# Undergraduate Colloquium

**Fall 2010**

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

**LCB 225**

Pizza and discussion after each talk

Receive credit for attending

Past Colloquia

- August 25 No Talk

- September 1 Jimmy Dillies
**From Lagrange to Cathay**

*Abstract:*We will see how a simple analogy between numbers and functions gives a direct solution to a very very old problem in number theory.- September 8 Dan Ciubotaru
**The Gamma Function**

*Abstract:*The gamma function was constructed by Leonhard Euler in 1729. It is defined by means of an integral, and generalizes the factorial of natural numbers. It has many beautiful, elementary properties. In this talk, we will discover some of its basic features, and we do this by using only Calculus.- September 15 Bryan Wilson
**Monkeys, Checkers, and Mathematical Invariance**

*Abstract:*One of the most important ideas in higher mathematics is the idea of mathematical invariance. When looking at objects which are different in some way, we like to examine how they can also be the same. More in general, what are necessary and sufficient conditions for these "invariants" to hold between different objects? In this talk we introduce the idea of invariants in the solution of two math-related games - one called "Monkeys and Rooms" and the other a checker-jumping puzzle by John Conway.- September 22 Peter Alfeld
**Hotel Infinity**

*Abstract:*You are the owner of Hotel Infinity. It has infinitely many rooms, and it's full. A new guest arrives and insists you give her a room. How do you accommodate her? The next day, a family with infinitely many members arrives, each of whom wants a private room. The next day infinitely many families, each with infinitely many members, arrive. Each family member insists on a private room. You can do it! Infinity is different.- September 29 Brendan Kelly
**Fermat's Little Theorem: Over and Over Again**

*Abstract:*This talk will present several proofs of Fermat's Little Theorem, a result in number theory. The statement is a now standard result with a notable application in the RSA cryptographic algorithm. By demonstrating an array of techniques to solve the problem we will investigate the question: if it only takes one proof to turn a conjecture into a theorem, why as mathematicians do we seek alternative solutions.- October 6 Greg Rice
**Random Walks, Mixing Times and Coupling Times**

*Abstract:*A frog, curiously named Marcel, wakes up each morning on a lily amongst his network of lily pads and plays a game. Long ago he found a coin at the bottom of his pond which he flips, if heads he spends the next night on the lily to his right, if tails he spends the night on the lily to his left. Problem!!! Marcel's girlfriend Jessica is coming to find him in a week! Can we help her by figuring out where Marcel will likely be? Or is he all mixed up amongst the lilies? During this talk, I will answer this question and further discuss the notion of mixing times for random walks on graphs. In particular, I will compute a bound for the mixing time on a strange graph of my own invention, the hexagonal torus. We will use the notion of coupling times to compute this bound.- October 13 No Talk - Fall Break

*Abstract:*- October 20 Aaron Wood
**Sneaky Segments**

*Abstract:*A segment between two lattice points in the plane is called sneaky if it doesn't meet any other lattice points. In this talk the question "What is the probability that a segment chosen at random is sneaky?" will be discussed and answered. The solution will involve some clever probability, some interesting number theory and some analysis of infinite sums and products.- October 27 Dylan Zwick
**The Mathematics of Democracy**

*Abstract:*When we talk about a democratic decision, what do we mean, and how can we make sure the result most accurately reflects the will of the people? In this talk we'll see that this simple question is surprisingly difficult to answer, and we'll discuss some of the problems and paradoxes that arise when we try to construct an optimal system for democratic decision making. The talk should be accessible to anybody with some mathematical background.- November 3 Brian Mann
**Groups of Intermediate Growth**

*Abstract:*Groups are the simplest algebraic objects widely studied by mathematicians. Finite groups are, in many senses, well-understood, but there is still much not known about infinite groups. One approach to studying infinite groups is to view them as geometric objects and look at their growth. It turns out that most examples of groups you might come up with have either polynomial or exponential growth. Can we construct a group with a growth rate which is not either? A background in some basic abstract algebra will be helpful but not necessary: I will attempt to explain everything from the ground up.- November 10 Stefano Urbinati
**Want to be a cryptographer? Start with finite groups!**

*Abstract:*I will introduce some arithmetic properties of finite groups, Euler's generalization of the famous Fermat's Little Theorem, and from these totally abstract objects I will try to show how the idea of a cryptographic algorithm developed, called RSA - one of the most famous algorithms for public-key cryptography.- November 17 Andrejs Treibergs
**Heat Equation & Curvature Flow**

*Abstract:*Recently, parabolic partial differential equation methods have had a profound impact on mathematics, such as in Hamilton - Perelman's resolution of the Poincare Conjecture using Ricci Flow. I'll discuss the Heat Equation on the circle and Curvature Flow of closed curves in the plane, the simplest of geometric evolution equations. The curvature flow unwinds any starting curve until it vanishes in a round point. These flows are studied by combining the maximum principle, integral estimates and geometric inequalities.- November 24 Movie

*Abstract:*- December 1 Andy Thaler
**The Effect of Dissipation on the Transformation-Based Cloaking Scheme**

*Abstract:*The transformation-based approximate cloaking scheme has been studied extensively. In the literature, however, the cloak was considered to be non-dissipative. This is unrealistic, since all materials are dissipative to some extent. In order to determine the effects of dissipation on the cloaking scheme, we performed numerical simulations with dissipative cloaks and found that the cloaking scheme is not drastically affected as long as the amount of dissipation in the cloak is not too large. In this presentation, I will introduce the ideas behind the transformation-based approximate cloaking scheme in the context of a dissipative cloak.- December 8 Chris Kocs
**Holiday Logic**

*Abstract:*Alone in a cave on the island of misfit toys, there are 50 plastic robot toys which all possess a serious defect--if one of these robots discovers its own eye color, it will explode within 24 hours! Fortunately, there are no reflective surfaces in the cave, and the robots, being rather simple in design, have no means of communicating with each other. All robots are aware of their shared defect and also know that the other 49 robots have red eyes. However, since the robots don't have enough information to logically conclude that they also have red eyes, they continue merrily coexisting and jostling each other in the dark until, one day, an armless teddy bear stumbles into their cave. The teddy makes the single following remark addressed to all of the robots (and none in particular): "I see a pair of red eyes." What are the exact consequences of this seemingly innocuous comment? Why?

To solve this riddle, I'll introduce a simple and very useful tool that all mathematicians should have at their disposal. Then we'll consider other riddles and apply the same approach. To find out what this tool is (and the answer to the above riddle), you'll have to attend the talk!