Undergraduate Colloquium

Fall 2007

August 22 No Talk

August 29 No Talk

September 5 Dennis Allison

The Method of Archimedes
Abstract: The greatest mathematician of the Ancient World, and one of the greatest of all time, was Archimedes (287-212 B.C.). By applying his law of the lever and the method of exhaustion from Eudoxus, Archimedes was able to compute volumes of certain solids using what is now known as "the Method." In this talk we will examine the Method and see how it anticipated the integral calculus by 2000 years.

September 12 Stewart Ethier

Parrondo's Paradox
Abstract: Parrondo's paradox, due to Spanish physicist Juan Parrondo (1996), states that a player may alternate regularly or randomly between two negative-expectation games to achieve a positive-expectation game. There are a variety of games that exhibit the Parrondo paradox, including capital-dependent games, history-dependent games, multi-player games, and cooperative games. We will describe each of these and explain why the "paradox" holds.

September 19 Dylan Zwick

How Far Can You Go With Calculus?
Abstract: So there's no question calculus is useful. In fact, it's probably the most useful idea in the history of mankind. But, what you might not be aware is that by using some pretty basic ideas from calculus you can actually prove some amazing things. Things you wouldn't expect, and that hint at much deeper mathematics. In this talk I will give three examples of this where we will use basic calculus ideas, especially sequences and series, to prove some astonishing results. We will, for example, derive a closed form formula for the Fibonacci sequence and figure out a proof of the infinitude of primes, discovered by Euler, that is completely different than Euclid's classic proof. It should be fun, and while the results will probably be new, the math used will not go beyond first year calculus.

September 26 Aaron Wood

The Hat Game
Abstract: In the hat game, the host puts either a blue or a red hat on each of the 2^k-1 participants so that each participant can see all of the hat colors except for their own. The host then takes them into a room one at a time and gives them the chance to either pass or guess the color of their hat. The host wins if a participant guesses wrong or if everybody passes. Otherwise the host loses. Supposing that the players can develop a strategy before the game begins, what should their strategy be in order to optimize their chance of winning? For 3 players, i.e. k=2, the strategy is fairly straightforward, but the extension of this strategy to higher values of k is not so obvious. We will develop some ideas in coding theory and introduce the Hamming code, which is capable of correcting a single error. This code will be utilized in developing the optimal strategy for the hat game.

October 3 Aaron Bertram

Why does pi keep popping up?
Abstract: The number pi is defined in terms of the circumference and diameter of a circle. After a course in calculus, it then comes as no surprise that pi comes up in the formulas for the area of a circle and volume of a sphere. With a considerable amount of effort, one can prove that pi is irrational, and even transcendental. With all the transcendental numbers to choose from, why then is:

1 + 1/4 + 1/9 + 1/16 + ... = pi²/6?
and why does pi show up in Stirling's approximation of n!? We'll ponder these and other mysteries of pi.

October 10 No Talk - Fall Break

Abstract:

October 17 Bill Casselman, University of British Columbia

How to read Euclid
Abstract: In spite of the fact that Euclid's Elements of Geometry is available in a cheap paperback edition from Dover and can also be found on the Internet, it is not widely read by mathematicians at any level, nor used in many courses. I'll try to explain why anyone interested in mathematics can and should read the Elements with pleasure, and what techniques one might use to make it more pleasurable.

October 24 Arend Bayer

"I bet you are wrong - probability paradoxa and more"
Abstract: Given an everyday math question, the odds are pretty high that it is a probability question. And many simple questions that start with "What are the odds..." have quite unintuitive answers, and are a lot of fun to discuss.

This talk will play with several such probability paradoxa, and explore Bayes' formula (likely THE most important mathematical formula - assuming the concern is to answer the everyday questions above...)

October 31 Dan Margalit

Juggling Braids and Permutations
Abstract: Besides being a lot of fun, juggling is a very mathematical endeavor. For instance, special sequences of numbers can give rise to juggling patterns. In this talk, I will demonstrate the juggling patterns known (in "site swap" notation) as 531, 5313, 5551, 40141, and others. We will investigate a connection to the permutations of the integers, and explain how to count the number of periodic juggling patterns of a certain length. Also, we will discuss the relationship between juggling and the mathematical theory of braids. We will see why your favorite hair braid is jugglable!

November 7 Paul Monsky, Brandeis University

Cutting up trapezoids into trianges all of the same area
Abstract: A square may be cut up into triangles all of the same area. Whether a trapezoid has such an "equidissection" depends on the ratio, r, of the trapezoid's parallel sides. (For example it's possible when r is rational, and impossible when r is pi). When ar2-br+c=0 with a, b, and c positive integers, Elaine Kasimatis and Sherman Stein showed it can be done. I'll explain how to get equidissections when the quadratic equation for r is replaced by a certain type of cubic or quartic equation. To do this, I look at 4 simultaneous quadratic equations in 6 variables, and try to find solutions in positive integers; miraculously this is often possible.

November 14 Amber Smith

Who Needs a Watch?! The Science of Biological Clocks
Abstract: Building a clock that could keep accurate time on a ship was a challenge for inventors in the 18th century. However, billions of years before, bacteria living in these same seas were already performing regular tasks on a 24 hour cycle. Biological clocks are apparent everywhere in nature, from the timing of your sleep-wake cycle to the firing of heartbeats. These cycles are a result of an enormous amount of complex interaction on both the molecular and cellular level. As such, these processes present exciting challenges for both biologists and mathematicians. This is the motivation for the Spring 2008 REU project where you could be investigating one such clock with a team of mathematics & biology students and professors. In this talk, I will introduce you to the many places we find biological clocks and give you more details about the REU project.

November 21 Movie: The Math Life

Abstract: "Why did a magician become a mathematician? How can a person see in four dimensions? What does a mathematical proof have in common with a Picasso portrait? This elegant program brings to life the human dimension of mathematics through lively interviews with Freeman Dyson, David Mumford, Ingrid Daubechies, Persi Diaconis, Michael Freedman, Fan Chung Graham, Kate Okikiolu, Jennifer Tour Chayes, Peter Sarnak, Steven Strogatz, and seven other mathematicians. These captivating luminaries vividly communicate the excitement and wonder that fuel their work as they explore the world through patterns, shapes, motions, and probabilities. Computer animations and analogies drawn from the visual arts are incorporated, to maximize accessibility to the fascinating concepts discussed."

November 28 Peter Alfeld

Multivariate Splines and the Bernstein Bezier Form of a Polynomial
Abstract: Splines are smooth piecewise polynomial functions. They are used ubiquitously for the numerical solution of univariate problems which involve one independent variable. Splines are well understood in that context. The structure of multivariate splines is much more complicated and a number of difficult problems are unsolved. I will describe the Bernstein Bezier form of a multivariate polynomial which is a major tool for spline research and will use it to describe and motivate a number of unsolved problems. The talk will include computer demonstrations.

December 5 Rex Butler

The Halting Problem and Rice's Theorem
Abstract: Suppose we have a collection of simple computer programs. 'Simple' meaning that each program does nothing but input an integer, compute, and if a result is obtained, output an integer and halt.

Perhaps the most essential question we can ask about our collection of programs is not which programs finish and output results as expected, but rather which programs finish, period. For example, a program which outputs the least prime greater than the input will always halt, because there are always more primes. But, given a polynomial, a program which outputs the least root greater than the input will not halt: there are only finitely many roots.

I will discuss the following problem: given a fixed input, can we determine which programs will eventually halt on that input and which will not? I will do so by first defining the partial recursive functions. These formalize the intuitive idea of tangible computation. Finally, time permitting, I will state Rice's theorem which determines exactly which properties, such as halting, we can detect.