2007 Mathematics Seminar
All talks are at noon in Hurst 101 (some exceptions apply)
Each year senior mathematics majors participate in a seminar series. The students are responsible for learning about an advanced mathematics topic not generally covered in the curriculum. Each student has a faculty adviser, writes a paper, and gives a presentation. The presentations are open to all.
The Ideal Half Pipe Curve
Casey Davis
Wednesday, 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 Smith
Friday, 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.
Scott, James, Adam, and Casandra
Determining Matrix Products from Vector States
Adam Forland
Monday, December3
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 Naillis
Wednesday, 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 Spotts
Monday, 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 Dahlberg
Wednesday, 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.
2005 Seminars
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