# 2008-2009 Seminar Series

## March 30

### A Most Marvelous Theorem in Mathematics

Marden's theorem states that the triangle formed in the complex plane with vertices at the roots of a third degree polynomial will form a boundary for the roots of the derivative of that same polynomial.  Remarkably, the roots of the derivative are the foci of a unique ellipse that is inscribed in the triangle, the unique ellipse being tangent to the midpoint of each side of the triangle. In order to prove this delightful result, existence and uniqueness of the ellipse must be shown.  This research spans many concepts in mathematics such as analytic geometry, linear algebra, complex analysis, calculus, and properties of polynomials.

## April 1

### Is Google Trustworthy? The Eigenvector Worth Billions.

Google has been providing searches for over ten years. The algorithm created to rank web pages, called "PageRank" by Google, was developed by two Stanford University graduate students, Larry Page and Sergey Brin. Much of Google's success has been credited to the PageRank algorithm. This algorithm, which uses linear algebra techniques, will be described and demonstrated. After understanding how PageRank works, we will explore how parameter choices affect ranking accuracy and performance.

## April 8

### Harnessing Randomness for Financial Gain: The (in)Famous Black-Scholes Equation

Pricing options involves the difficult task of predicting future stock prices. We will suggest the stochastic model $\Delta V_i=a \Delta t + b \sqrt{\Delta t} \xi_i$, where the change in value, $\Delta V$, depends on $a$, the expected rate of change, $b$, the expected volatility, and $\xi$, a uniform random variable. Through simulation, we will examine the long-term behavior of this model, and find that the distribution of predicted prices is normal. This regularity allows us to calculate the expected value, or price, of the option. We use this intuition to derive the Black-Scholes equation to price options: a Nobel Prize-winning partial differential equation.

## April 13

### A Mathematical Approach to Rubik’s Cube

Rubik’s cube, the game craze of the 1980s, has been well studied mathematically.  In this talk we will formalize the moves used to solve the cube.  These moves are understood in terms of permutations of the “cubies”.  To study these permutations, we will use some basic results from the theory of groups.  Moreover, we will provide a glossary of some useful moves to solve the cube.

## April 15

### The Center of Gravity - Will the Cup Tip Over?

The center of gravity for a two dimensional object is the point on which a thin plate in the shape of the object balances horizontally. This notion can be extended to a three dimensional shape. We will be looking at how the center of gravity changes as liquid is added to or removed from a cup. We will look at two different shapes, a cylindrical cup and a frustum shaped cup. Calculus will be used to relate the liquid level and the center of gravity.

## April 22

### The Red Scare: A Model of Plankton Blooms

"Red Tide" is the common name for any number of explosions of plankton populations within a body of water. Called "blooms", these events can be harmful to the aquatic environment, as well as to external organisms eating, drinking or even just in the water. Though the exact causes of the blooms are not fully understood, in phytoplankton it is reasonable to associate the event or bloom with changes in nutrient availability. In this study a model of phytoplankton (prey) and zooplankton (predator) populations will be entertained. The model itself consists of a nonlinear system of differential equations. Local and phase plane analysis will be presented as well as numeric simulations to better understand the dynamics of the model, and thus the interaction of the plankton populations and its possibly underlying cyclic behavior.

## April 29

### Montmort’s Matching Problem: The Probability of Coincidences

A group of women arrive at a party and each
checks her hat at the door. As each woman leaves she takes a hat
at random. We will look at the probability that at least one woman
receives her own hat back. If no woman receives her own hat back,
we will look at the probability of a 2-way swap. If no 2-way swaps
occur, we will look at the probability of a 3-way swap, etc. The
general formula for these probabilities and its limit will be
discussed, as well as a proof for the probability of at least one
match.