Undergraduate Colloquium Spring 2004

- January 13 NO COLLOQUIUM

- January 20 Andrejs Treibergs
**Squeezing Elastic Tubes**

*Abstract:*What shape is the cross section of a rubber hose if there is more air pressure on the outside than on the inside? The same equations answer a related geometric question: what closed curve in the plane with a given length and surrounding a given area has the least bending energy? These are examples of optimization problems from the calculus of variations. The Euler-Lagrange equations will be introduced and used to study the problem.- January 27 Sandra Spiroff
**Unique Factorization**

*Abstract:*Starting from the familiar factorization of integers into primes, we extend the concept of unique factorization to polynomials and beyond. In particular, we will discuss how unique factorization, or lack of it, influenced early attempts to prove Fermat's Last Theorem, and we will explore how it can be used to determine the probabilities associated with rolling a pair of dice.- February 3 Gordan Savin

**The Lucas-Lehmer Test for Mersenne Primes**

*Abstract:*Primes of the form 2^{n}-1 are called Mersenne primes. The first three Mersenne primes are 3, 7, and 31, which correspond to n = 2, 3, and 5. There are only 40 known Mersenne primes. The last, discovered on November 17, 2003, is 2^{20,996,011}-1. (Sorry I cannot write it down as it has 6,320,430 decimal digits. Assuming that we can print about 6,000 digits per page, we would need about 1,000 pages to accomplish this task.) This number has been found through The Great Internet Mersenne Prime Search (GIMPS) which uses the combined computing power of thousands of personal computers worldwide. There is also a prize of $100,000 available for those involved in finding the first Mersenne prime with over 10,000,000 digits. Computers aside, testing is based on a simple test discovered by Lucas in the 1870's and simplified later by Lehmer in the 1930's. The Lucas number, 2^{127}-1, remained the largest Mersenne prime until the 1950's when 2^{521}-1 was shown to be the next Mersenne prime.- February 10 Stewart Ethier
**Seven Shuffles Suffice**

*Abstract:*Rarely is a mathematical discovery of such general interest that it is the basis of a feature article in The New York Times, but on January 9, 1990, the newspaper of record described the research of Dave Bayer and Persi Diaconis showing that seven riffle shuffles are both necessary and sufficient to adequately mix a deck of 52 cards. The goal of this lecture is to formulate and prove the theorem of Bayer and Diaconis that leads to this conclusion. The key idea in the proof is the concept of a rising sequence, first exploited around the beginning of the 20th century by magicians C. O. Williams and C. T. Jordan. As an added bonus, we will perform a card trick due to Jordan that is based on this concept.- February 17 Davar Khoshnevisan
**"What Does a Typical Number Look Like?"**

and Other Enchanting Tales

*Abstract:*A number is normal if the fraction (more properly, the asymptotic density) of 0's, 1's, 2's, ... , 9's in its decimal expansion is one-tenth each. Even though it is very difficult to know if any given number is normal, a 1904 discovery of Émile Borel implies that "most" numbers are normal. To understand this we will re-examine what we know of the decimal system. Time permitting, I will discuss further connections to entropy, dimension, and symbolic dynamics on the line.- Friday, February 27 SPECIAL COLLOQUIUM - Piotr Kokoszka (USU)
**Statistical Self-Similarity: Applications to market and network data**

*Abstract:*The last decade has seen the emergence of two important sources of statistical data: high frequency market data and digital network data. After reviewing the nature of these data, we will introduce the concept of statistical self-similarity and show how it reflects the structure of the data. We will then outline statistical methods for detecting self-similarity and sicuss mathematical theories constructed to explain the observed self-similarity. We will conclude with some recent challenges to these theories and describe the relevant research effort currently under way.

The talk will be accessible to undergraduate math, science, business and engineering majors. It will focus on ideas rather than technical arguments. Wherever possible, suitable visualizations will be used to explain the concepts.- March 2 Fred Adler
**Can Ants Do Calculus? and if not, can they fake it?**

*Abstract:*Animals can be thought of as using sophisticated analogue computers to solve complex optimization problems. For examle, colonies of ants must decide where workers should look for food. Sending too many to one place might be a waste of effort, while sending them over the entire desert might be a waste of time. Solving this problem requires thinking about the difference between what is good for an individual ant and what is good for the colony, which corresponds somewhat surprisingly, to the difference between secant and tangent lines in calculus.

In solving this problem, realants have two major drawbacks. First, they have only limited means to communicate information to each other. Second, they are stupid. I will show that ants can do precalculus and find a pretty good solution to the problem, and then extend this idea to how they can fake their way through calculus and find an even better solution.- March 9 Don Tucker
**On the Infinitude of Primes and Twin Primes**

*Abstract:*I will first introduce the idea of remodulization of congruence classes and use that to characterize the complementary sets of systems of congruences. This will be followed by characterizations of the primes and the twin primes in terms of the complementary sets of systems of congruences and a proof of the existence of infinitely many primes followed by an effort at a proof of the existence of infinitely many twin prime pairs. The listeners are enoucraged to complete the proof and become famous.

**Note: This talk will be continued on Thursday, March 11 at 12:55 PM in JWB 333.**- March 16 NO COLLOQUIUM - Spring Break
- March 23 Henryk Hecht
**Regular Polyhedra**

*Abstract:*We can construct regular polygons with an arbitrary number of sides. However, there exist only five distinct regular polyhedra. Why is it so? We investigate this, and related questions, about polyhedra. In particular, we outline a beautiful construction of a regular polyhedron with twenty sides (icosahedron) due to Luca Pacioli, who was a friend of Leonardo da Vinci.- March 30 Jim Carlson
**The Mathematics of Google**

*Abstract:*Two Stanford graduate students, Sergei Brin and Larry Page, developed a way of searching the world wide web that was far more effective than the methods used by other search engines. Thus Google was born. Their method was based on an elegant piece of mathematics which we will explain.- April 6 Mike Woodbury

**Congruent Numbers and Elliptic Curves or How Harder Math is Sometimes Easier**

*Abstract:*An elliptic curve is a set of solutions to an equation of the following form

*y*^{2}= x^{3}+ bx + c

A congruent number is a whole number that is the area of a right triangle that has sides whose lengths are fractions. We will see that determining whether*n*is a congruent number is (almost?) equivalent to finding rational solutions on the elliptic curve

*y*.^{2}= x^{3}- n^{2}x

You might ask, "So what?" Well, as is often the case in math, by changing the context of the problem a previously unobtainable solution becomes relatively simple. Understanding elliptic curves involves geometry, algebra and topology, and has many applications to cryptography and number theory. Motivated by the congruent number problem, I will introduce this theory. Only a basic knowledge of linear algebra and calculus is needed.- April 13 Tom Robbins

**Biological Invasions: from critters to mathematics**

*Abstract:*Biological invasion, defined as the introduction and spread of an exotic species within an ecosystem, is an ever increasing problem. It has been estimated that the cumulative losses in the United States to harmful non-indigenous species has exceeded $100 billion by 1991. In addition, biological invasion is a main contributor to the loss in biological diversity in the Earth's most sensitive regions and is one of the greatest threats to endangered species. In this talk, we will consider only one part of the invasion process, the spread of a biological species. Several case studies and mathematical models will be presented.- April 20 David Dobson

**Smelborp Esrevni: backward thinking and its applications**

*Abstract:*Inverse problems encompass an extremely wide variety of applications and mathematical techniques. Application areas include medical imaging, geophysics, astronomy, nondestructive testing, and microscopy. Relevant mathematical techniques involve differential equations, optimization theory, functional analysis, numerical analysis, and scientific computation. Problems generally involve obtaining information about inaccessible or "hard to see" quantities (such as the detailed subsurface structure of a certain part of the earth) from indirect measurements (such as recordings of reflections of acoustic waves directed into the earth). The field is rich with fascinating open problems and important scientific applications.

This talk will attempt to describe some of the basic difficulties encountered in inverse problems and what kinds of ideas can be used to define, and obtain approximate solutions. The talk should be accessible to everyone who has seen calculus and a little linear algebra.- April 27 Nelson Beebe

**Pseudo-random numbers: mostly a line of code at a time**

*Abstract:*Random numbers have an amazing range of applications in both theory and practice. Approximately-random numbers generated on a computer are called pseudo-random. This talk discusses how one generates and tests such numbers, and shows how this study is related to important mathematics and statistics - the Central Limit Theorem and the X^2 measure - that have broad applications in many fields.Come and find out what the Birthday Paradox, Diehard batteries, gorillas, Prussian cavalry, and Queen Mary have to do with random numbers.