Various people outside the class have expressed an interest in following along with the course as it progresses. You are welcome to do so and to download copies of linked documents. It would be helpful to me if you'd send comments or otherwise let me know that you are doing this.
F 2/10: Richard talked about a number theory homework problem, that there is only one prime p for which p, p+2 and p+4 are all prime. Ian completed his topological discussions with a discussion of the very strange and general Urysohn Univeral space. I gave a quick proof of Pick's Theorem, passed out Greed is good and brought some math books for class members to borrow.
W 2/8: Joel talked more about counterfeit coins and wighings and Ian talked about metric spaces and Hyperspace. I mentioned the example of error correcting codes, which moves metric spaces from topology to the more comfortable (for me!) realm of discrete mathematics. There was a handout about how the power of p that divides the binomial coefficient C(n,m) is equal to the number of carries when m and n-m are added in base p. Also, if N is a positive integer and x = p/q is a positive rational, then x can be written as a sum of distinct unit fractions 1/m, with m >= N. Handout on Friday.
M 2/6: Several more presentations. Ben Walt talked about the five Platonic solids and a proof that they are only five, which led to a discussion of various other geometric topics, including an old theorem of mine. Ben's source was From geometry to topology, by Graham Fegg. I suggested as a further reading book, Proofs and refutations by Imre Lakatos. I also gave as a handout a humorous piece: A game theoretic approach to the toilet seat problem. I talked at the end about sequences, and the wonderfully useful On-Line Encyclopedia of Integer Sequences.
F 2/3: Another day where the class spoke and the prof listened. Jo Nelson continued her discussion of some exotic topologies. Richard Carr gave the proof of a number theory homework problem he'd liked: when (precisely) does 2^b - 1 divide 2^a + 1. Dylan Roeh gave a proof that pi is irrational. (He e-mails me that the source of his proof is Pi is irrational. The (same) proof can also be found in section 6.3 of the textbook An introduction to the theory of numbers by Niven, Zuckerman and Montgomery, which is often used as a 453 text.) Then, Joel Tadmor gave Bendixson's Theorem about cyclic solutions to differential equations and also a weighing problem. Finally, Ian Shipman presented a theorem from commutative algebra on finitely generated prime ideals.
W 2/2: Before the bell: Prof. John D'Angelo's stirring rendition of "Traces" by the Classics IV. Handouts on Monty Hall and JSTOR, accessible through UIUC machines only, or by subscription. Steven Canning gave a proof of the Bertrand Postulate on the distribution of primes and Jo Nelson started talking about topology.
M 1/30: Lots of speakers. We talked about the two "homework" problems given on Friday. Dylan Roeh solved the corner snip porblem and Ian Shipman proved his Putnam problem. (Basically the greedy algorithm. Find the largest power of a <= n, say a^r, and then take the largest 2^k*a^r <= n. Subtract. Repeat.) Ben Walt talked about the Monty Hall problem. Joel Tadmor gave a variation thereon. Andrew Reder talked about why 1/1 + ... + 1/n is never an integer.
F 1/27: Nobody wants to speak besides Ian, who does something involving moving triangles around. He also gives the following Putnam problem as "homework". Suppose r is a positive odd integer. Show that every integer n has a representation in the form n = x_1 + x_2 + ... + x_k, where each x_k = 2^(a_k) * r^(b_k), and no two x_k's divide each other. His earlier presentation inspired me to offer another Putnam problem as homework. (This can be found on pp.343-4 of the handout "Some thoughts on writing for the Putnam", which is a good thing, because it would be hard to express in .html. I give a proof of a special case of Hilbert's theorem on finite generation; namely, if p(x) and q(x) are two non-zero polynomials, then there is a non-zero polynomial F(u,v) in two variables with the property that F(p(x),q(x)) = 0. I finished the hour by finishing a discussion of "resources for research", and then a handout on the way that mathematics is sliced up into areas by the ArXiv, Math Reviews, the NSF and various mathematical departments.
W 1/25: People start speaking in great numbers. David Hovorka starts with the Cantor diagonalization and winds up with, essentially, arrays of integers which might create primes in juxtaposition when read up, down and diagonally. (I suggest base 2 to make the numbers smaller.) Jo Nelson talks about Evolutionary game theory and the Prisoner's Dilemma, and Joel Tadmor follows it up with a discussion of dominant strategies and how they can lead to peculiar conclusions. Ian Shipman starts to talk about fractals, but it's really our friend, the Cantor set. (Look at the Cantor set in ternary notation to prove the theorems.) Goal: everyone will have spoken by next Monday.
M 1/23: A handout on TeX, taken from the math department website -- see TeX for more, a handout Resources for Research, taken from this website. Ian Shipman talked about the Mazur-Ulam Theorem, isometries of the plane, and fixed points of finite groups of isometries. Richard Carr talked about Pythagorean triples extended to n variables. I was inspired to talk about a direct proof of the Mazur-Ulam Theorem in the plane: if f; R^2 -> R^2 preserves distance, then it can be expressed as a composition of a translation, a rotation and, possibly, a reflection.
F 1/20: Many people turned in the first assignment; I hope the rest will do so on Monday. Nobody besides me wanted to speak at first. The results of the questionnaire showed a preference for number theory, geometry and "good" proofs. I spoke a little more about the Fibonacci numbers and passed out Fibonacci Notes and More Fibonacci Notes. Discussion of lattice points, an old Putnam problem, brush with greatness with Richard Feynman and the one lattice point in the triangle theorem. At the very end, Ian Shipman made the first student presentation of the semester -- on Boolean algebra and characteristic functions.
W 1/18: First day of class; ten people present. The introduction,
including Class
Organization handed out, along with
Questionnaire,
One introduction to mathematical research (to appear in
Math Horizons,and
How to write for the Putnam. This last paper is excerpted, without
permission, from the MAA Book The William Lowell Putnam
Mathematical Competition, 1985-2000 .
First two class presentations: can you use dominos to cover an 8 x 8 checkerboard with antipodal corners missing? This was followed by many variations. An introduction to the Fibonacci numbers, together with some general approaches that work for more generally defined recurrences. There will eventually be notes.
First assignment: take a ``favorite" problem you did in a previous course, solve it, then change it and solve the new problem. Then change it in a way that makes you think you can't solve it.