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Darin Stephenson's Research

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Noncommutative Algebraic Geometry: There is a strong connection between the theory of commutative rings and classical algebraic geometry. The starting point for this theory is the connection between polynomials and the geometry of their solution sets in space. For example, in two variables x and y, one can study an algebraic object such as the polynomial p(x,y) = y - x2, or the associated geometric object given by the solution set of y - x2 = 0 in the xy-plane. This idea leads to many interesting parallels between the study of ideals in the polynomial ring k[x,y] and the associated geometry of the plane.

In essence, my research objective is to understand noncommutative versions of this picture. In the classical example above, the variables x and y are understood to "commute" with respect to multiplication, that is, yx = xy. But what if this were not the case? Perhaps, instead, x and y could satisfy some other multiplication rule, like

yx = qxy,
where q is a number that is not equal to 1, or perhaps
yx = xy + x2.
Would there still be a natural geometry associated to the new algebraic structure, and, if so, what would it look like?

The task, therefore, is to understand noncommutative versions of commutative polynomial rings in n variables (such as the so-called Artin-Schelter regular algebras of global dimension n) that are amenable to the methods of algebraic geometry. Currently, my work is focused on

  • Understanding the geometry of Artin-Schelter regular algebras of global dimension 3 which are not generated by elements of degree 1.
  • Constructing examples of Artin-Schelter regular algebras of global dimensions 4.

Geometric Probability: Lately, I've also been interested in questions related to geometric probability. The following classical problem has served to motivate my study. Consider a unit circle in the plane, and randomly (uniformly and independently) choose three points on its circumference. What is the average area and perimeter of the triangle formed by these three points? This question has many interesting variants and extensions that provide nice opportunities for undergraduate research in mathematics. In the summer of 1999, my students studied methods of randomly generating triangles of fixed perimeter. In the summer of 2005, my students studied methods of generating triangles by randomly choosing three vertices of a regular polygon.

Curriculum Development: I am currently writing a textbook to be used in a one-year course covering the topics of linear algebra, differential equations, and multivariable calculus.