# MathCS Seminar 2005

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

## Fall 2005

All seminar talks take place in Argyros Forum 211.

### Thursday, December 7, 2005, 12-1 pm

Speaker: Professor Achim Jung, University of Birmingham

Title: Extending valuations to measures

Abstract: Many theorems in measure theory are about extending a "proto-measure" to a full measure, where "proto" means that the given function assigns a volume only to some measureable sets. Computer scientists became interested in one particular instance of this where the proto-measure is defined on the open sets of a topological space. Such proto-measures had been looked at before under the name "valuations".

In my talk I will try to explain why valuations are particularly interesting from a computer science point of view, and I will prove an extension theorem using classical theorems from analysis. This proof works only for compact ordered spaces, and so I will get an excuse for explaining why these are of interest to computer scientists, too.

The talk is based on joint work with Klaus Keimel.

### Thursday, November 30, 2005, 12-1 pm

Speaker: Dr. Mihaela Vajiac

Title: Harmonic Maps and Loop Groups on SU(2).

### Thursday, November 29, 2005, 12-1 pm

Speaker: Arek Goetz, University of San Francisco

Title: Multimedia online learning environment as a tool in teaching calculus and in research in geometric dynamical systems

### Thursday, November 9, 2005, 12-1 pm

Speaker: Dr. Mihaela Vajiac

Title: Quantum Cohomology in Symplectic Geometry

### Thursday, October 26, 2005, 12-1 pm

Speaker: Dr. Adrian Vajiac

Title: Equivariant Cohomology

### Thursday, October 12, 2005, 12-1 pm

Speaker: Dr. Adrian Vajiac

Title: The Global Aspect of the Parallelism Axiom

### Thursday, October 5, 2005, 12-1 pm

Speaker: Dr. Peter Jipsen

Title: Complex algebras of groups and Boolean algebras

Abstract: Given a group $G$, the complex algebra Cm$(G)$ is defined by $(P(G),\cup,\sim,\emptyset,*,^{-1},\{e\})$, where

$X*Y=\{xy : x \in X, y \in Y\}$ and $X^{-1}=\{x^{-1} : x \in X\}$

for any subsets $X,Y$ of $G$. Likewise, for a Boolean algebra $B$, the complex algebra Cm$(B)$ is defined by $(P(B),\cup,\sim,\emptyset,+,-,\{0\})$, where $+,-$ are the lifted versions of the Boolean join and complementation of $B$. It is well-known that the class of all complex algebras of groups is nonfinitely axiomatizable and undecidable. The corresponding problems for the variety HBA generated by complex algebras of Boolean algebras have not yet been resolved. As a contribution in this direction, we find a number of identities that hold in all complex algebras of Boolean algebras, and we prove that if a Boolean algebra with additional operations $+,-,0$ has $\le 24$ elements then it will be in the variety HBA iff it satisfies these identities.

### Thursday, September 29, 2005, 3 pm

Speaker: Prof. Atanas Radenski

Title: Can Introductory Computer Science Be Relieved From the Complexity of Commercial Languages? A "Python First, Java Second" Approach to CS1/CS2

Abstract: In this talk, I will advocate the need for, and the benefits from, a dual-language approach to the introductory computer science course sequence, commonly referred to as CS1/CS2. I will describe my implementation of a "Python First, Java Second" approach that has been used at Chapman University since the fall of 2004: A gentle CS1 course builds core knowledge using the Python language, while a subsequent comprehensive CS2 course upgrades and extends this knowledge using a mainstream language, Java.

To support "Python first, Java second" courses, I have created comprehensive online study packs that provide complete coverage for all course activities. Both study packs feature detailed self-guided lab assignments and honor lab report system. The study packs reduce the need for face-to-face activities thus facilitating students with busy class and work schedules. Both study packs are installed on a Moodle server.

## Spring 2005

All seminar talks take place in Beckman Hall 402.

### Thursday, April 14, 2005, 3 pm

Speaker: Dr Adrian Vajiac

Title: Topological Invariants of 4-manifolds

### Thursday, April 7, 2005, 4 pm

Speaker: Dr Scott Baldridge (Louisiana State University)

Title: On symplectic 4-manifolds with prescribed fundamental group

Abstract: In this talk I will discuss minimizers of the function $f=a\chi+b\sigma$ on the class of all symplectic 4--manifolds with prescribed fundamental group $G$ ($\chi$ is the Euler characteristic, $\sigma$ is the signature, and $a,b\in \BR$). The values of $a,b$ for which the function $f$ has a lower bound have some surprising features. I will discuss these features and describe some examples of manifolds which minimize $\chi$.

### Thursday, March 31, 2005, 3 pm

Speaker: Dr Andrew Moshier (Chapman University)

Title: Synthetic Topology and Semantics of Exceptions. Part II

### Thursday, March 17, 2005, 3 pm

Speaker: Dr Andrew Moshier (Chapman University)

Title: Synthetic Topology and Semantics of Exceptions.

Abstract: In the late 1980's, Mike Smyth developed a series of analogies between computational and topological ideas that have come to be known as "Smyth's dictionary". Recently the dictionary has been extended along various lines leading to research programs called "Synthetic Topology" (pursued especially by Martin Escardo and Andre Bauer) and "Abstract Stone Duality" (pursued by Paul Taylor). In this talk, we will introduce the basics of Smyth's dictionary, outline some of the newer directions of this research (borrowing heavily from Escardo and Taylor), and mention a further recent development that brings bitopological concepts into the story and sheds some promising light on how computational exceptions fit into Smyth's analogies.

### Thursday, February 17, 2005, 3 pm

Speaker: Dr Peter Jipsen (Chapman University)

Title: On congruences in Residuated Kleene algebras

Abstract: Kleene algebras have a long history in Computer Science, with applications in formal foundation for automata theory, regular grammars, semantics of programming languages and other areas.

However the algebraic structure of Kleene algebras has not been studied as extensively. This is partly because the class of all Kleene algebras is a quasivariety but not a variety, i.e. it can be defined by strict Horn formulas but not by identities. If we add the quite natural operations of left and right residuals for the Kleene product, we get the variety of residuated Kleene algebras (also called action algebras by V. Pratt, 1990 and D. Kozen, 1994). Until recently it was not even known if the variety of residuated Kleene algebras is congruence distributive. Since this property has many useful algebraic consequences, it is certainly of interest that (a noncommmutative version of) a result in algebraic logic by C. van Alten and J. Raftery, 2004, based on a description of congruence filters by W. Blok and J. Raftery, implies that the congruence lattice of any residuated Kleene algebra is distributive.

We give an overview of this result in the present setting and explore some of its consequences for the lattice of varieties of residuated Kleene algebras. In particular we show that this lattice contains uncountably many varieties covering the Boolean variety.

### Thursday, February 10, 2005, 3 pm

Speaker: Dr. Mohamed Allali

Title: Interpolation and Equidistribution on the unit sphere, part II.

Abstract: The problem of generating a large number of points on the sphere has many applications in various fields of computation such as quadrature, placing grids on S2, tomography, coding theory, etc.