Maverick Science mag 2013-14

Michaela Vancliff thrives
while focusing her research
and teaching on a branch
of algebra that even many
other mathematicians find
to be mystifying. By Greg Pederson
Vancliff discusses a problem with students in her
Graduate Algebraic Geometry class last fall.
Brandon Wade
Abstract artist
T
he subject of non-commutative algebraic geometry is likely to be baffling or even slightly intimidating to those
unfamiliar with advanced concepts in algebra. For Michaela Vancliff, it’s the subject she has devoted her career
to studying. Vancliff has been at the forefront of research in non-commutative algebraic geometry since it first
came on the scene in the late 1980s.
So, what is it, exactly?
First, a bit of a primer on some mathematical terms
might be in order. Addition is an example of a “commutative”
operation, because a + b equals b + a. Subtraction
is an example of a “non-commutative” operation, because
generally, a – b does not equal b – a.
Mathematics, in general, is the study of patterns and,
frequently, such patterns are described via systems of
equations, Vancliff explains. For instance, systems of
polynomial-style equations and their solutions play a
critical role in almost every scientific field, including elementary-particle
physics, quantum mechanics, robotics,
crystallography and more.
“Often, the solutions cannot be found by experimentation,
and often they are not numbers but are functions
and so, in general, they do not commute,” said Vancliff,
a UT Arlington professor of mathematics. “Non-commutative
algebra has application to fields such as
physics and chemistry. The science of seeking methods
42 Maverick Science 2013-14
that find all solutions to a system of polynomial-style
equations in non-commuting variables is non-commutative
algebra.
“The problem of solving a system of equations in
non-commutative algebra may be translated to one involving
an algebra over a field, and the representation
theory (or module theory) of that algebra. My research
is in the subarea of non-commutative algebraic geometry,
which is about using geometric methods to understand
the algebra and its representation theory that
arise in this way.”
The main idea to finding solutions to a system of
polynomial-style equations is to associate an algebraic
object, called a ring, which encodes all the properties of
the original equations, Vancliff said. Associated to this
ring are modules, which encode all the properties of the
solutions to the equations. So, in order to find all of the
solutions, one should find all the modules for the associated
ring. In many of the applications, the rings tend
to share certain properties satisfied by commuting polynomials.
Such rings are called AS-regular algebras and
are the main focus of Vancliff’s research.
“One of the goals of the study of AS-regular algebras
and their modules is to use geometric techniques to find
certain modules of the AS-regular algebra, and then to
use those modules to find the modules that give the solutions
to the original system of equations,” she said.
“My underlying goal throughout my research career has
been to improve on these geometric techniques. Being
an algebraist, I don’t work on the physics that generates
the equations that need to be solved; nor do I work on
the equations themselves.
“Typically, mathematical physicists translate the
quantum physics into algebraic problems, and then an
algebraist picks up the problem at that stage. In my case,
I work on techniques that solve types of equations, in