which one can assign a positive integer called the \emph{Castelnuovo--Mumford regularity} to a ring, or to its associated variety. Roughly speaking, the Castelnuovo--Mumford regularity of a ring or variety measures how complex it is, in a sense that can be made precise computationally. While informative, the Castelnuovo--Mumford regularity of a ring or variety can be very hard to determine. Recently, there has been an explosion of combinatorial strategies for computing Castelnuovo--Mumford regularity. At the same time, but in a separate mathematical community, there is right now a great deal of interest in a refinement of standard graded rings to what are called multigraded rings, which come equipped with a multigraded Castelnuovo--Mumford regularity. The purpose of this workshop is to bring together established experts and rising stars from the two communities: commutative algebraists and algebraic geometers who would make use of multigraded Castelnuovo--Mumford regularity on the one hand and algebraic combinatorialists who have tools for computing standard graded Castelnuovo--Mumford regularity on the other. The goal is to jumpstart inquiry into combinatorial models for multigraded Castelnuovo--Mumford regularity.