Undergraduate Colloquium Fall 2002

Sept 3     Brandon Baker
Boolean Algebra and Virtual Reality
Abstract: This is the first talk in a two-part series on the mathematics behind 3D computer graphics hardware. We will cover binary arithmetic and boolean operations, integer representation and computation, floating point computation, and applications to texture mapping in 3D graphics.

Sept 10     Klaus Schmitt
Iterative vs Continuous Newton Methods: Similarities and Differences
Abstract: Newton's methods for solving equations are iterative methods which provide good approximation schemes for solutions once the iteration scheme starts close enough to a solution. On the other hand, by starting the iteration scheme at an arbitrary point and trying to judge whether the resulting sequence converges, decomposes the set of starting values into subsets, some of which have fractal structures (the bad starting values). By introducing a so-called relaxation parameter it is often the case that the sets of bad starting values shrink in size with the relaxation parameter. As the relaxation parameter tends to zero, Newton's method approaches a differential equation (the continous Newton Method) whose solution orbits in some sense approximate the interation sequences for the relaxed method. The lecture will discuss the similarities and idffeneces between the discrete and continuous methods. The lecture will be accessible to students who have completed Calculus.

Sept 17    Brandon Baker
Linear Algebra and Virtual Reality
Abstract: This is the second in a two-part series on the mathematics behind 3D computer graphics hardware. We will cover linear algebra, 3D graphics rotations, 3D graphics linear translations, and lighting and fog calculations.

Sept 24    Hugo Rossi
Linear Recursive Sequences, Part I
Abstract: Given n numbers a1, ... ,an, form the sequence {ak}, where the kth term is the average of the preceding n terms. Does the sequence converge? If so, what is the limit? If that is too easy, consider instead of "average", the weighted average with positive weights µ1, ... ,µn with the sum of the µ's equal to 1. The answer to these questions results from a judicious use of the basic tools of linear algebra. Definition. A linear recursive sequence, of degree n is given by specifying the initial n terms, a0, ... , a(n-1), and defining the remaining terms by a recursive relation of the form ak = C1*a(k-1) + ... + Cn*a(k-n), k greater than or equal to n. In the first week, I shall discuss a variety of recursive sequences, in particular, that sequence Archimedes studied in his determination of a value for pi. Between the first and second weeks, students are urged to experiment with various sequences defined by averages; hopefully leading to a conjecture for the answer. In the second week, I shall introduce the methods to solve the problem, and provide the solution.

Oct 1     Hugo Rossi
Linear Recursive Sequences, Part II
Abstract: A continutation of the previous week's talk.

Oct 8    Kree Cole-McLaughlin
Scalar Field Topology and Contour Trees
Abstract: A scalar field is a real valued function over a manifold. Understanding the topology of a scalar field gives one qualitative information about the behavior of the field. This can be very useful in the analysis of many physical systems that are represented by the field. One way to understand a field's behavior is by studying its level sets (sets of constant field value). An important question is how the topologies of level sets evolve as one sweeps through the field values. The contour tree is a graph that represents the topological evolution of level sets. In this talk I will present an algorithm for computing contour trees of piece-wise linear functions over R3.

Oct 15    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 Clay Mathematics Institute in Cambridge, MA. The award for solving it is $1 million.

Oct 22    Gordan Savin
On Primes in Arithmetic Sequences
Abstract: A famous theorem of Dirichlet says that there are infinitely many primes in every arithmetic progression a, a+d, a+2d, ..., where a and d are two relatively prime positive integers. Take, for example, d=3. Then a can be either 1 or 2, and the corresponding arithmetic progressions are (with primes in bold face)

1, 4, 7, 10, 13, 16, 19, ...
2, 5, 8, 11, 14, 17, 20, ...

In this lecture we shall present a complete and simple proof of the theorem of Dirichlet in this case, based on the fact that the alternating series

1 - 1/2 + 1/4 - 1/5 + 1/7 - 1/8 + ...

is convergent and non-zero. (Note that the denominators of fractions in this series are precisely the terms of the two arithmetic progressions given above.) The general proof follows the same idea, but is technically a bit more involved, and requires the use of complex numbers.

Oct 29    Fletcher Gross
Rubik's Cube
Abstract: How many different configurations can a Rubik's Cube have? What is the relationship between this number and the number of configurations obtained by disassembling and reassembling the cube? What does all this have to do with group theory?

Nov 5    Nat Smale
Fractals, Metric Spaces, and the Contraction Mapping Theorem
Abstract: In this talk, we will first look at some fractal type structures. This will be followed by a short discussion of some basic ideas from Real Analysis, such as complete metric spaces, and the contraction mapping theorem. Finally, we will see how fractals can be constructed by using the contraction mapping theorem on an appropriate metric space, a result due to John Hutchinson.



Nov 19   Brad Peercy
Experimentation with Acid
Abstract: Hydrogen, H+, is an important biological ion. Many proteins in a cell behave differently in the presence of a high rather than a low concentration of hydrogen, [H+]. Along with normal fluctuations of [H+], the acidity or [H+] can change dramatically under abnormal conditions. During a heart attack [H+] can increase by an order of magnitude inside of cardiac cells.

Until recently, little has been done to quantify even normal H+ movement in cardiac cells. In this talk I will discuss experiments which have been performed to quantify H+ movement in rabbit cardiac cells. I will also discuss the role mathematical modeling had in aiding the experimentalists and derive the diffusion based model.

Nov 26   John Zobitz
Pascal Matrices and Differential Equations
Abstract: As any graduate student knows, solving differential equations can be a difficult task. Even the "simpler" ones with constant coefficients become challenging when nonhomogeneous equations arise. Unfortunately, methods to solve these equations (variation of parameters, annihilator method) are not very "user-friendly". This talk develops a novel method to solve nonhomogeneous differential equations with constant coefficients using matrices. I will show how to reduce any differential equation of this type into a simple non-singular matrix equation. What is interesting about the solution is the mixed bag of tricks one needs to arrive there: fundamental calculus ideas, linear algebra, and a touch of combinatorics. Along the way we will encounter "Pascal Matrices"--lower triangular matrices with entries that correspond to Pascal's triangle--and prove a nice result about such matrices.

Dec 2   Aaron Bertram
Pythagoras vs the Elliptic Curves
Abstract: A Pythagorean triple is three integers satisfying a2 + b2 = c2 or, in other words, a right triangle whose sides all have integer lengths. Examples abound, including: 32 + 42 = 52, 52 + 122 = 132, 72 + 242 = 252 There are infinitely many of them. How do we know this? Because we can plot Pythagorean triples as points on a circle, and generate new ones by adding old ones to each other (using angle addition). An elliptic curve is another curve in the plane that has an addition law. Examples include y2 = x3 + c for any (rational) constant c. This addition is much more interesting and subtle than angle addition, but still starting with a point P on the elliptic curve, we can generate more points by adding P to itself repeatedly. If P has rational numbers as coordinates, this generates a sequence of points whose rational coordinates have been used to crack RSA codes by factoring large numbers. I'll touch on this and other issues in this talk, which is a preview of the REU I'll be running with Jim Carlson this coming summer.