Explaining the Work: The Mathematicians Describe Their Research
Ellie Dannenberg: geometric topology
Some questions in my field of geometric topology are quite easy to state but surprisingly difficult to answer. For example, here is a question from Chris Leininger and Benson Farb: How many curves—that (pairwise) intersect each other at most once—is it possible to draw on a given surface? Note that there are some restrictions on the types of curves we're considering. (For the mathematically inclined, we want the curves to be pairwise non-isotopic simple closed curves.) For a torus (a surface that looks like the outside of a doughnut) the answer to “how many?” is known to be three. In general, however, the answer is still unknown. We can also ask related questions such as: What if we require the curves to intersect each other exactly once? How about if we let the curves intersect at most (or exactly) some other number of times?
Yen Duong: geometric groups
Geometry, as you might remember, has to do with shapes and measuring things like angles, areas, and volumes. Groups are a fundamental object of study in the very large field of algebra, and for the past few centuries mathematicians have been thinking about groups and coming up with theorems about them. Many groups can be related to geometric objects—for instance, one kind of group might be the collection of ways you can rotate or flip a square (called the symmetries of a square). I study geometric group theory, so I look at shapes and try to learn things about groups from them, and I look at groups and try to learn things about their associated geometric objects.
Jessica Dyer: topological dynamics
My research area is topological dynamics, which studies changing mathematical systems as if they were physical moving systems. The change happens as time goes on, and we study the time evolution of these systems. For example, a physical system that can be studied in the real world is water moving in a current, and we would want to know what happens to the molecules of water and to the overall system as time goes on. In topological dynamics, the “water” is an abstract topological space, such as a sphere, and the “current” is a function on that space that we keep applying over and over again. We use a lot of pictures to represent what is going on in these systems, but we also must describe them precisely using equations and proofs, since the pictures can't show all the detail of what is being studied.
Cara Mullen: arithmetic dynamics
My research area is arithmetic dynamics, which is in the intersection of number theory and complex dynamics. In particular, I study what happens to 0 under repeated iteration of specific quadratic polynomials, in a non-traditional number field setting: Does 0 ever come back to itself (is it periodic)? Does 0 ever enter a cycle not including itself (is it eventually periodic)? Does it escape to infinity? I hope to be able to state a connection between the behavior of 0 in these non-traditional settings and certain arithmetic properties of the original polynomial.
Janet Page: algebraic geometry
Algebraic geometry is an area of math that studies, among other things, zero sets of polynomials. In high school, people often study solutions to equations like “y = x2”, or in other words “y - x2 = 0”. By the "zero-set" of the polynomial “y-x2,” we just mean all the pairs (x,y) that satisfy this equation. As many high school students learn, this particular equation makes a picture called a parabola when plotted on a two-dimensional graph. This is an example of a polynomial with two variables (x and y) of degree 2 (the highest power of any of the variables that shows up is 2). Algebraic geometers study zero sets of polynomials in any number of equations and any degree. They are curious about how different polynomials produce different shapes, and study when these shapes are similar, when they change the polynomials slightly, and when changing a polynomial slightly produces a fundamentally different shape.
Photos by Joshua Clark