|Do you have neighbors? Are there people that are close to you? Are there areas that are closed off to you, and others that are open to you? Isn’t it comforting at times to be completely surrounded by family and friends that you know and love? Do you sometimes think that the boundaries that parents impose upon you are too strict?
These notions that I have just described are so commonplace, yet they form such an important part of your life. In fact, these very same ideas play an important role in all of modern mathematics. Mathematicians have taken these ideas and constructed a way of looking at and doing mathematics. It is called topology. Topology is the mathematician’s way of dealing with points being close to each other and what it means for points to be in a neighborhood. It is natural to try to associate a boundary to a neighborhood, something that “closes in” the neighborhood. For example, in the early 1960’s, the Berlin Wall was constructed to separate West Berlin from East Berlin, which were both in what was then East Germany. The Berlin Wall made a way to define two different cities within what was once a single city (Berlin) by means of a boundary. Very curious things happen when you get close to a boundary, just as when people used to get too close to the Berlin Wall. The concept of a boundary is so fundamental in mathematics that whole books are devoted to the study of the behavior of functions on boundaries and boundary points.
Topology is such a fundamental subject in mathematics that mathematicians can spend their entire lives working on it. In fact, I wrote my doctoral dissertation on topology. My choice of topology for a topic not only shows what my interests in mathematics are; it also shows how stubborn I can be. After I earned my bachelor’s degree in mathematics I decided to go on for a master’s degree in mathematics at The University of Notre Dame. I began my first year of study by taking a graduate level course in topology. Three weeks into the semester I was pulled out of this course because the instructor felt I would fail the material. This made me very angry and my pride was so hurt that I decided not only to take this course, but also to write my doctoral dissertation on topology! I completed my Ph.D. in 1961 at University of California, Los Angeles with a thesis entitled “Minimal Topological Spaces.”
I get to do many things having a Ph.D. and being a faculty member at a university. Part of my life has been spent creating new mathematics; however, I also train teachers. As a professor of mathematics, it might be assumed that I teach many classes on mathematics. This is not exactly true in my case. I teach one class per semester, and this class deals mainly with issues in mathematics education in elementary, middle, and secondary schools, as well as in community colleges. Teachers from all of these levels take my course. We discuss ways to teach mathematics more effectively at each level of schooling, an issue about which I deeply care.
Earning a Ph.D. is very enriching and it prepares you to do so many other things. I wasn’t trained to be a specialist in mathematics education, but here I am. My study of mathematical neighborhoods and boundaries provided me with the tools to break down the boundaries that have kept many diverse students out of the scientific workforce.