# Undergraduate Colloquium

**Spring 2011**

**Wednesdays 12:55 - 1:45**

**LCB 222**

Pizza and discussion after each talk

Receive credit for attending

Past Colloquia

- January 12 No Talk
- January 19 Fernando Guevara Vasquez
**How to Make Objects Invisible**

*Abstract:*What does it mean for an object to be invisible? We will find together what are the requirements for a cloak that can make objects placed within the cloak invisible. Then we will actually design a cloak to hide an object from a static electric field using networks. This construction actually extends to other physical situations. You only need to know what a derivative is for this talk.- January 26 Peter Alfeld
**What is a sliderule?**

*Abstract:*There was a time when calculators did not exist. That did not stop us from building the Boeing 747, or going to the moon. In those days engineers, scientists, and students used sliderules on a routine and daily basis in place of calculators. I will show several sliderules, explain how they work, and describe what kind of mathematical expressions can be evaluated with a slide rule. (There are tens of thousands.) We'll also have a drawing. The lucky winner will get a sliderule to keep.- February 2 Aaron Wood
**Archimedes' Cattle Problem**

*Abstract:*In 1773, a librarian and art critic named G.E. Lessing found a problem which Archimedes communicated to Eratosthenes. The problem asks for the smallest number of total cattle (white, black, dappled, and yellow) which satisfy 7 linear equations and 2 extra conditions. According to Archimedes, one who fully solves the problem "shalt depart crowned with glory." In 1880, A. Amthor gave an exact solution to the problem, computed the first 4 digits of the smallest solution and stated that the number of Cattle of the Sun has 206,545 digits! In this talk, we will solve Archimedes' Cattle Problem by reducing it to a Pell's equation and discussing the solutions to Pell's equation.- February 9 Sarah Cobb
**Of Cubes and Counting**

*Abstract:*The Rubik's Cube is a puzzle introduced by Erno Rubik in 1980. Since then, people have put in huge quantities of time and brain power trying to extract the solution from the cubes advertised 43 quintillion possible configurations. While this talk won't teach you how to solve a cube, it will discuss the mathematics of the cube, providing a very gentle introduction to group theory along the way.- February 16 R. Michael Range, State University of New York at Albany=
**Calculus: Have We Been Teaching it Wrong?**

*Abstract:*A method introduced in the 17th century by R. Descartes and F. van Schooten for finding tangents to classical curves is combined with the point-slope form of a line in order to develop the differential calculus of all functions considered in the 17th and 18th centuries based on simple purely algebraic techniques. This elementary approach avoids infinitesimals, differentials, and similar vague concepts, and most importantly it does not require any limits in the study of algebraic functions. It naturally leads to continuity and to the modern definition of differentiability - in an elegant formulation introduced by C. Carathéodory - which needs to be considered when studying the elementary transcendental functions. This approach suggests new ways to teach calculus in the 21st century.- February 23 Fred Adler
**Why aren't we all dead?**

*Abstract:*Some disease, like the plague, kill pretty much everybody they infect. But after they sweep through a population, usually there are a few survivors left to repopulate the earth. We'll write down some simple ordinary differential equation models that might explain this important phenomenon, and talk about other mechanisms that might explain our continued existence.- March 2 Tony Lam
**Game theory and The El Farol Bar Problem**

*Abstract:*The Prisoner's Dilemma is often described as a two-person general sum game. Such games have strategic equilibriums - each player can do better by anticipating the best response of the other. What happens when there are one hundred players, however? As humans we are bounded by our rationality and require a substitute in order to choose a best response, and then we modify it. I will present the Prisoner's Dilemma with two players and how to find optimal strategies with a general technique, and then present the El Farol Bar problem: If a popular bar is enjoyed by its customers only when it is at or below 60% capacity, how can individuals decide if they should attend given the attendance history?- March 9 No Talk
- March 16 Aaron Bertram
**Happy Pi Day!**

*Abstract:*Pi is the ratio of the circumference of a circle to its diameter. It is also one half the period of the sine and cosine functions, and makes an appearance (along with e) in Stirling's formula for the factorials of large numbers. But suppose you were stranded on a desert island (with a lot of paper and pens) and asked to calculate the first 20 digits of pi. Could you do it? How did people do it in the 19th century? How did Archimedes approximate pi? Inquiring minds want to know.- March 9 No Talk - Spring Break
- March 30 Robbie Snellman
**P-adic Numbers: What are they and why are they useful?**

*Abstract:*In order to do analysis on**R**we need the notion of "size." This notion of size is usually determined by the standard absolute value, |x - y|. Upon styding the close relationship between**Z**and**C**[x], Hensel came up with another generalization of**Q**which involved completing**Q**with respect to particular absolute value, called the p-adic absolute value. This talk will focus on the construction of the p-adic absolute value along with proving some nonintuitive results to the topology of the p-adic numbers.- April 6 Alla Borisyuk
**Cobwebs, Maps and a Route to Chaos**

*Abstract:*I will introduce one-dimensional maps (you can simulate some by pressing the same calculator button many times). They can easily be defined and analyzed and are used as models in many applications. Looking deceptively simple, the maps give rise to complex behaviors, including chaos. As we follow the route to choas in these systems, we will find universal constants (as basic to this route as π is to circles): δ = 4.669..., and α = -2.5029...- April 13 Jing Tao
**The Farey graph: A Geometric View of Numbers**

*Abstract:*It is common to represent a real number using decimals. However, a flaw with this system is that many simple fractions do not have a finite decimal expansion. This is essentially because decimal expansions favor those numbers which relate well with the integer 10. In this talk, I will explain another system of representing numbers, called continued fraction expansion, which is in some sense more intrinsic and natural. We will also see how this system is encoded in a beautiful geometric object, called the Farey graph. If we have time, we will try to explore this interaction between arithmetic and geometry via solving some Diophantine equations.- April 20 Distinguished Lecture Series: Wilfried Schmid, Harvard University
**Riemann's "Nowhere Differentiable" Function**

*Abstract:*According to second hand reports, Riemann once presented an explicit example of a continuous, nowhere differentiable function. Various mathematicians tried to prove the non-differentiability, but the matter was not completely settled until 1971. In fact, Riemann's function is differentiable at certain rational points, after all. I shall describe the history of Riemann's function and outline a proof of its properties.- April 27 Movie Day!
**The Math Life**

*Abstract:*"Why did a magician become a mathematician? How can a person see in four dimensions? What does a mathematical proof have in common with a Picasso portrait? This elegant program brings to life the human dimension of mathematics through lively interviews with Freeman Dyson, David Mumford, Ingrid Daubechies, Persi Diaconis, Michael Freedman, Fan Chung Graham, Kate Okikolu, Jennifer Tour Chayes, Peter Sarnak, Steven Strogatz, and seven other mathematicians. These captivating luminaries vividly communicate the excitement and wonder that fuel their work as they explore the world through its patterns, shapes, motions, and probabilities. Computer animations and analogies drawn from the visual arts are incorporated, to maximize accessibility to the fascinating concepts discussed."