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 ...
-
Secret sharing: an innovative use of
polynomial interpolation; Lagrange
interpolation
- Modular arithmetic and a little bit of number
theory (Fermat's Little Theorem, Fermat-Euler); information about
primes
- The idea of crypto systems: secure repeated communication, perhaps
of large amounts of information; one-time pads; random bitstreams.
- RSA; Diffie-Hellman: these are
interesting topics and served to introduce various modern and
classical mathematical ideas.
- The factoring and discrete log problems
- Computational complexity; introduction of the
P vs NP problem.
- An introduction to probability, including
independence and expectation and ...; Von Neumann's exercise on producing unbiased bits from
biased bits.
- Various models for the reliability of a
network; binomial
distribution; the simplest change of
phase: rigorous proof of percolation in a complete binary
tree.
- Ramsey theory: mandatory small-scale structure
(monochromatic complete graphs)
in large two-colored complete graphs; existence of Ramsey
numbers; underestimate via a probabilistic argument; the idea of
probabilistic algorithm.
- A short lesson on coding theory: Bell Labs,
Shannon, Hamming; error-detection (parity-checking, casting out 9's), error-correction
(best 2 out of 3, the Hamming (7,4) code).
Guest speakers
- Professor Paul Leath, Physics Department
Discussion of experimental determination of a phase change.
- Professor József Beck, Math Department
Discussion of games with complete knowledge ("games of no chance"),
where the kinds of games considered were based on coloring complete
graphs.
- Professor Jeffry Kahn, Math Department
Discussion of correlation inequalities. What "information" can be
transmitted from one vertex of a big graph to
another? (Previous knowledge of bipartite graphs and conditional
probability was needed.) To appreciate his work totally, a background in
some physics problems (statistical mechanics) is nice.
- Professor Doron Zeilberger, Math Department
Discussion of computer-aided proofs, and the future of
mathematics. What is a "deep" proof>
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?
- Mathematics is a human activity/ an art form (?!)/ a stringent mental
exercise. There are social rewards of various types.
- Mathematics and mathematicians may be motivated by accidents and
intentions of applicability. Is mathematics discovered or invented? Is
mathematics "natural"? This consideration is connected to the anthropic principle:
mathematics is part of how human beings understand the universe and
maybe how universe "accommodates" human beings. The following
well-known essays are interesting and relevant:
The future of mathematics (some versions)
- The more complicated stuff usually studied by
mathematicians. The material is hard to understand (more like theology
or philosophy or ...): who can predict how worthwhile or relevant
things may turn out to be, and "Beauty is in the eye of the beholder", maybe.
- The century of biology broadly interpreted: data analysis of
genes, dynamic analysis of epidemics, control of gene expression
RNA/DNA, so inspiration coming from bio rather than physics or CS or
... Applications driven by biological considerations.
- Computer-aided proof: see some of Professor Zeilberger's
opinions. He believes that human participation in
mathematics must be supported and perhaps eventually supplanted by
machine-aided mathematics. Certainly now there are very few branches
of mathematics which have been unaffected by computer help. What will
happen?
Disclaimer
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
greenfie@math.rutgers.edu and last modified 11/29/2003.