This is the schedule for the rest of the semester:
W 4/20 No class. I'll be in my office 327 AH.
F 4/22 We meet in the Rare Book Room at 2pm. Ask if you need directions.
M 4/25 One meeting (at 2) Summing up. I'll have some things to say and
some last minute encouragement, or what I think will be encouraging.
W 4/27 Speakers: Wing Ho Ko and Abbey Rechner.
F 4/29 Speakers: Sabreena Ashraf and Nana Arizumi.
IMPORTANT NOTE: "DRAFTS" OF YOUR FINAL REPORT ARE DUE IN CLASS. I WILL COMMENT ON THEM AND HAVE THEM BACK BY 5/2. IF YOU ARE really STUCK, YOU CAN TURN THEM IN 5/2 AND I'LL HAVE THEM BACK BY 5/4!
M 5/2 (2pm) Speakers: Seth Case and Jim Freitag.
M 5/2 (5pm) Speakers: Greg Stanton and make up time if any of the first
six missed their slots.
W 5/4 Speakers: Mike Salwan and Holly Krieger.
If you, as speaker, want me to have any material reproduced for the rest of the class, please get it to me by noon of the day you will speak. Please plan your presentation to cover no more than 20-25 minutes. This schedule is tight.
Final report due Thursday 5/12 1:30 at my house, 308 W. Delaware Ave, U. This is an easy 25 minute walk from Altgeld, and is close to both the Red and the Orchard Downs lines. There will be a party from 1:30 -- 4:30, the official Final Exam period for this course.
Please let me know if I have made any errors in the schedule or if you have any questions. I will also be posting this on the webpage.
My intention is to provide, at the very
least, an archive for all of the TeX-d handouts in the course and a guide to
the semester, class-by-class.
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.
First class presentation: can you use dominos to cover an 8 x 8 checkerboard with antipodal corners missing? This was followed by many variations.
First assignment: take a 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.
F 1/21 -- A presentation by Wing Ho Ko of his solution to the first assignment, involving the cardinalities of infinite sets, Cantor diagonalization, and the differences between N and R. Class members: talk in class, and you can see your name here, too! I started to apply the methods of Now what to the following situation. As usual, let {F_n} denote the Fibonacci sequence, defined by F_0 = 0, F_1 = 1 and F_n = F_{n-1} + F_{n-2} for n \ge 2. There is a ``standard" addition formula; namely F_{m+n+1} = F_{m+1}F_{n+1} + F_mF_n. We are encouraged to find variations, questions and things suggested by this. Among them: is there a formula if we change the initial conditions? what if we change the coefficients of the recurrence? is there a connection to the addition formula for the trig functions? what are the sequences which satisfy this addition formula? can the Fibonacci numbers be extended to negative indices? fractional indices? Note that F_{2n+1} = F_{n+1}^2 + F_n^2 is a sum of two squares, the characteristic of an important class of integers. On the other hand, it's not hard to show that if m = 4 (mod 6), then F_m = 3 (mod 4), so F_m is not a sum of two squares. Further investigation seems useful. And, it turns out that there is a very strong connection between the addition formulas. Let F denote the set of sequences which satisfy the Fibonacci recurrence. Then dim(F) = 2, and a basis for F consists of the sequences whose n-th terms are F_n and F_{n+1} respectively. Thus, for fixed m, F_{m+n+1} must be a linear combination. Similarly, the space of solutions to y" + y = 0 is spanned by cos[x] and sin[x], and for each fixed t, cos[x+t] and sin[x+t] must be a linear combination of cos[x] and sin[x]. This is what I mean by a "rhyme".
Homework for Monday: Is it possible to cover an 8x8 checkerboard plus one other square with 13 1x5 "dominos"? If so, where can the extra square be placed?
M 1/24 -- Two enthusiastic presentations filled the day. Holly Krieger talked about several uniqueness theorems about analytic functions of one complex variable, based ultimately on the idea that if f is analytic on a domain D and z_0 in D is such that f(z_0) = f'(z_0) = f''(z_0) = ...., then f is identically 0 on D. Mike Salwan gave a logic/strategy problem involving the proper allocation of gold coins to a large group of rational and well-behaved pirates. The main idea was working backwards from what the last person would do, in a deterministic way, until you figured out what the first person would do. Answer to homework question: "no". If you tile the checkerboard with an ever-repeating pattern of "12345123..." both vertically and horizontally, you'll see that the colors appear 12, 13, 14, 13 and 12 times. This means that if the checkerboard is covered with 1 x 5 dominos, then there must be at least 14 used.
W 1/26 -- Three enthusiastic presentations filled this day. Nana Arizumi talked about the infinitude of primes of the form 3n+2. Greg Stanton talked about entire functions (functions of a complex variable which are analytic on the entire complex plane) and the values they may or may not omit conditions which lead to them being constant. Abbey Rechner talked about some proofs that the square roots of 2 (or 3), are irrational. I gave a quick proof of the Binet formula for the Fibonacci numbers, and expect to spend most of Friday on some Fibonacci variations. The class consists of the six people who have given talks, plus Sabreena Ashraf, Seth Case and Jim Freitag, who were in last year's class, and will at some point talk about their projects, or something else. There was a handout on REUs, consisting of printouts of two webpages, the links being NSF REU list and UIUC REU's in 2005.
F 1/28 -- Two handouts on the confusing way that mathematical
knowledge is organized. This was pieced together from the description
of graduate areas at UIUC (individually linkable from Research Areas) and
text cut-and-pasted from the descriptions of mathematical areas
at the NSF, Harvard, Michigan and Berkeley. The selection is random,
and the point is just that every place organizes things
differently. This is neither good nor bad, but it shows that to a
large extent, the organization into areas is idiosyncratic and not
always inherent in the topic. As I like to say, if you ask a
polynomial if it is an object in algebra, an object in analysis, an
object in number theory or an object in combinatorics, it will not
answer you.
The mathematical content of Friday's class was a fantasia on the
Fibonacci theme, establishing how we can define F_n for negative n,
but not so easily for half-integer n. (For example, the addition
formula, F_{m+n+1} = F_{m+1}F_{n+1} + F_mF_n, with m = n = -1/2
implies that F_{-1/2} = F_{1/2} = 0, so F_{k+1/2} = 0 for all integers
k. But with m=n=1/2 in the addition formula, we now get a
contradiction. The answer is to ignore the question or else allow
complex non-integral Fibonacci numbers. The Binet formula for the
Fibonacci numbers was derived by observing that in the two-dimensional
structure discussed earlier, (1,a,a^2,a^3,...) is in the vector space
of Fibonacci sequences if a^2-a-1 = 0; ie a = (1 + \sqrt[5])/2 or
(1 - \sqrt[5])/2.
We also discussed, in the manner of "now what", the
addition formula. Suppose a_n is a sequence and a_{m+n+1} =
a_{m+1}a_{n+1} + a_ma_n for all m,n \ge 0, what can we say about
a_n. We saw quickly, via m=0, that there are 2 cases. If (a_0,a_1) is
not (0,1), then a_n has to be a geometric series and a_n =
c^{n+1}/(c^2+1) for some number c. On the other hand, if a_0 = 0 and
a_1 = 1, then a_n = a_2 a_{n-1} + a_{n-2}. Writing a = a_2, we see a
nice pattern.
a_0 = 0
a_1 = 1
a_2 = a
a_3 = a^2 + 1
a_4 = a^3 + 2a
a_5 = a^4 + 3a^2 + 1
Your job is to make this pattern into a theorem and prove it. We'll talk about this on Monday, and Jim Freitag will talk about his project from last year.
M 1/31 -- We agreed that a "fourth hour" would be scheduled on
Mondays at 4, to accomodate my being out of town in March and to give
room at the end of the semester for the students to concentrate on
their projects. Handout: an interview with two famous mathematicians,
M. Atiyah and I. Singer, from the latest Notices, link to
follow, and then material on the Atiyah-Singer Index Theorem, mostly
from mathworld, link also to follow.
Mike Salwan and Abbey Rechner worked out the pattern, involving
binomial coefficients, and then I gave the proof. Here's roughly
how it goes: suppose a_0 = 0, a_1 = 1 and a_n = a * a_{n-1} + a_{n-2}
for n \ge 2. Construct the generating function
A(x) = a_0 + a_1 x + a_2 x^2 + a_3 x^3 + .....
The recurrence implies that (1 - a x - x^2)*A(x) = 0 + 1*x, and so A(x) = x/(1 - a x - x^2). If a were numerical, this would be expanded by partial fractions, and you still can do that, but there's also the idea of expanding this as a geometric series
A(x) = x \sum_{i=0}^\infty (a x + x^2)^i.
If you look for the coefficient of x^n on both sides, you get the closed form for a_n.
Jim Freitag began to talk about his REU experience with evolutionary game theory, and will continue on Wednesday.
W 2/2 -- Jim continued to discuss his evolutionary game theory problem from his REU. The key names are Peyton Young, Markov Chains and Schilling's Neighborhood Game. I passed out a copy of notes on the Fibonacci sequence (link to come) and then used the addition formula to prove that if k and n are positive integers then F_n always divides F_{kn}. This can also be proved via the formula for F_n involving phi and phi bar. The next question is to look at F_{2n}/F_n and F_{3n}/F_n, which are both sequences of integers, and try to see what recurrences *they* solve.
F 2/4 -- I began by discussing what REU organizers might be looking for; most of the class is graduating in May, however, and is not eligible for an REU any way. I also announced that the extra hour next Monday will be in 141 Altgeld. Seth Case spent most of the class period developing group theory to the point that he could start to talk about his REU last summer on Coxeter groups.
M 2/7 -- I passed out a packet of links to interesting material on the web. (Actual links to come!) Seth Case finished up talking about Coxeter groups. His talks illustrated one of the misleading aspects of this course. Usually, research is not as accessible as what I present here and you need to go through much preparatory material.
M 2/7 -- (2nd hour) I talked about the old switcheroo. What are the recurrences satisfied by the Fibonacci sequence? More generally, if a sequence is given in closed form, how do you find the recurrences it satisfies. This led to questions like determining a closed form for phi^n in terms of the Fibonacci sequence, and a version of the Conway Peg problem.
W 2/9 -- Handouts on the Conway peg problem and the paper I wrote with Mike Bennett on the equation x^y = y^{mx}, inspired by an earlier incarnation of this course. Both will have links to follow. I botch of the proof of Brown's Theorem; this will be fixed on Friday.
F 2/11 -- Sabreena Ashraf was finally able to get away from her internship to attend class and introduce herself to the class; she was in the last version of this class. I went through several free associative leaps, starting with the correct proof of Brown's Theorem on necessary and sufficient conditions on a sequence of positive integers a0 \le a1 \le a2 ... so that every positive integer is a sum of a subset of the aj's. (Also handed out: a copy of an old paper of mine related to this subject.) One variation is the Putnam problem posed by your instructor back in the early 1980s: how many ways can an integer m be written as \sum_{j=0}^\infty a_j*2^j, if we allow "digits" of {0,1,2,3}, rather than the usual {0,1}. The correct answer, [m/2] + 1, can be derived in many different ways: by induction, through writing a_j = 2b_j + c_j, with b_j,c_j \in {0,1}, and through a generating function. This led to the question of the method of parital fractions, which is not explained adequately before it's used in the techniques of integration part of calculus.