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Spring Semester 2004

Organized by Radhika Ramamurthi
date Room speaker title (click for an abstract)
May 3 SCI2 337A Liz Fenwick Mesh Generation Techniques
Apr. 26 SCI2 337A Jessica Jones Factoring polynomials mod p
Apr. 19 SCI2 337A Marshall Whittlesey An introduction to complex analysis of several variables
Apr. 12 SCI2 337A Amber Puha An introduction to the Mathematics of Pricing Stock Options
Feb. 9 SCI2 337A Jessica Jones NumberTheory and Galois Theory


Abstracts from Spring 2004


 Mesh Generation Techniques by Liz Fenwick, UCSD Math.
Mesh generation is an important part of Numerical P.D.E. computation. The efficiency of the P.D.E. solver depends in part on the (shape) quality of the mesh. Some of the current approaches to 2D and 3D mesh generation will be discussed.

Number Theory and Galois Theory by Jessica Jones, CSUSM Math.
Galois sought to distinguish the polynomial equations which can be solved by a formula from those which cannot. Galois theory later developed into a study of the behavior of field extensions. I will discuss Galois theory and reveal its usefulness to other areas of mathematics, particularly Number Theory. For example, Galois theory can be used to prove Quadratic Reciprocity and portions of Fermatbs Last Theorem. We will focus our attention on extensions of the rational numbers and I will introduce a significant result, the Kronecker-Weber Theorem, which is obtained when the extensions are Abelian.

An introduction to the Mathematics of Pricing Stock Options by Amber Puha, CSUSM Math.
In 1997, Robert Merton and Myron Scholes received the Nobel prize for their work on pricing securities derivatives. An example of a securities derivative is European call option, which is a contract that gives the owner the right, but not the obligation, to purchase a specific security (e.g., a particular stock) at a set time for a set price. By trading such an option, an investor can hedge against the risk associated with market volatility. For example, suppose that an investor believes that the price of a certain stock will be high at some time in the future. Rather than purchasing the stock, the investor could instead purchase a call option. Then, the investor can decide whether or not to exercise the option based on the actual stock price.
Pricing securities derivatives emerged as an important concern in the late 1960's and early 1970's when a relaxation in the regulations allowed insurance companies and banks to invest in the derivatives market. At about that time, two economists, Fisher Black and Myron Scholes, developed a mathematically based pricing strategy, which Robert Merton later simplified and expanded. This work had a profound impact on the trading practices in financial markets worldwide. Sadly, Black passed away prior to 1997, and therefore was ineligible to be a co-Nobel prize recipient.
In the talk, we will examine a simplified version of this body of work. In particular, we will analyze the single period Cox-Ross-Rubenstien (CRR) model using some of the key principles and methodologies that Black, Scholes, and Merton developed to construct their derivatives pricing theory. Despite the fact that this model is very simple, the analysis will illustrate certain essential aspects of their Nobel prize winning work such as dynamic hedging and the risk neutral probability measure.

An introduction to complex analysis of several variables by Marshall Whittlesey, CSUSM Math.
Students of complex analysis learn that differentiable (i.e. analytic) functions with respect to a complex variable possess certain exotic features not typically present in functions of real variables. In this talk I will describe how even more exotic features lie unexpected in several complex variables. Among these are Hartogs' 1906 discovery that analytic continuation is much more common in several variables, and Poincare's discovery the same year that the Riemann Mapping theorem does not hold in higher dimensions.
This talk requires some knowledge of complex analysis; students who have taken Math 536 will find that knowledge sufficient to understand the talk.

Factoring polynomials mod p by Jessica Jones, CSUSM Math.
Students are often first exposed to Quadratic Reciprocity in a Number Theory course. We will discuss how Quadratic Reciprocity is directly related to factoring quadratic polynomials mod p. This will lead us to investigate polynomials of higher degree. We will discuss Kummer's Theorem, which reveals a connection between the factorization of polynomials and the factorization of primes. I will also introduce some of the tools, including the Kronecker-Weber Theorem, which allow us to make predictions as to how particular polynomials will factor mod p.



Questions/Comments: (ramamurt(at)csusm(dot)edu)
Last modified: May 9, 2006