Diary for 01:090:101:21, Experimental Math
First meeting, September 8
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 completeed 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
Links to other material mentioned in this lectureBernhard 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
apparantly were important in his discovery of the conjecture. Some
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
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
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.
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
email@example.com and last modified 9/3/2008.