Two members of the Department of Mathematical Sciences — associate professors Ilie Ugarcovici and Bridget Tenner — received five-year Collaboration Grants for Mathematicians from the Simons Foundation, which was started by James Simons, a mathematician and the founder of Renaissance Technologies, and his wife, Marilyn.
The Foundation’s division for mathematics extends the frontiers of basic research by focusing on theoretical sciences radiating from the discipline. The grants support the “mathematical marketplace” by enabling face-to-face, collaborative research. They’re given to “accomplished, active researchers who do not otherwise have access to substantial research funding that supports travel and visitors.”
In 2012, approximately 140 Collaboration Grants for Mathematicians were awarded. All funding decisions are made through a peer-reviewed proposal process.
“Having one member of the department receive such an award would be an honor, but having two is an outstanding achievement,” says Ahmed Zayed, professor and chair of the department. Here, the professors talk about their work.
My area of research is dynamical systems. Think of a process that changes with time: I want to see the changes, quantify them, and understand — maybe — some essential aspects of long-term behavior.
For example, changes from one year to the next in a population of fish or insects can indicate something important, either intrinsically or extrinsically. In density-dependent populations, individual members of the group will behave one way when there aren’t many and another way when there are a lot. For several decades, scientists have tried to understand this type of dynamic mathematically; this is where my collaborators and I come into the picture. We try to provide insights about why some populations die out and others, by changing some parameter just a little bit, oscillate or grow fast. A mathematician wants to understand or identify the exact moments that define the change, one way or another.
In my more abstract research, I study how particles move on a curved surface. Think of an inside-out sphere — that’s called a surface of constant negative curvature. Let me point out that the notion of negative curvature plays an important role in the theory of relativity. A particle that moves freely on this surface can exhibit complex behavior; in fact, it might never return to its starting point, but rather just wander around somewhat randomly. I’m trying to understand, statistically or quantitatively, this behavior.
The Collaboration Grant provides wonderful opportunities for me to share ideas, especially with people doing research in fields that interact with dynamical systems, including geometry, statistical analysis, number theory, and chaos theory.
Mathematicians are a small community: If we don’t share insights with peers, each of us can become very isolated. And because mathematical research is so complex and time-consuming, with lots of trials and lots of failures, collaboration enables change and growth. I’ve worked with peers from Penn State, Georgia Tech, University of Colorado, Northeastern Illinois, and University of Chile.
My field is combinatorics, which includes counting (understanding the size of complex objects) and sorting (determining some sort of criteria for ranking or ordering things). Combinatorics bridges applied and pure mathematics, but I lean toward the latter because I ask combinatorial questions about theoretical objects within algebra.
I’ve become an expert in two things — permutation patterns and reduced words — both of which are aspects of a particular family of algebraic objects.
Here’s a simple example of permutation patterns. Inside a string of numbers — say a telephone number — you can look for patterns. If you were reading from left to right and saw a 1, and then an 8, and then a 5, that would form a 1-3-2 pattern, because the relative order of 1, 8, 5 is small, large, medium. One could then try to count the number of telephone numbers containing (or avoiding) a 1-3-2 pattern.
Reduced words are related to algebraic objects called Coxeter group elements. These words are decompositions of the group elements into fundamental pieces called generators; they’re "reduced" because they are the shortest such decompositions.
In my work, I’ve made a connection between permutation patterns and reduced words. Previously, it was known that these two things referred to two different ways of describing the same set of objects, but there had been no dictionary for translating between the two languages. Now, I’ve made the two speak to each other.
In mathematics, collaboration is the norm: It’s hard to imagine that a single perspective would be enough to answer hard questions and solve complex problems. That’s why the Collaboration Grant for Mathematicians is so outstanding. Collaboration starts (and continues) because of face-to-face meetings. Sitting around a table with a big pot of coffee and a chalkboard, spending hours poring over problems — that’s ideal, and that’s what the grant enables.