# MathCS Seminar 2002

*Seminar Organizer:* Mihaela Vajiac, *Webpage maintained by:* Peter Jipsen

## Contents |

## Fall 2002

**All seminar talks take place Wednesday mornings in Beckman Hall 402 (corner of One University Drive and N. Glassell, Orange, CA) at 11 am.**

### December 11, 2002

*Speaker:* Dr. Adrian Vajiac (Chapman University)

*Title:* Topological Quantum Field Theory and Invariants of 4-manifolds (Part 2)

### December 4, 2002

*Speaker:* Dr. Adrian Vajiac (Chapman University)

*Title:* Topological Quantum Field Theory and Invariants of 4-manifolds (Part 1)

*Abstract:* We investigate the relationship between Donaldson and Seiberg-Witten invariants of smooth 4-manifolds,using equivariant localization techniques applied to the corresponding TQFTs that generate these invariants.
This approach is intended to lower the gap between the mathematics and the physics of this subject, possibly leading to a rigorous interpretation of the physics of the generating TQFTs.

### November 13, 2002

*Speaker:* Dr. Mohamed Allali (Chapman University)

*Title:* A Covering of the Sphere by a Special Set of Rotations

*Abstract:* We will present a method of covering the unit sphere by means of spherical caps of fixed radius. The method based on a set of rotations provides an explicit formula for the number of spherical caps that cover the whole unit sphere and the exact positioning of their centers. To find the optimal number (the minimal number of spherical caps) is still an open problem.

### November 6, 2002

*Speaker:* Dr. Mihaela Vajiac (Chapman University)

*Title:* "Harmonic Maps and Integrable Systems"

*Abstract:* The theory of harmonic maps on Lie groups has
aquired a modern interpretation through integrable systems. We will outline the structual background on harmonic maps into a (compact) Lie group and talk about the arising loop group action and the implications in the theory of integrable systems.

### October 30, 2002

*Speaker:* Dr. Andrew Moshier (Chapman University)

*Title:* Stably Compact and Related Spaces

*Abstract:* Compact Hausdorff spaces, which are well-known to classical mathematicians, have a useful generalization to the setting of non-T1 spaces. The resulting spaces, known as Stably Compact, enjoy many of the useful properties of compact Hausdorff spaces and in fact help to tease apart some concepts that accidentally coindice in the Hausdorff setting. In this talk, we will discuss various topological ideas that lead to characterizations of stable compactness. We will introduce the key result in this area: the Hofmann-Mislove Theorem. Time permitting, we may also discuss interesting topological constructions under which stable compactness is preserved, particularly the construction of the space of Borel measures on an underlying space.

### October 23, 2002

*Speaker:* Dr. Peter Jipsen (Chapman University)

*Title:* An introduction to residuated KATs and RATs and related algebraic models of computation

*Abstract:* In this seminar I will examine some algebraic theories that have been developed in the past half century to reason about algorithms and automata. In particular I will look at Kleene algebras with test and relation algebras with transitive closure. Adding residuation to Kleene algebras gives the algebraic form of Pratt's action logic. It has many of the useful features of relation algebras and is still decidable. I will give a number of finite examples and indicate how to calculate the models on an n-element set.

### October 9, 2002

*Speaker:* Dr. Adrian Vajiac (Chapman University)

*Title:* The Geometry and Physics of Fermat's Last Theorem