# 2007-2008 Seminar Series

## The Ideal Half Pipe Curve

### Casey DavisWednesday, April 30

Imagine two points in the plane connected by a wire with a bead sliding from one end to the other. Changing the shape of the wire would alter how long it takes for the bead to reach the end of the wire. These changes in the wire are referred
to as variations. We will examine the Brachistochrone problem, which motivated a new study: the Calculus of Variations. The problem is to find the curve that minimizes the time it takes for the bead to reach the end of the wire (assuming the bead is only influenced by gravity). The problem dates back to the founders of Calculus who did not have the techniques we have today. We will solve this problem using a theorem in the Calculus of Variations in an attempt to find the ideal curve for a half pipe. The solution to the Brachistochrone has interesting properties that we will also animate using models of motion.

## Constructible Numbers

### Meryn SmithFriday, April 25

Euclid's books The Elements have been the foundations of geometry for over 2000 years. During Euclid's time three famous constructions were considered: trisecting an angle, doubling the curve and squaring the circle. For centuries, people attempted these constructions using a compass an straightedge, but it wasn't until the late 1800s that they were proven impossible. Come see a modern presentation of these proofs.

## Determining Matrix Products from Vector States

A very powerful technique in mathematics is to study an object by studying mappings on that object. In this talk we consider a set of functions mapping 2x2 matrices to the complex numbers. These functions, known as vector states, are constructed from the set of unit vectors in the comples plane. By carefully studying this space of functions, information about the product structure of the group of 2x2 unitary matrices can be deduced. Using complex numbers, linear algebra, and calculus this presentation will outline the background and proof of a construction that recovers the product structure.

## The Hyperbolic Pythagorean Theorem

### Casandra NaillisWednesday, December 5

The Pythagorean Theorem is one of the best known theorems from Euclidean geometry. There is a similar result in hyperbolic geometry. It expresses the hyperbolic length of the hypotenuse of a hyperbolic right angled triangle as a natural“sum” of the squares of the hyperbolic lengths of the other two sides. The proof of the Hyperbolic Pythagorean Theorem will be presented using Mobius transformations and Mobius addition.

## Weierstrass's Factorization Theorem

### James SpottsMonday, December 10

The fundamental theorem of algebra tells us that a polynomial equation of degree n has n zeros. This tells us that any polynomial can be expressed in terms of its zeros {x_1, x_2,..., x_n} as P(x)=a\prod_{k=1}^n (x-x_k). This leads to the more general question: can we factor functions other than polynomials in terms of their zeros? If so, then what kind of functions can be factored and how? Weierstrass's factorization theorem provides a construction that addresses these questions.

## Group Law on Elliptic Curves

### Scott DahlbergWednesday, December 12

The points on an algebraic curve do not form a group, but elliptic curves are special because points on this curve can be used to define a group. For this to work out nicely we will not only explore the group law of rational points on elliptic curves, but also the projective plane in which we will be working. The proof of the group law will be shown both geometrically and analytically.