A: Counting associative binary operations on a finite set.
The number of associative binary operations on a finite set is only known for sets of order <= 8. I'll give an explanation of why it is so computationally intensive, as well as some suggestions to drastically speed up a brute force counting algorithm.
B: Using computers in proving theorems.
I'm doing my masters paper on this topic. I'll be explaining ways of integrating computers into advanced mathematics education to help clean up proofs. I'll also briefly address why it is that computers can't already do all proofs if they are so fast (p vs. np), and offer my own suggestions as to how computers can work together with people to prove theorems.
C: A plan for inexpensively getting objects into space in two stages.
The plans for building such a device will be explained in detail.
Classical Cauchy problem for first order partial differential equations
I will describe, more or less geometrically, how to solve the noncharacteristic initial value problem for first order partial differential equations. I will try to point out the invariant features of the theory. This material is fairly elementary.
I will start by introducing the notion of jets, so we can view a differential equation as living on a jet bundle, and in special cases, on the cotangent bundle. This sounds pretty fancy, but its just a way of viewing Taylor's theorem, so Calculus 2.
A stochastic model of a frictionless security market
A general stochastic model of a frictionless security market is presented. The modern theory of contingent claim valuation (option pricing) provides a framework for this model. In particularly the option pricing formula of Black and Scholes is in focus here.
Stability Analysis of a 2-level Timestepping Method Applied to the Linearized Shallow Water Equations
The main idea I'll be discussing involves deriving the stability condition of a particular numerical method used to discretize the linearized shallow water equations as proposed by Bob Higdon. I'll be using standard numerical discretization schemes as well as familiar linear algebra results.
Methods for computing entropy for 2-dimensional symbolic shift systems
Outlined is a method for computing the topological entropy for 2-dimensional shifts of finite type.
Bisecting pairs of bounded polygons in the plane. The Borsuk-Ulam Theorem.
Given a single bounded polygon in the plane showing there is a line that bisects it is a strait forward application of the intermediate value theorem; try it. However given two bounded polygons in the plane it is less clear that there must be a single line that bisects both regions. In fact such a line does exist as shown by a corollary to the well known Borsuk-Ulam theorem. I will develop this theorem in terms of introductory algebraic topology.
Hecke groups and modular embeddings
We are considering subgroups G of the group of automorphisms of the upper half plane H that act discontinuously on H. In the case of Hecke groups we have 3 generators with a given signature. If G is arithmetic the quotient H/G will parametrize isomorphism classes of abelian varieties. Paula Cohen (Lille, France) and Juergen Wolfart (Germany) have studied the case of non arithmetic Hecke groups (1990).
Factoring with elliptic groups
I will discuss the basics of elliptic curves, and their use in factoring algorithms.
How Students Understand Limit
This talk will focus on the views students have of limit, the language which may cause incorrect views of limit and the representational preferences of students when determining limits. The previous research in this area will be discussed and will be followed by a discussion of my specific research study and presentation of the results.
Exploring the Principle of Duality and it's applicatons in Projective Plane Geometry, Linear Programming and Boolean Algebra
This talk will look at a very general definition of Duality and explore the parallels in the above areas of mathematics. The general definition used will be:
"Duality is defined as the existence of two logical systems characterized by certain interrelationships. The essence of a dual system is a correspondence between concepts in one logical system and concepts in the other which allows us to derive a correspondence between results in one system and results in another" (Russel and Wilkinson, 1979)
Investigations on the theory of the brownian motion
I will be presenting one of Einstein's articles on the topic of Brownian Motion written in 1905. We will be looking at how Einstein derived the Diffusion coefficient by using the probability distribution function of particles suspended in liquid. From the probability distribution we can calculate the average displacement of the suspended particle in a given time interval.
Image Reconstruction
Computed tomography entails the reconstruction of a function f from line integrals of f. This mathematical problem is encountered in medicine, science, and technology. There are many types of reconstruction methods. I will discuss 2 of those, the Filtered Backprojection Reconstruction, and Non-equispaced Fast Fourier Transform Reconstruction.