F 2/23 -- I gave part of the solution to the cubic equation. Wanrong continued his discussion of Golomb's conjecture. Handouts included an article by Amir-Moez on Omar Khayyam's solution of certain cubic equations back in the 11th century, a long historical article on the Cardan-Tartaglia controvery by Nordgaard and an article on De Bruijn sequences by Ralston: all from Mathematics Magazine via JSTOR.
W 2/21 -- Wanrong started a discussion of recurrence sequences and their associated polynomials over the finite field Z_2, which will lead to Golomb's conjecture. I gave handouts on number theory for the millenium and spoke a bit about Felkel's ill-fated factorization table, a biography of Maria Agnesi, an article about the curve she discovered, and a German website "Mathematische Basteleien" with many recreational mathematics links.
M 2/19 -- Field trip to the Rare Book Room. Amy was in last year's class and found a helpful link "The Game of Logic": Lewis Carroll's "The Game of Logic" Online as an ebook. Here is a link to a larger exhibition that I co-curated back in 2000: Number Theory for the Millenium.
F 2/16 -- David spoke about the irrationality of some numbers and started by giving some numbers which you might think were irrational but aren't. Handout: a new article on "Undecidability in Number Theory" by Bjorn Poonen, from the latest AMS Notices, as well as two articles from my own website: "Chalking It Up" on teaching, and "How to write for the Putnam".
W 2/14 -- Wanrong gave a proof of Beatty's Theorem, and I gave another one. Handouts included Beatty's original problem in the Monthly and its solutions, an article by Buck on mathematical induction and recursive definitions, and another article on Pick's Theorem, this one by Grunbaum and Shepherd. All from the Monthly, via JSTOR. Also, a new article on "Undecidability in Number Theory" by Bjorn Poonen, from the latest Notices.
M 2/12 -- Dan made a presentation on rapidly growing functions. Handouts were the two magnificent articles by Richard Guy (via JSTOR) on the Strong Law of Small Numbers. I hinted at Beatty's Theorem.
F 2/9 -- Robert gave a presentation on push-down automata and what they can and cannot do. I distributed a copy of the article Lisa had talked about "The dark side of the Mobius strip", by Gideon Schwartz, a new (not on-line) article from Mathematics Magazine "On Infinitely Nested Radicals", and copies of articles from my website on how to teach ("Chalking It up"), and how to write questions for the Putnam. I also spent a good deal of class time on "war stories" on how things actually get done in universities.
W 2/7 -- Lisa gave a presentation on the Mobius strip. I distributed some handouts on the front page of the MathSciNet search engine, which is accessible from UIUC machines, and on the 2000 AMS Classification scheme and the way this department divides its faculty by area. I then made a presentation about Pick's Theorem and some results of mine which relate to it.
M 2/5 -- Dan gave another version of the proof of Ramsey's Theorem, in one special case. I actually talked about the Fibonacci example a bit more, gave the proof that the sum of (m^2n^2)^(-1), taken over all pairs of relatively prime positive integers, is 5/2. I talked about how, if x[n] = c1*a^n + c2*b^n, then linear algebra shows that there is a polynomial P satisfying P(x[n],x[n+1]) = 0 for all n if and only if there is a polynomial Q satisfying Q(a^n,b^n) = 0, and past work shows that this is true if and only if there are integers r and s so that a^r*b^s = 1. I distributed an old (1941) article on Fourier Transforms by Robert Cameron from Mathematics Magazine.
F 2/2 -- Michael finished his presentation on Sturm Liouville and Sturm Comparison and Sophia continued her presentation on Ramsey's Theorem. The handouts were two articles from Mathematics Magazine: one by Solomon Golomb on polyominoes and covering checkerboards and one by Martin Erickson on combinatorial existence theorems, including Ramsey's Theorem. I will be talking more next week.
W 1/30 -- Wanrong finished talking about Golomb's Conjecture. Michael started a presentation on the Sturm-Liouville Equation and on the Sturm Comparison Theorem. Sophia began her presentation on Ramsey's Theorem. The handouts were both found from Mathematics Magazine via JSTOR: a nice short article by Golomb (a visual proof of an identity usually proved by induction) and two songs of mine: "Hooray for Calculus" (written with Hal Fredericksen, a student of S. Golomb, and R. McEliece) and "A set is a set", a version of Russell's Paradox set to the "Mr. Ed" theme.
M 1/28 -- Lisa gave a solution to the Car Talk Puzzler, as did Wanrong, who also solved the polynomial problem, and started his presentation on the sequence shift problem that will be his project. The handouts included a page on "Knitting One-Sided Surfaces" by Nate Berglund, the essay "How Euler Did it" by Ed Sandifer from MAA Online and the essay on experimental mathematics by Doron Zeilberger which inspired Dan's presentation on 1/25.
F 1/25 -- Dan talked about the existence of a polynomial P(x,y) in two
variables with the property that P(F_n,F_{n+1}) = 0 for all n, where (F_n) is
the Fibonacci sequence. I elaborate and show that there is no
polynomial P(x,y) so that P(2^n,3^n) = 0 for all n. (Notes and
significance to follow.) I give Hilbert's proof that if f(t) and g(t)
are two polynomials in one variable, then there exists P(x,y) so that
P(f(t),g(t)) = 0 for all values of the variable t. (Notes to follow.)
Two problems to work on, and possibly present, on Monday. The first is
to generalize Dan's proof. Suppose (a(n)) is a sequence which satisfies the
Fibonacci recurrence: a(n) = c1 * phi^n + c2 * barphi^n, with phi = (1
+ sqrt{5})/2, barphi = (1-\sqrt{5})/2, find a specific polynomial P(x,y) so
that P(a(n),a(n+1)) = 0. The coefficients of P will depend on c1 and
c2.
The second problem is the car-talk puzzler from last week:
Palindromic Odometer .
The day's handouts include an on-line available
paper by Robin Chapman, giving many proofs of David's result from
1/23, an article by Brian Conrey on the Riemann Hypothesis, from the
March 2003 issue of the Notices of the AMS, and an interview of Bernd
Sturmfels from the January 2008 issue of the MAA Focus.
W 1/23 -- David talked about the evaluation of \sum_{n=1}^\infty^2 1/n^2 as pi^2/6 by elementary methods. This takes most of the period. I can now sort of use the bottom quarter of the board as I elaborate on a few of his points. Two Fibonacci handouts: Fibonacci handout I and Fibonacci handout II
M 1/21 -- University Holiday
F 1/18 -- Robert talked about the halting problem
and Turing machines and self-referential mathematical proofs.
I am recovering from a fall and cannot write on the blackboard. I fill
time by distributing a large number of printouts of the front of many
useful webpages, and talking about them:
UIUC Math Department LinkPage,
JSTOR Home Page (available from UIUC
machines),
Math Professional Society
Combined Membership List,
Mathematical Genealogy,
Univ. St. Andrews Math History Search,
Earliest Uses
of Mathematical Terms,
Mathworld,
Springer Encyclopedia of
Mathematics,
Plouffe's Inverter,
The Prime Pages, and quite a bit
of material
from The
On-Line Encyclopedia of Integer Sequences.
W 1/16 -- Dan is the first student presenter, talking about the halting problem and Turing machines. Handouts on The Erdos Number Project, including a nod to Bacon numbers, and a selection from the writings of UIUC alum Paul Halmos, taken from the October 2007 issue of the Notices of the AMS. Halmos was one of the most famous UIUC PhD students. His advisor was my late colleague Joe Doob, a member of the National Academy of Science and a contemporary of Halmos was David Blackwell. A fascinating interview with Prof. Blackwell is linked here. The first 1/8 of these 200 pages covers his childhood and his education at UIUC. Biographies of all three of these
M 1/14 -- First day of class. Four handouts:
Course
Organization, plus the questionnaire, and
Problem-solving template.
. Here's the version of
"One
Introduction to Mathematical Research" with all the jokes.
Homework assignment -- taken from the article: