Mathematics
D epartment
H ighlight
Srikanth Iyengar – What is Commutative Algebra?
What is commutative algebra? Can it help us solve
M ichele
S waner
to study these objects. One of them is to study functions
motivated by questions in physics.
real-world problems?
on the space of solutions, and algebra is a good way to
begin. These functions form a mathematical structure
began under the name “ideal theory” with the work of
solve linear and quadratic equations, generally a single
called a commutative ring. Commutative algebra
mathematician Richard Dedekind, a giant of the late
equation and in one variable,” said Srikanth Iyengar,
is the study of commutative rings and modules, or
19th and early 20th centuries. In turn, Dedekind’s work
Professor of Mathematics at the U. “But most real-world
algebraic structures over such rings.
relied on the earlier work of Ernst Kummer and Leopold
“When we first study advanced math, we learn to
problems aren’t quite so easy—they often involve multiple equations in mul-
“One of the things I
As a subject on its own, commutative algebra
Iyengar’s research
Kronecker. The mathematician responsible for the
focuses on understanding
modern study of commutative algebra was Wolfgang
these structures, which
Krull, who introduced concepts that are now central to
have links to different
the study of the subject, as well as Oscar Zariski, who
tiple variables.”
enjoy about my research is
how commutative algebra
areas of mathematics,
made commutative algebra a foundation for the study
has so many connections
particularly topology and
of algebraic varieties.
representation theory.
Iyengar joined the
how commutative algebra has so many connections
Mathematics Department
to other things,” said Iyengar. “It makes for rich and
in 2014. He grew up in
lively research. Commutative algebra is continually
Hyderabad, India, and
reinvigorated by problems and perspectives from
received a master’s degree
other fields.”
and Ph.D. from Purdue
University. Before joining the
Science Foundation. The Humboldt Foundation and the
U, he taught at the University
Simons Foundation have also provided support.
of Nebraska-Lincoln.
Finding explicit
solutions to such equations is generally not feasible nor useful—it’s
to other things; it makes
much more helpful to look
for rich and lively research.
for overall structure in the collection of all possible
Commutative algebra is
solutions. These solution
continually reinvigorated by
sets are called algebraic
problems and perspectives
varieties. The word algebraic indicates their
from other fields.”
origin is from polynomial
10
b y
“One of the things I enjoy about my research is
Funding for Iyengar’s research is from the National
Commutative rings arise in diverse
equations, as opposed to equations involving things
like trigonometric and exponential functions. Over the
the work of 20th century German mathematician
and computer science,
centuries, mathematicians have developed various tools
David Hilbert, whose work on invariant theory was
among other
The foundation of commutative algebra lies in
contexts in mathematics, physics,
