An informal restrospective summary of Newer Math, fall 2003

My ambition was to show mathematics as a continuing human endeavor, discussing current mathematics and, to a certain extent, contemporary mathematicians (where the qualification was that the mathematics is less than 50 years old).

I restricted my choices of topics a great deal. I needed to cover material that could be understood by students having a minimal background. Since much of mathematics is relentlessly cumulative, this was a major difficulty. For example, a large group of people in the Rutgers Math Department (graduate students and faculty) study non-linear partial differential equations related to problems in differential geometry. I could not realistically discuss their work, nor could I discuss, for another example, the work of people in mathematical logic who are trying to understand the consequences of various assumptions about different types of infinities. I'd estimate that 90 to 95% of contemporary mathematics was essentially unreachable in a course of this type. I also applied a rather stiff "filter": I wanted to teach and discuss the actual math, not "appreciate" it. Also, I had to restrict to mathematics I knew or could learn well enough to teach.

We studied the topics mentioned below, which don't need huge prerequisites. Many of them were recognizably inspired by problems of physics and computer science. Topics in blue I consider newer math, while those in green were part of the necessary background.

What the lecturer said ...

Guest speakers

The applications to computer science and physics were included to show how some math in the twentieth century was "inspired" by these new concerns, or at least related to them. Some mathematicians declare that their work is inspired only by ideas of beauty and intricacy structure. Thus funding needs, social pressures of their community, and student attendance don't matter. (?!)

We still note the "unreasonable effectiveness" of mathematics (see below for more about this phrase). For example, Erdos and Renyi invented the theory of random graphs in the 50's and 60's, and it just happened that random graph theory was quite important in many applications during the next two or three decades. (Although it turns out that "random graphs" don't seem to be the correct models for, say, the Internet or for the network of chemical interactions in a cell or for scientific collaboration networks -- these all seem to be modeled better by the random "scale-free networks".)

What do mathematicians do? Why do mathematicians do it?

The future of mathematics (some versions)

This webpage not a finished essay on the topics of the course or the intent of the course. It is just some notes (11/25/2003) about the course.

Maintained by and last modified 11/29/2003.