Diary for Experimental Math, 01:090:101:21, fall 2008

The first meeting, 9/8/2008

I will introduce myself, and then discuss the idea of mathematical creation (perhaps contrasting it with, say, poetic or divine inspiration) and the reality of trying lots and lots of examples.
Students in the course could have a voluntary sort of "term paper" assignment, which if completed will make them immortal (sort of). I'll try to explain this later. And students will be able to help ME with my research program. This is not likely to be an income-producing opportunity.
The major goal of this meeting is to begin students' familiarization with Maple. This program will be our primary tool for experimentation. Seven pages were distributed.

Links to other material mentioned in this lecture

  • Bernhard Riemann lived in the nineteenth century and created a large amount of really significant math. An unwary student can suppose that this was all done with magnificent intuition or inspiration or ... A long-standing problem associated with Riemann, a problem which is certainly one of the outstanding targets of current math research, concerns the Riemann zeta function. Riemann conjectured (means: "guessed", but "conjectured" is a neater word used in academic communities) some results about where this function equals 0. Consequences of this conjecture, if true, would be facts about prime numbers (primality testing, cryptography, etc.) and even facts about some models in theoretical physics. I always thought the conjecture was done with pure thought. Only a few years ago, deep in the mass of material left after Riemann's death, a huge number of numerical computations were discovered, and these computations apparently were important in his discovery of the conjecture. Some references:
    Biography of Bernhard Riemann, 1826-1866. He was born in Germany and died in Italy at age 40 of tuberculosis.
    How to make a million bucks!. The link is to a collection of math questions. Any person solving one of them would become very famous and rapidly rich. Here is very specific (complicated) information about the Riemann hypothesis, which is one of these problems.

  • Srinivasa Ramanujan lived a bit later than Riemann. He is the ideal example of a (seemingly) naive person from way outside the mainstream of mathematics whose genius permitted him to make fantastic contributions. Again, his life looks like fiction (there's actually been a recent novel about him). Many of his notebooks were found after his death (the last one, the lost notebook, was discovered as recently as 1976). These notebooks contain a collection of intricate formulas and statements which were verified after great effort by others. Ramanujan also made incredible experiments in computation. In his society, paper was expensive, so very little evidence survives. But there is ample testimony to his sitting and making many computations on a slate, hours and hours of work every day! He is probably the patron saint (pardon me if this phrasing is offensive!) of experimental math. Some references:
    Biography of Srinivasa Ramanujan, 1887-1920 He was born and died in India at age 33, probably due to the combined effects of tuberculosis, dysentery, and hepatic amoebiasis.
    Ken Ono's article about a visit to India includes some information about Ramanujan's life and his daily hours of experimentation.

  • A general experimental math website. This page has many links to courses and activities at other universities and research centers.

  • Rutgers is a major research university. It has an active experimental mathematics community. I'll try to get a few of these people to come and talk to us about their work. One aspect of this community is an experimental math seminar. In general, the talks in this seminar are expositions of advanced scholarly work, and I would not expect many of our students to understand them. But you could always attend, sit in back, and then do your own work (I do that frequently!). On occasion, you may enjoy and understand a significant amount of a presentation. The web page of the seminar has brief descriptions of the talk. And you can discuss with me how understandable a talk might be!

    I handed out a request for student information. During class students worked on the pages below. At the end of class, I handed out the first homework assignment.

    Maple handouts
    Page 0:
    Goals of showing Maple to students;
    GUI compared to command line interface
    Page I: playing with arithmetic Page II: playing with algebra Page III: playing with calculus Page IV: playing with graphs

    Maintained by greenfie@math.rutgers.edu and last modified 9/3/2008.