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 understanding. 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.