Undergraduate Colloquium

Spring 2007

January 10 No Talk

Aspects of Wireless Communications - Analog to Digital Converters
Abstract: Although we increasingly think of digital data, most transmission modalities such as cell phones still send analog (continuous) signals. How then does one transfer between the analog and digital domain? We will discuss the classical Shannon Sampling Theorem which is the underpinning of analog to digital converters, and I will present a recently developed algorithm to allow for greater transmission rates as are necessary for wireless video.

January 24 Gordan Savin

Construction of a Regular Pentagon using Straightedge and Compass
Abstract: The problem of constructing a regular pentagon by compass and straightedge looks like a geometric problem. However, ideas from algebras are needed to solve this problem. These ideas will also explain why it is possible to construct a regular 17-gon and 257-gon, but not a regular 7-gon.

January 31 Mladen Bestvina

On the Total Curvature of Knots
Abstract: By the (total) curvature of a polygon in, say, 3-dimensional Euclidean space I mean the sum of all exterior angles. For example, for convex planar polygons the curvature is 2*pi. I will discuss a theorem of Fenchel and Borsuk that for all other polygonal curves the curvature is greater than 2*pi. Borsuk raised a question whether for knotted polygonal curves one could show that the curvature is greater than or equal to 4*pi. This problem was solved in 1949 by a Princeton undergraduate student named John Milnor. I will discuss Milnor's solution. If there is time at the end, I will discus the relationship between this "piecewise linear" concept of curvature and the more traditional curvature of smooth curves in differential geometry.

February 7 No Talk

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February 14 No Talk

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February 21 No Talk

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February 28 No Talk

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March 7 No Talk

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March 14 No Talk

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March 21 No Talk (Spring Break)

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March 28 Dylan Zwick

The Brachistochrone - How One Problem's Solution Rewrote Classical Mechanics
Abstract: In 1696 Johann Bernoulli posed the following problem -

Given two points A and B in a vertical plane, what is the curve traced out by a point acted on only by gravity, which starts at A and reaches B in the shortest time.

The problem was solved by Newton, Leibnitz, de L'Hospital, and Johann's brother Jacob. It led to a new branch of analysis called the calculus of variations, and eventually to a deep reformulation of Newton's laws of motion by Lagrange. In this seminar I will explain the problem and its solution, introduce some more general concepts from the calculus of variations, and discuss its applications to classical mechanics. Along with the mathematics, I will cover some of the history and the characters behind this problem and its subsequent development. It's a great story, and it culminates with some amazing and surprising facts about how our universe works.

To understand everything you should know multivariable calculus, to understand most of it you'll need to know single variable calculus, and to learn something wonderful you'll just need to show up.

April 4 Mike Purcell

Intuition and Conditional Probability: an examination of the Monty Hall Problem
Abstract: Probability is a field rich in paradox. There exist many well known problems, the solutions to which are highly counterintuitive. Perhaps the most famous of these is the so-called Monty Hall Problem. This problem has appeared repeatedly in popular culture (including mention in a recent episode of the T.V show Numb3rs, and a much publicized appearance in Parade magazine in September of 1990) and is an excellent illustration of how our intuition can lead us astray when dealing with conditional probabilities. In this talk we will examine several problems whose solutions seem (at first) to be paradoxical, including the aforementioned Monty Hall Problem, the Three Card Problem, and (time allowing) the Waiting Time Paradox. Because these problems can be addressed with only the most basic of probabilistic tools, we will take the time to build the necessary machinery, and the talk will be self contained.

April 11 Bill Casselman, University of British Columbia (Distinguished Lecture Series) This talk will be held at 2:00 PM in LCB 121 - please note the change in time and room

All About n!
Abstract: The factorial function n! = 1*2*3* ... *n is ubiquitous in mathematics, playing both theoretical and practical roles of great importance. It grows very fast as n grows, and for many the n! button on their calculator is well known to produce garbage for large enough n. This talk will cover various methods for calculating n! for large n, including some remarks on modern techniques for fast computer arithmetic. But I'll concentrate on a method due originally to the Scottish mathematician James Stirling in the 18th century, but developed by Euler, Legendre, Gauss, Poisson, and other major mathematicians over a period of many years.

April 18 Karim Khader

Rademacher Functions and a Sequence of Independent Coin Flips
Abstract: We will construct a concrete mathematical model for a sequence of independent coin flips. We will see how the notion of independence comes into play in this particular model and will prove the strong law of large numbers. There is a wonderful connection between the strong law of large numbers and normal numbers which we will also investigate briefly. The talk should be self contained and will include a very brief description of the important ideas related to Lebesgue measure.

April 25 Peter Trapa

Mysteries of the McKay Correspondence
Abstract: We consider two problems, one combinatorial and one geometric. The combinatorial one asks for a classification of all so-called harmonic graphs. Such a graph is finite, has no self-loops, possesses only single undirected edges, and has a distinguished vertex. In addition, each edge is labeled by a nonzero positive integer and the distinguished vertex has label one. Finally we require that twice the label of each vertex v is equal to the sum of the labels of the vertices adjacent to v.

The second, seemingly unrelated problem asks for the classification of all finite subsets S of rotations in three dimensions which are closed in the sense that if r1 and r2 are in S, then their composite is also in S. For instance, one could fix a polyhedron in three-space and take S to be all rotations that preserve it.

About thirty years ago, McKay discovered a remarkable correspondence between the answers to these questions. In many ways, the McKay correspondence even today is still mysterious and poorly understood. This talk will offer a taste of these mysteries.