Ask Dr. Silverman 4 — Embedment: A Tale of Matrioshka
Herb Silverman is the Founder of the Secular Coalition of America, the Founder of the Secular Humanists of the Lowcountry, and the Founder of the Atheist/Humanist Alliance student group at the College of Charleston. Here we talk about mathematics and its fields.
Scott Douglas Jacobsen: To simplify the universes of discourse for disciplines, one automatic maneuver comes from segmentation of the the natural sciences into further specialties, sub-disciplines, and so on. What have been the staples of mathematics? How have segmentation help organization ideas and research into mathematics? What fields comprise fundamental areas but remain more fringe as they’re a more niche discipline or sub-field of mathematics?
Professor Herb Silverman: After receiving my master’s degree in mathematics from Syracuse University in 1965, I passed the qualifying exam for my PhD, which I received in 1968. Students at that time took a qualifying exam in four mathematics specialties in which they could earn a PhD at the university –Algebra, Topology, Real Analysis, and Complex Analysis. Remember there were no computers back then, so the fields of investigation were somewhat limited.
Algebra at the university level is quite different from what you learned in high school. It investigates topics called groups, rings, fields, and other entities too complicated to define here. Topology is a branch of mathematics concerned with the properties of space that are preserved under continuous deformations such as stretching or twisting, but not tearing. An old joke is that a topologist does not know the difference between a doughnut and a coffee cup (because the coffee cup can be topologically transformed into its handle (a doughnut). Real Analysis studies the behavior of real numbers, sequences and series of real numbers, and real-valued functions. Complex Analysis deals with the study of complex numbers and their functions.
Whichever of these branches of mathematics people chose, they always specialized in a subfield because the whole field was too large. I did my PhD thesis in Complex Analysis, doing research in a subfield called Geometric Function Theory.
Today many more fields with related subfields exist in which to do mathematical research, especially in applied mathematics. When a new PhD mathematician applies at a university, he or she usually gives a research talk to the math department before being hired. The applicant is often warned by an advisor, not completely in jest, to divide the talk into thirds. The first third of the talk should be understood by just about all the mathematicians in the department to show that you are a good teacher. The second third should be understood by about half. And the last third should be understood by almost nobody, indicating that your research is deep.
It’s not easy to say which areas of mathematics are fringe and which are not. Beautiful mathematical results have often been found in areas not considered part of the mainstream. Some so-called fringe areas today will likely become more mainstream when people see how they might be useful in solving a host of other problems. So I wouldn’t denigrate any research area in mathematics, no matter how fringe-like they might appear to some.
As with research in science, I would say regarding what should be considered mainstream in math, “Follow the money.” Areas in which there are a lot of substantial research grants can be called mainstream.
