Mathematical Sciences Faculty Seminar

This seminar is a series of talks aimed primarily at a (math) faculty audience. However, graduate and undergraduate students are welcome to attend. We plan to meet approximately once a week to discuss a topic of interest. Please feel free to suggest a topic (or be brave and volunteer to present!)

Series I: An Introduction to Differential Algebra (Bill Cook)

Roughly, a differential algebra is an algebra equipped with some kind of derivative. These algebras are fundamentally important for the studyof symbolic integration.

We plan look at the definition of a differential algebra and the basic related algebraic concepts (field of constants, differential ideals, subalgebras, homomorphisms etc). Then discuss a little bit of differential Galois theory, Picard-Vessiot extensions, and the solvability of differential equations.

Next, focusing in a different direction, yet using the same framework, we wish to discuss what elementary functions are and then look at various types of extensions of function fields such as: algebraic, logarithmic, and exponential extensions.

Finally, we would like to take a crack at Liouville's principle, the Rothstein/Trager method, and the Risch algorithm (the algorithm which integrates *any* elementary function which has an elementary antiderivative or shows why it no such antiderivative exists).

Our inital meeting place/time is 10am on Fridays in Walker 302.

• March 30: Classical vs. Differential Galois Theory. We will compare and contrast classical Galois theory (used to study solvability of polynomial equations) and differential Galois theory (used to study solvability of linear differential equations). Answering questions like: What is a solvable group? What is an exponential extension? What are elementary functions?
• February 24: Differential algebra basics. Defining algebras, derivations, differential ideals, subalgebras, homomorphisms. Basic examples as well as Picard-Vessiot extentions. Maybe some differential Galois theory (time permitting).
  

Series II: Wavelets (Bill Bauldry)