Photo by Shubechhya Mukherjee | Mercury Staff
Take an inside look at how UTD’s mathematical scientists operate, solve problems using high dimension math
When people think of research, they imagine a scientist in a
white lab coat pipetting chemicals or culturing cells on a petri dish. But when
asked to consider “math research,” what comes to mind? What does pure math
research entail?
This fall, the School of Natural Sciences and Mathematics
inducted seven new tenured and tenure-track faculty. Among them, Baris
Coskunuzer and Stephen McKeown conduct research in math, while Qiwei Li
conducts statistics research.
“Pure mathematicians are not really interested in the
real-life applications,” Coskunuzer said. “They are trying to solve nice
puzzles, which give you interesting relationships between objects.”
Coskunuzer got his first taste of math as a high schooler in
Turkey, where he enjoyed classes in abstract math. He majored in math in
college, and as a Ph.D. student at Caltech, his advisor introduced him to
geometric topology. Given his knack for visualizing objects in high dimensions,
he decided to enter the field, investigating the shapes and classification of
high dimensional objects.
“In the pure math, after (the) third or fourth year of PhD,
you start to have a sense of some beauty in the subject … you can think of pure
math like an art,” he said. “ You go deep in this sea, and you discover
treasures.”
Daily tasks in math research include studying the
publications of other mathematicians, attending conferences and collaborating
with colleagues, Coskunuzer said.
Like Coskunuzer, McKeown said that a large part of math
research involves reading the work of others.
“A lot of the time is (also spent) sitting at your desk with
paper and pencil and doing calculations or staring at the wall for that matter,
thinking really hard,” McKeown said.
McKeown was first drawn to math when he read a book about
geometry and relativity in high school.
“In college I started taking calculus for the first time,
and I really loved it,” McKeown said. “I was surprised by how beautiful math
could be, so I started reading popular math books … I got some sense of the
field of math … and I found that very exciting.”
At the same time, McKeown was also interested in becoming a
lawyer. He finished law school and passed the bar exam before returning to the
field of mathematics.
“At some point I realized that I missed math and wanted to
do that instead,” McKeown said.
McKeown concentrates on conformal differential geometry in
his research. Conformal differential geometry is the study of spaces where
angles, but not lengths, are well defined.
In contrast with pure math research, Li said, statistics
research centers on application, especially in medicine or biology.
“Statistics is the science about data, because the data can
reveal lots of interesting things about the body and the world,” Li said.
Daily tasks in statistics research include collaboration
with biologists, chemists and doctors to collect and analyze data using
computer models and statistical analysis software.
In college, Li majored in electronic engineering. As a
master’s student, Li took a data mining course, where he classified the quality
of photos based on different metrics. Li’s interest in photo classification led
him to change his major to statistics.
“I didn’t like the electronic engineering research, I wanted
to do something new (and was) interested in data analysis,” Li said.
Li uses Bayesian statistical tools to draw conclusions in
two areas of application, digital pathological images and microbiomes, which
are collections of microorganisms, using both data and prior knowledge. High
resolution images of pathological tissues can be analyzed by a deep learning AI
to identify different types of cells, Li said. The patterns of cells are
statistically quantified and used to predict patient survival outcome.
“If you are suspicious that you might have cancer … I want
to confirm that,” Li said.
Li also identifies biomarkers that can be used to predict
colorectal cancer by quantifying bacterial abundance in the human microbiome.
Research is often rife with challenge, as all three
professors can attest to. In math research, one can get stuck on an aspect of
the problem for months, even years. In statistics research, a major challenge
is locating datasets that can prove the merits of one’s methodology.
“You find yourself at this side of the chasm looking across
and you just don’t know how you’re going to get across, and I think that’s the
most frustrating thing,” McKeown said. “Our students feel frustration because
they don’t know how to solve a problem, but we do too.”
McKeown said that he had once been stuck on one problem for
three months, another for nine to ten months, and is still grappling with a
third problem, which he set aside a year ago because he did not see a solution.
“One of my mentors in undergrad said, ‘the mathematician
spends 90% of his time depressed and 10% of the time elated,’” McKeown said.
To bridge that chasm in a problem, McKeown may write out
arguments or read papers that reference a similar concept. This helps him gain
a new perspective on the problem and adapt other techniques to his
situation.
“The actual process
of sitting down and doing math, I enjoy,” McKeown said. “I think that moment of
when you actually understand something you didn’t before is … in some sense why
you’re a mathematician, because that’s really rewarding.”
He also appreciates how complex math ideas are built around
simple fundamental concepts.
“The biggest, most beautiful proofs in the subject are
reducible in some sense to (the) little things in the most pleasing way,”
McKeown said.
As advice to aspiring students, McKeown suggests taking a
variety of math classes and participating in an undergraduate research project
to find out whether math is a good fit.
“Challenge yourself and put yourself in classes where you’ll
have to prove things, (to experience) a small taste of getting stuck on
something (and of) the frustration and the excitement and joy of getting past
it,” McKeown said.