Recent Sunday Speakers
Sunday, January 12, 2003:
Eric Matsen of Harvard University treated us to a topological proof of the "Fundamental Theorem of Algebra:"
Introducing "i," the square root of minus one, into the number system is all that is needed to make the solution of any kind of polynomial equation possible.
This is surprising. The number "i" certainly solves x^2 + 1 = 0. It is amazing that adding i to our number system (to create the "complex numbers") is enough to also solve something like: x^732 + 534x^23 + 45x^2 + 9773 = 0.
Sunday, January 5, 2003:
Leo Goldmakher of Princeton University gave a spectacular talk proving that the following result of Sophie Germain:
Suppose p is an odd prime such that q = 2p+1 is also prime. Then any integer solution to the equation
x^p + y^p + z^p = 0
has one of the integers x, y, or z a multiple of p.
This result is closely linked to Fermat's Last Theorem, and Germain used it to show that there are no nontrivial solutions to x^n + y^n = z^n for 2<n<100.
Question: Are there infinitely many primes p so that 2p+1 is also prime? No-one knows!
Sunday, December 15, 2002:
Friend of the Math Circle, Andrei Zelevinsky, of Northeastern University, gave a wonderful presentation counting the number of ways to get things wrong - at least in "hat-return" problems: What is the probability that N people attending a party each take home the wrong hat when guests' hats are distributed randomly? In a random distribution, what number of guests do you expect to receive the right hat, on average? The answers to the first question approaches 1/e as N grows large. Surprisingly, the answer to the second question is "1" irrespective of N.
Andrei asked us to ponder upon three questions:
1. Let D(n) be the number of ways to arrange the numbers 1, 2, ..., n so that no number is in its correct place. Is there an easy way to see that: D(n) = n*D(n-1) + (-1)^n ??
2. Andrei showed: n! = D(n) + (n choose 1)*D(n-1) + (n choose 2)*D(n-2) + ... + (n choose n)*D(0). Use this and induction to prove: D(n)/n! = 1 - 1/1! + 1/2! - 1/3! + ... +/- 1/n!.
3. Prove: n! = 0*D(n) + 1*(n choose 1)*D(n-1) + 2*(n choose 2)*D(n-2) + ... + n*(n choose n)*D(0) [Andrei used this to show that the average number of guests to get their correct hats is always 1.]
Andrei would be delighted if you shared your thoughts and answers with him at: firstname.lastname@example.org.
Sunday, December 8, 2002:
Sonal Jain of Harvard University treated us to the delights and mysteries of writing numbers as sums of squares. For example, he showed us that a number is a sum of two squares (e.g. 20 = 2^2 + 4^2) if it is composed only of primes that leave a remainder of 1 when divided by four (and the converse is also true), that, on average, a number can be written as a sum of two squares in pi/4 different ways, and that every number, if not a sum of two squares, is certainly a sum of at most four squares.
Sundays, November 17 and November 24:
Alas and alack I cannot give proper credit here to two fabulous speaker's who came to Math Circle these two days: Moses Liskov of MIT, and Dana Rowland of Merrimack College. Due to illness on my part I missed out on seeing these spectacular speakers in action. I hear that Moses did a wonderful job presenting ideas and extrapolations of the Euclidean Algorithm, and Dana treated our students to the amazing delights of the Prime Number Theorem.
Sunday, November 3, 2002:
Aaron Dinkin lured us into the realm of dark numbers. We proved that the counting numbers, the integers, the rationals, the algebraic numbers and even the computable numbers only take up a countable portion of the uncountable reals. This means, the majority of the numbers that exist cannot even be conceived.
Sunday, October 27, 2002:
Jake Abernethy introduced us, by way of physical demonstration, and intellectual pursuit, to a world that unites mathematics and juggling. Not only did Jake demonstrate for us the juggling patterns dictated by beautiful mathematical sequences, he also attempted to juggle those patterns that we proved to be impossible - and indeed they did not work!
Sunday, October 20, 2002:
Dan Zaharopol, of the MIT Educations Studies Program gave a super talk introducing to the power of generations and the derivation of Binet's formula for the Fibonacci numbers. He also showed some surprising connections to matchings in graphs.
Sunday, October 6, 2002:
Here's a honeycomb. The numbers show the number of ways to walk to the chosen cell from "Start" only ever moving one cell to the right, one cell diagonally down, or one cell diagonally up. Notice that these are the Fibonacci numbers!
Question 1: Let be the number of ordered partitions of n if the number “1” can be coloured one of two colours, red or blue say. (All other digits can be written only in black.) For example,
“3” = 3, 1(red) + 2, 1(blue) + 2, 2 + 1(red), 2 + 1(blue),
1(red) + 1(red) + 1(red), 1(red) + 1(red) + 1(blue), etc.
These turn out to be the numbers on the top row of the honeycomb - namely every second Fibonacci number. Speaker Jim Tanton showed why.
Question 2: The language "ABEEBA" has just three letters: A, B and E. Every combination of letters is word except those that have an A followed by an E. How many n letter words are there in this language?
Surprisingly, the answers are the numbers on the bottom of the honeycomb!
After exploring these curiousities we went on to explore the representation of numbers "base Fibonacci" (which turned out to be base phi, the golden ratio.)
Sunday, May 19, 2002:
Leo Goldmakher (mailto:email@example.com) gave a spectacular research talk presenting a solution to a problem that stumped James Tanton:
Let 1, 2, 3, 5, 6, 7, 10, 11, 13, 14, .... be the sequence of "square-free" numbers. If S(n) denotes the n-th element in this list, what's n/S(n) as n becomes large?
It turns out the answer is 6 divided by pi squared! The fact that answer is not zero has dashed James' immediate hopes of finding an elementary proof of the Prime Number Theorem. Well done Leo!
Sunday, May 12, 2002:
The product of two non-zero real numbers is never zero. The product of two non-zero complex numbers also is never zero. As every complex is really an ordered pair of real numbers, we now have the natural question: Is there a way to define the product of ordered triples so that non-zero triples never multiply to produce zero? Ordered-quadruples? Daniel Biss (mailto:firstname.lastname@example.org) brought us to the verge of discovering quaternions!
Sunday, May 5, 2002:
William Stein of Harvard University (email@example.com) gave a wonderful presentation on the Diffie-Hellman Key Exchange Protocol, describing the seminal algorithm that got public-key cryptography going.
Sunday, April 28, 2002:
Mira Bernstein (mailto:(firstname.lastname@example.org) gave a dynamic presentation on the "hat problem" that caused a tizzy of excitement amongst mathematicians across the nation a few years ago over the e-mail circuit. The problem has deep connections to Hamming codes. See her website www.math.stanford.edu/~mira/hats.pdf for details.