# Undergraduate Colloquium

**Spring 2009**

- January 14 Sam MaWhinney, Colorado School of Public Health
**A Varying-Coefficient Method for Analyzing Longitudinal Data with Nonignorable Dropout**

*Abstract:*Dropout is common in longitudinal studies and is nonignorable when the probability of dropout depends on unobserved outcomes even after conditioning on available data. In the presence of nonignorable dropout, mixture models can be used to account for the relationship between a longitudinal outcome and dropout. We propose a varying-coefficient mixture model using natural cubic B-spline basis functions (Vncs) to semiparametrically model the outcome-dropout relationship. This method is computationally stable, highly flexible and relatively simple to implement. We apply the Vncs and comparable methods to an HIV/AIDS clinical trial that shows evidence of nonignorable dropout. In addition, we conduct simulation studies to evaluate performance and compare methodologies in settings where the longitudinal trajectories are linear over time and the time of dropout is observed for all individuals. The simulation studies suggest that the Vncs is an improvement over existing methods when dropout has a nonlinear dependence on the outcome.*** Bonus: Dr. MaWhinney will bring gift certificates to the Dairy Keen up in Heber! ***

- January 21 Jay Mace, Meteorology
**Toward Improved Understanding of Clouds in the Climate System**

*Abstract:*Since the first fully operational climate models were implemented in the middle to late 1980's the complicated role of cloud feedbacks have continued to be one of the major sources of uncertainty regarding anthropogenic climate change. In fact, these uncertainties have decreased very little in the last 30 years despite great effort on the part of the atmospheric science community. I will discuss why this is the case and summarize our current and emerging strategies for solving this problem.Dr. Mace will also discuss graduate study in meteorology.

- January 28 Suresh Venkatasubramanian, Computer Science
**Zero Knowledge Proofs, or "I know it, and you know I know it, but you don't know what I know or how I know it."**

*Abstract:*Imagine you're a VP at Lehman Brothers going down in the elevator on Sep 15. At the 15th floor, in walks another VP. The elevator descends, and as the two of you stare morosely at your scuffed patent-leather shoes, but one thought goes through both your heads: "Did this other joker lose more money than I did?"Can you both figure this out WITHOUT either of you knowing how much the other lost, and without relying on anyone else's help? The answer, along with a helping of Sudoku and maybe some Ali Baba and the 40 thieves, can be yours for the price of one highly encrypted bit...

- February 4 Aaron Wood
**Binary Gray Codes**

*Abstract:*An n-digit binary Gray code is an ordering of all 2^n strings of 0's and 1's of length n so that consecutive "binary numbers" in the list differ only in one digit. In 1947, Frank Gray developed such an ordering of binary n-strings, called the binary reflected Gray code, that was used in encoding analog signals into strings of binary digits. Today, binary Gray codes have many applications, including digital error correction, circuit testing, and puzzle solving. In this talk we will discuss a few methods for generating Gray codes, including a method for generating "balanced" codes.- February 11 Ben Trahan
**Gödel's Incompleteness Theorems**

*Abstract:*One of the major problems of the 19th and early 20th centuries -- and to a lesser extent, the past 2500 years -- was to find a set of axioms that described all of mathematics. A good set of axioms should not only be consistent, but also answer every question in mathematics. In 1931, Kurt Gödel surprised everyone by proving that it couldn't be done -- for any set of axioms, there will always be sentences that are true but can't be proved. In this talk, I will outline the proof of this surprising but extremely accessible result.- February 18 Peter Alfeld
**What Can You Do With A Slide Rule?**

*Abstract:*Back in the days when people first went to the moon, electronic calculators did not exist. Instead we used slide rules. They were indispensable for professionals, and students were required to own one and know how to use it. There were courses on the proper use of a slide rule. Just like calculators today, slide rules were mostly everyday and commonplace instruments, but some were fancy, expensive, and treasured by their owner.I'll describe how slide rules work, why they work, and what you can do with them. A typical slide rule has anywhere from ten to thirty scales, rather than just two, and there are thousands of mathematical expressions that you can evaluate just as easily as you can multiply or divide two numbers. On the other hand, you can't use a slide rule to add or subtract two numbers, and you need to understand your problem well enough to be able to figure out on your own the location of the decimal point in your answer.

You'll be able to examine several slide rules, and I'll tell you what's involved in being a slide rule collector.

Here's a couple of home work problems. You can do them before or after the talk. Let me know your answers:

Why is it so hard to find slide rules that can be used for addition and subtraction?

What's the base of the logarithm used for the design of any specific slide rule?- February 25 Dylan Zwick
**Voting, Elections, and Group Choice**

*Abstract:*In this talk I'll discuss some curious aspects about how we vote in the United States, and present some general mathematical concepts about voting theory and group choice, including the famous Arrow's impossibility theorem.- March 4 Gordan Savin
**Diffie Hellman Key Exchange**

*Abstract:*Have you ever wondered why your internet purchase is "safe and secure?" Although the need to make information secret or unreadable to the general public is as old as human civilization, some recent ideas have revolutionized the subject. Come to this talk to see for yourself.- March 11 Allen Moy - Distinguished Lecture Series
**Ostrowski's Theorem - Canonical Completions of the Rationals**

*Abstract:*The real numbers are constructed from the rational numbers by a completion process based on the familiar absolute value. A consequence is the archimedean property of the real numbers. There are other absolute values on the rational numbers based on prime integers leading to non-archimedean number systems. Ostrowski's theorem gives a characterization of all absolute values.- March 18 No Talk - Spring Break

- March 25 Derek Hacon, Pontificia Universidade Catolica, Brazil
**Topological Invariants of Knots**

*Abstract:*Topological invariants (in particular knot invariants) have a long history, going back to Gauss and his (integral) linking numbers. Relatively recently (1980´s) Jones discovered his famous knot invariant, subsequently recast in an absurdly simple fashion by Kauffman. We will describe Kauffman´s invariant as the expected payoff of a topological slot machine and do some calculations.- April 1 Dragan Milicic
**The Riemann Hypothesis**

*Abstract:*The zeta function appears for the first time in Leonhard Euler's book "Introductio in Analysin Infinitorum" published in 1748. Among other results, Euler used it to prove that the number of primes is infinite.Bernhard Riemann studied the properties of the zeta function as a function of a complex variable. He observed that it has zeros at -2, -4, ...; which are known as trivial zeros. Also, in 1859, he conjectured that all other zeros have real part equal to 1/2. This is known as "the Riemann Hypothesis." Although the numerical evidence is overwhelming in favor of the hypothesis, nobody was able to prove (or disprove it).

The Riemann Hypothesis has deep consequences in number theory. This is probably the most important open problem in pure mathematics. It is one of seven Millennium Prize Problems listed by the Clay Mathematics Institute in Cambridge, MA. The award for solving it is a million dollars.

- April 8 No Talk

*Abstract:*- April 15 Gordan Savin
**Elliptic Curve Primality Test for Fermat Numbers**

*Abstract:*The problem of factoring a large integer, or verifying that it is a prime number, has gained interest in recent years due to applications in cryptography. Some efficient tests are based on an observation that the problem of factoring is similar to the problem of determining the order of an element in a finite group. I will explain this idea and show how elliptic curves can be used to test Fermat numbers.- April 22 Mladen Bestvina
**Inflexibility of Convex Polyhedra**

*Abstract:*In 1813 Augustin Cauchy proved that convex polyhedra are inflexible. This means that if you make one by gluing faces made of cardboard, you will not be able to deform it so that the edges act as hinges. I will present Cauchy's proof, which requires knowledge of the Euler characteristic and some trigonometry. Surprisingly, there exist flexible (non-convex) polehedra, as discovered by Robert Connelly in 1977.- April 29 Enrico Rogora
**Biological Sequences from a Geometric Viewpoint**

*Abstract:*Hidden Markov Models are probabilistic models used to solve such important problems in biological sequence analysis as sequence alignment and gene recognition. These models allow for efficient dynamic programming algorithms, which can be seen as a simplification of polynomial computations known as "tropicalization." On the other hand, a natural algebraic variety can be associated with any Hidden Markov Model (and more generally to many graphical models). When the process of tropicalization is carried over to algebraic varieties, it produces a polyhedral complex which retains some important aspects of of the original object, at the same time reducing its complexity. In this seminar we shall introduce these ideas in a simple and hopefully painless way.