Math and Oil: do they mix?
Sylvia Forman
St. Joseph's University

If you have been reading the newspaper or watching the news, you may have noticed that oil and gas prices have been in the news a lot. Of course, if you have a car, you have definitely thought about oil as you monitored the cost of filling up your tank. What is going on with the world oil supply? Why all the concern now?

In 1956, the geologist Marion King Hubbert predicted that U.S. oil production would peak between 1965 and 1972. He was widely ridiculed until 1970, when he was proven correct. Now, many experts using similar techniques predict that we are currently experiencing the peak in world oil production, or will reach the peak in the next 10 years. How do they make these predictions? Why don't they all agree? This talk will explain some of the mathematics behind developing a model for and making a prediction about when world oil production will peak.

Looking at Perspective
Leah Berman
Ursinus College

How do we see what a camera sees? How can we make a drawing in proper perspective? What happened in the Renaissance to change the way people looked at the world? This talk will give a very brief introduction to one-point perspective.

Geometry in the Game of Set®
Liz McMahon
Lafayette College

The game of Set® turns out to be deeply geometric. We'll explore some of this geometry, and even get to discuss Magic Squares and a new geometric result that arose from looking at the game.

Iterated sharing
Gordon Williams
Ursinus College

Goods are frequently distributed unevenly. Today we will be exploring some ways of distributing the wealth.

For example, suppose a group of kids are given candy, but the number of pieces given to each kid varies over a fairly wide range. If we further suppose that each kid starts out with an even number of pieces, one approach to making the distribution fairer would be for the kids to sit in a circle, and have the kids divide the pile in front of them in two equal parts, giving half to the person on their right and keeping half. Now the kids all check to make sure they have an even number of pieces, and if not, take another candy from a reserve pile. If we keep repeating this process, what will eventually happen?

Making the Cut
Ethan Berkove
Lafayette College

It's a common elementary school art project to fold a piece of paper, make a bunch of cuts, and unfold something like a snowflake, or dolls with joined hands. We'll look at a variation of this problem--the paper can be folded in any way we like, but we're only allowed to make one straight cut. What shapes can be cut out of the paper now? The answer may be more general than you think!

Monocromatic Points
Leah Berman
Ursinus College

Consider a collection of straight lines in the plane, and look at all the points where they cross. Can you draw the lines in such a way that each point contains three lines? Suppose the lines are colored either red or green. Does each point have a line of each color passing through it? Come find out!

Math and Magic of Financial Derivatives
Klaus Volpert
Villanova

Are Financial Derivatives the `Engine of the Economy`, as declared by Alan Greenspan, or `Weapons of Mass Destruction', as Warren Buffett views them?

Over the last 30 years, financial derivatives have overtaken stocks and bonds as the investment vehicle of choice for many large investors. Derivatives are often behind the spectacular profits of investment banks as well as the mind-boggling losses (e.g. at Citigroup) that we read about in the papers. While CEO’s and hedge fund managers profit handsomely when things are going well, the losses are mostly born by shareholders and small investors. Pension funds, even school districts and townships have suffered from disastrous deals in derivatives.

It is therefore no exaggeration to say that taxpayers and investors can no longer afford to not understand derivatives.

So what are derivatives? Simply put, they are contracts between two parties that stipulate some cash flow over a certain period of time. The size of that cash flow depends on what happens to some underlying asset, such as a stock prices, interest rates, currency exchange rates or commodity prices.

The uncertainty in the development of the underlying creates the key difficulty, which is to properly evaluate the price and the risk inherent in a derivative.

In this talk I will give an overview of the three main methods to price derivatives:

1. The analytic method by Black and Scholes, based on PDE’s

2. The discrete approach by Cox-Ross-Rubinstein, based on binomial trees.

3. Monte-Carlo Methods, which average information obtained from simulating a large number of random walks of the underlying.

I will also discuss the advantages and short-comings of each method.

Students will be presenting their projects for this semester's Discrete Mathematics course during a poster session. Please join us to see all the fascinating things our students are up to.

Score sequences for tournaments
Garth Isaak
Lehigh University

Teams from Ursinus, Villanova, West Chester, Xavier and Yale compete in a round robin tournament, with a winner declared for each match.

Results are U beats V,W,Y; V beats W,X; W beats X,Y; X beats V,Y and Y beats V. The score sequence 3,2,2,2,1 records the number of wins for each of the teams. Can a tournament with 25 teams end with the score sequence:

22,22,20,20,20,20,19,19,18,1616,13,13,10,8,6,6,6,5,4,4,4,3,3,3?

Why or why not?

We discuss the mathematics related to this question and show

connections to two other problems:

Does the pair of equations

2x + 3y + 4z = 3

x - y - 2z = -2

have a non-negative solution (and its generalization to more equations and variables)? When can we find a circulation in a network of pipelines?

History of π (pi)
Alan Levine
Franklin and Marshall

It has been known for 4000 years that the ratio of the circumference to the diameter of a circle is constant. Many attempts have been made to determine the "value" of that constant. We will look at a variety of those attempts, ranging from ancient Egypt to modern times. Along the way, we'll see many places in which π (pi) makes and unexpected and fascinating appearance.

Bouncing balls, Jenga blocks, and infinity
Robert Styer
Villanova

If a ball bounces an infinite number of times, will it bounce an infinite amount of time? We use some elementary physics of a bouncing ball to lead to geometric series. Next, can you pile up Jenga blocks so the top one overhangs the bottom one? Some elementary physics leads to the harmonic series. Along the way we enjoy some paradoxes of calculating with infinite series.

Robert Styer is a graduate of the University of Pennsylvania and received his Ph.D. in analytic number theory from MIT.

(Really) Big Numbers
Tom Carroll
Ursinus College

A moment's thought about exponential population growth or the lottery should convince you that big numbers play an important role in the real world. Physicist Albert Bartlett has said, "The greatest shortcoming of the human race is our inability to understand the exponential function."

In my own research with interacting quantum systems, numbers that grow even faster than exponentially pose a problem. What are the biggest numbers yet described? Are they useful? And, can we really use them to prove or disprove any mathematical conjecture?

The hidden graphics of associativity.
Jim Stasheff
U. Penn.

The associative law: (ab)c = a(bc) has an obvious consequence that any

meaningful way of inserting parentheses gives the same result. But to verify that leads to some interesting graphical representations in term of:

1) (mathematical) trees

2) triangulations of polygons

3) convex polytopes.