Monica VanDieren's research specialty is about as close as a professor can get to Rodin's "The Thinker," except that she doesn't sit on a rock and wears a lot more clothing.
Dr. VanDieren is a pure mathematician based at Robert Morris University, and she works in an area so rarefied that she and her husband, fellow researcher Rami Grossberg, can gather most of the discipline's leading thinkers in their Franklin Park home for marathon sessions of high-level cogitation.
Their specialty is called model theory, and unlike such fields as molecular biology and astronomy, it doesn't depend on computers or hi-tech equipment. Instead, Dr. VanDieren and her colleagues sketch their ideas and equations with pen and paper.
"There may be 10 people in the world who work on this," she said. "We'll do our work around our kitchen table, and with a lot of coffee and a lot of wine, theorems will come out."
Position: Assistant professor of mathematics, Robert Morris University; co-director, University Honors Program.
Residence: Franklin Park
Education: Bachelor's in mathematics, University of Illinois, 1996; master's and Ph.D. in mathematical sciences, Carnegie Mellon University, 1998, 2002.
Previous positions: Assistant professor of mathematics, University of Michigan, 2003-06; Stanford University, 2002-03.
Outreach activities: Director, Expanding Your Horizons Pittsburgh conference for junior high girls; head judge, Pittsburgh Regional Elementary Science Olympiad; Science summer workshops for junior high and high school students, Robert Morris.
Publications: Ten articles in refereed journals.
To carry out her research, which is funded with a grant from the National Science Foundation, she sets aside blocs of time during school vacations and the summer.
That's partly so she can concentrate and meet with fellow researchers, and it's partly because she's too busy with teaching and other activities the rest of the time.
Besides carrying a significant teaching load at Robert Morris, Dr. VanDieren also is co-director of the school's honors program, mother of a young daughter and active in outreach programs to encourage girls to pursue studies in science and math.
This month, for instance, she led Saturday workshops for junior high and high school students on origami and math, and she has spearheaded the Expanding Your Horizons program, where teenage girls can explore math and science careers.
It was not until she got into graduate school that Monica VanDieren realized what a rarity she was.
As an undergraduate math major at the University of Illinois, not only was she surrounded by more fellow women math majors, but she often took other classes where women predominated.
"I didn't even think of it as a problem," she recalled. "When I saw that there was an Association for Women in Mathematics group on campus, I thought, 'Who needs that? Why do we have to have our special group?' I just thought they were a bunch of whiners, at that point."
But in graduate school at Carnegie Mellon, she suddenly found "I could be the only woman, or maybe there'd only be two or three of us" in a class, and then she saw the numbers drop even further when she became a math professor.
National statistics bear that out. While the number of women pursuing graduate degrees in math has risen, only eight of 180 tenured professors at the nation's top five mathematics departments, as ranked by U.S. News & World Report, were women, a study last year said.
The lack of role models may be one reason more women don't seek graduate degrees in math, Dr. VanDieren said. In other cases, women may be tied to spouses or other relationships that restrict them to a certain geographic area.
In her case, she said with a laugh, "I stuck with it because I didn't think about it. I was good at math and I just stayed in school until they kicked me out with a Ph.D."
She was drawn to math early on in college -- the more abstract, the better. In a theoretical calculus class she took as a freshman, she remembered learning about ideas like Gabriel's Horn, a shape that had a finite volume, but an infinite surface, "so you can fill it up, but you can never paint it."
Concepts like that fascinated her, and it's what she tries to convey to younger students in her workshops.
Math, she tells them, "isn't all about equations and computations -- that can be pretty boring, actually. But there's a huge beauty to math, there are lots of symmetries and patterns, and when you try to visualize them you almost need an artistic sensibility to get your head around it."
She earned her master's and Ph.D. in mathematical sciences at Carnegie Mellon, and ended up marrying one of her professors, Dr. Grossberg, who still teaches there.
In her field of model theory, Dr. VanDieren said, she constructs special mathematical universes that exist when a certain set of propositions are all true.
One particular problem to which she now devotes a lot of energy is Shelah's Categoricity Conjecture, devised by Israeli mathematician Saharon Shelah, considered one of the world's top five mathematical theorists.
The mathematical universes she works with come in different sizes, and Shelah's Conjecture proposes that if a set of axioms produces one unique model of a certain size, or "cardinality," then "once you understand how that model is built up and how you can deconstruct it, you can carry that over and say every other model of other cardinalities is built up by that same set of rules."
She is the first to acknowledge that such descriptions leave many people nonplussed. "I don't even try to describe what I do" at social gatherings, she said.
On the other hand, she and Dr. Grossberg said, pure math often has a major impact on the world when other scientists take its lessons and apply them to various problems.
One example they cited is the antennas now embedded inside cell phones. Much of that work started when astronomer Nathan Cohen wanted to put a ham radio antenna on a rooftop, but needed to make it smaller to meet neighbors' objections.
To do that, he developed an antenna based on Sierpinski's Triangle, a set of triangles within triangles developed by Polish mathematician Waclaw Sierpinski.
"It turns out," Dr. VanDieren said, "that if you were to trace around all the little triangles you actually would have an infinite perimeter," which gives an antenna based on that design a lot of surface area in a very compact space.
Other examples of ideas that originated in pure math: iPods, digital cameras, computers and encryption.
In Dr. VanDieren's pure math world, there are some ideas that seem counterintuitive.
For instance, there are several versions of infinity. While many of us probably think of infinity as something that can't be measured, she said mathematicians have shown that the infinite size of all the decimal numbers between zero and 1 is larger than the infinite size of all the whole numbers such as 1, 2 and 3. They've also shown that the infinite size of even numbers is just as large as that of even and odd numbers combined.
For Dr. VanDieren, combining her work as a researcher, mother, teacher and mentor requires her to have her own infinity of energy.
That may explain why her Web site at Robert Morris includes this anagram of her full name, Monica Marie VanDieren:
"I AM ONE DRIVEN AMERICAN."
Mark Roth can be reached at firstname.lastname@example.org or at 412-263-1130