Teaching
Because of the large variety of courses that I have taught, I do not have a singular teaching philosophy--these various courses have different expected learning outcomes, after all. In courses such as a Transition to Higher Mathematics, Topology and Real Analysis, I believe in a bottom-up approach in which every theorem or technique must be rigorously verified by previously established truths before it is used. Here, I have found that the Moore Method (or a variation of it) is very effective. On the other hand, in courses such as Differential Equations or Elementary Probability and Statistics, I use a top-down approach to solving problems, in which we take a step back and look at the problem at a distance and then go find or develop the tools that we need. In these classes, I feel that written projects are a useful tool. In other classes, the best approach is a blend of the bottom-up and top-down approaches.
One thing that I have discovered in all my courses is that in order for the student to learn and grow, the professor cannot and should not show the student how to do everything. The student must discover most of the ideas for her/himself. This situation can be very frustrating for the student sometimes, but that is okay. Mathematicians are always frustrated. Learning to deal with the frustration and persevere through the difficult material on one鈥檚 own is how you begin to truly transform yourself into a mathematician.
When it comes to courses that I teach, I am a jack of all trades. The following are the course that I have taught so far: Linear Methods, Applied Calculus, Calculus I, II and III, Elementary Probability and Statistics, Transition to Higher Mathematics, Differential Equations, Linear Algebra, Mathematical Probability and Statistics I and II, Complex Analysis, Real Analysis I and II, Topology, and Junior and Senior Seminar.
Research
Most of my research is in the area of topology, which is the study of the "shape" of space. In particular, I have been exploring dynamical systems on continua, a certain type of topological space. Some examples of continua are circles, line segments, and spheres; these continua have simple topological structure. More complex continua include objects such as the pseudo-arc, the Sierpinski curve and the solenoid, which have complicated, fractal-like qualities.
A homeomorphism is an important type of function. A dynamical system on a continuum is created by iterations of the same homeomorphism on the same continuum so that points will move about the continuum. For example, rotate a circle by three degrees. If we continue to iterate that rotation, points on the circle will move around the circle. However, this dynamical system is boring. "Chaos" occurs when points that are very close to each other eventually move far apart. Chaos prevents accurate computer estimation as is the case in, for example, weather prediction. Much of my research interests are in chaotic dynamical systems--in particular, expansive homeomorphisms and entropy.
Mathematicians such as myself who study the relationships between chaotic dynamics and topology are interested in where chaos can and cannot occur. The use of topology helps to describe the "shape" of the space where the chaos occurs. In a sense, we try to describe the shape of chaos. A clear picture of this shape allows order to be distilled from seeming disorder. When a particular type of chaotic function exists on a certain space, we say that space admits that type of chaotic function. Much of my work is focused on the classification of continua that admit or do not admit expansive homeomorphisms or positive entropy homeomorphisms.
I am also interested in the topology and dynamics of Julia sets, symbolic dynamics, dynamics group actions and classifying properties of commuting maps.
Outside the Classroom
I live in Midtown with my cat Simon. I am a former rugby player. I was the captain of the rugby team at my alma mater, Lafayette College, and I founded and coached the women鈥檚 rugby team there. I now enjoy playing disc golf. Before graduate school, I taught high school mathematics and coached the wrestling team at Paulding County High School in Georgia. I have taught college mathematics in five different states over the past 17 years, and I have always been able to get to campus by bike or on foot.
