Cognitive Science | Seminar in Cognitive Science Topic: Group Theory and Galois Theory Visualized
Q700 | 11729 | D. Hofstadter

Q700  Seminar In Cognitive Scineces
Section:  11729
2:30PM 3:45PM  TuTh   Room: HD TBA

Topic:  Group Theory and Galois Theory Visualized

Douglas Hofstadter

Quadratic equations can easily be solved by formula; cubics,
too, yield, but the formula is much subtler to find.  Even quartics
yield, yet quintics are completely resistant.  What lies behind this?
In the late eighteenth and early nineteenth centuries, many
European mathematicians grappled intensely with this mystery, and it
was finally resolved by the young French genius Evariste Galois in
1830.  To crack the nut, the teen-ager had to invent both the theory
of groups and the theory of fields from scratch, and to make
profound discoveries in them, discoveries that would launch all of
modern mathematics.
Galois' ideas were extremely abstract then, and even today,
after nearly 200 years of deepening understanding, they are still
difficult to grasp, and yet they constitute one of the most central
pillars of all of mathematics.  Can Galois' elusive insights be made
concrete, visualizable, intuitive?  To do so is the purpose of this
seminar, which will attempt to get across the key ideas of group
theory, field theory, and their poetic fusion in Galois theory in a
nontraditional fashion, using such techniques as Cayley diagrams
(for groups) and, wherever possible, analogies to familiar ideas.
Proofs (usually informal rather than extremely rigorous) will be
important, but far more crucial will be the buildup of intuitive
The level of mathematics required is roughly one solid year of
calculus (calculus itself is not used, but one year of it testifies
to a minimal kind of "mathematical maturity"); in addition, I will
presume a fairly strong interest in mathematics, since this seminar
is aimed at people who are genuinely motivated to seek understanding.
The grade will be based on class participation (I insist on a
constant dialogue with students, in which they let me know where my
ideas or their own seem blurry, thus helping us to keep sharpening
the images), and on a term paper of some five to ten pages, on a
topic chosen in the last few weeks.
Readings used will include material taken from:

Ian Stewart: Galois Theory (Third edition).
Grossman and Magnus: Groups and Their Graphs.
Birkhoff and MacLane: A Survey of Modern Algebra.

We will also use Nathan Carter's wonderful Website called "Group
Explorer", which renders so much of group theory palpable and
learner-friendly.  Lastly, there will also be handouts of my own
materials.  I have long toyed with the idea of writing a book on
these ideas, and this seminar may lead to that, although I am not
sure yet.