Undergraduate Colloquium Spring 2002

Jan 15     Bob Guy
Linear and Non-linear Circuits: oscillations, bifurcations, excitability, and chaos
Abstract: I will begin by reviewing how simple circuits made up of resistors, capacitors, and inductors give rise to systems of differential equations. I will analyze a simple system and demonstrate with the circuit. Interesting behavior occurs when a bit of non-linearity is introduced via a non-linear system. The circuit was designed to model the electrical impulse in a nerve cell, and it provides an excellent example of an excitable system. I will demonstrate the results with the actual circuit. Finally I will discuss how a simple modification can give rise to chaotic dynamics.

Jan 22     Gordan Savin
Continuous Fractions and Pell Equations
Abstract: In this talk I will define continuous fractions and give some examples. Then we shall write down quadratic irrational numbers as continuous fractions, and apply it to solve the Pell Equation.

Jan 29    Andrej Cherkaev
Damage and Failure: A Mathematical Perspective
Abstract: Even if a structure fails, most of its material stays undamaged. The failure is mostly attributed to an instability of the stress distribution. A structural morthology can increase the stability of the loading process. The talk deals with the corresponding mathematical principles, problems of description and control of the damage, and simulation. Videoclips of destructing construction, made by Liya Zhornitskaya, will be demonstrated.

Mar 5    Paul Bressloff
What Visual Hallucinations Tell Us About the Brain
Abstract: Geometric visual hallucinations are seen by many observers after taking hallucinogens such as LSD or cannabis, on viewing bright flickering lights, on waking up or falling asleep, in "near death" experiences, and in many other syndromes. We present a dynamical theory of the origin of hallucinations in the visual cortex, based on the assumption that the form of the eye-brain map and the architecture of the cortex determine their geometry.

Mar 12     Jingyi Zhu
First Exit and Company Defaults
Abstract: Given a finite region, we can define the first exit for a Brownian motion starting from any interior point, and the problem is related to the diffusion equation. In financial applications, we would like to model possible company defaults in the future, and one of the approaches is to track some financial index as an indicator of the health of the company. When the index fall below (or above) certain threshold, we may declare that the company defaults. I will show in this talk how this approach can be formulated, the partial differential equation problem arises, and how it can be solved.

Mar 19    Mladen Bestvina
On the Total Curvature of Knots
Abstract: By the (total) curvature of a polygon in, say, the 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 (a good friend of John Nash). I will discuss Milnor's solution. If there is time at the end, I will discuss the relationship between this "piecewise linear" concept of curvature and the more traditional curvature of smooth curves in differential geometry.

Mar 26    Nick Korevaar
Making Minimal Surfaces with Complex Analysis
Abstract: In 1866 Karl Weierstrass discovered an amazing connection between the shapes of soap films and the field of complex analysis. Starting with the microscopic characterizations of minimal surfaces and of complex differentiable functions, we will derive Weierstrass' representation formula. It encodes the local differential geometric information of a minimal surface in terms of a pair of analytic functions. Concrete applications of Weierstrass' formula blossomed in the early 1980's, when David Hoffman realized that computer graphics would allow one to actually visualize the minimal surfaces corresponding to particular Weierstrass data. We will compare some computer creations to what we can make with a bucket of soapy water and wire frames.

Apr 2    Jesse Ratzkin
Geometric Models for Crystal Growth
Abstract: One can determine the growth of a crystal by tracking how the boundary between it and its environment moves, which amounts to determining how a surface in three-space evolves.If the motion of this surface (i.e. growth of the crystal) is determined by geometric properties (e.g. lengths, areas and volumes) then one can use geometry to track the motion of the surface. I will discuss some of these techniques, including motion by mean curvature, weighted mean curvature and crystalline curvature.

Apr 9    Ken Chu
The Good and Evil of the Axiom of Choice
Abstract: The Axiom of Choice (AC) is undeniably the most (well, the only) controversial axiom of Set Theory, the foundation of modern mathematics. In this talk, there are three points I would like to make:

1) AC is intuitively appealing (a lot of people will disagree).

2) AC is everywhere; in particular, a great deal of powerful results in functional analysis and PDEs rely on consequences of AC.

3) AC has certain "absurd" consequences for which one might almost want to reject AC as an axiom.

Personally, I find it mildly amusing to be "AC-alert," yet I am starting to feel that I am being perceived as "AC-lunatic" instead. I am planning on using this talk to vindicate myself (it might very well do the exact opposite). Come see for yourself; it is going to be fun.

Apr 16    Stewart Ethier
The Gambler's Ruin Formula
Abstract: Suppose you bet $1 on 'red' on each spin of a 38-number roulette wheel. What is the probablility that you double your $1,000 bankroll before losing it? The answer, 1.75 x 10^(-46), comes from the famous Gambler's Ruin Formula, which dates back to the time of Pascal and Fermat (1656). We discuss how the latter two mathematicians likely dervied the formula, as well as other aspects of the history, a modern derivation, and applications and extensions of the formula.

Apr 23    Jim Keener
Arrhythmias by Dimension
Abstract: Abnormalities of function of the cardiac conduction system are the cause of death of hundreds of people every day. For that reason, the study of cardiac arrhythmias is of great interest from a medical and scientific perspective. However, cardiac arrhythmias are also interesting for mathematical reasons because the cardiac conduction system can be viewed as a dynamical system and the variety of its behaviors can be studied from the viewpoint of dynamical systems theory. In this talk, I will give a classification of cardiac arrhythmias that is based on spatial dimension, and is therefore useful for mathematicians, but probably not (as much) for physicians. I will describe examples of zero dimensional arrhythmias (abnormalities of single cells), one dimensional arrhythmias (Wolff-Parkinson White Syndrome), two dimensional arrhythmias (atrial flutter), and three dimensional arrhythmias (ventricular tachicardia and fibrillation).

Apr 30   Peter Alfeld
Infinity is Different
Abstract: The natural numbers are 1, 2, 3,..., the integers are ...-3,-2-1,0,1,2,3... Since all natural numbers are integers but some integers (all the negative ones and zero) aren't natural numbers, there seem to be more integers than natural numbers. Not so! Indeed, by the end of this talk you will be fully and rightfully convinced that there are no more rational numbers (fractions) than integers, even though there are infinitely many rational numbers between any two consecutive integers (like 3 and 4). But it's far from true that infinity is infinity is infinity.... Indeed there is an amazing hierarchy and structure to infinity, and the talk will conclude with a bizarre fact that may change your life. The talk makes no assumptions about your mathematical background, indeed, you don't even need to understand Calculus.