# MathCS Seminar 2006

## Fall 2006

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

### Thursday, November 30, 2006, 12:30-1:50 pm

*Speaker:* Dr. Mihaela Vajiac

*Title:* Harmonic Maps and Loop Groups

*Abstract:* Harmonic maps have been the intense subject in differential geometry for some time now. We will talk about the theory of harmonic maps in Lie groups and the loop groups associated to these spaces of maps. We will then move to describing how this can be thought of as a Riemann-Hilbert Factorization Problem and conclude with the newly found Virasoro action on this space .

### Wednesday, November 15, 2006, 12:30-1:50 pm

*Speaker:* Halina Goetz

*Title:* Challenges and success stories in teaching preparatory courses

*Abstract:* The speaker summarizes her experience teaching remedial courses in various settings, she will outline misconceptions about tutoring, will and some challenges in teaching Mathematics for Elementary Teachers by comparing and contrasting the delivery of the course at other institutions. She will conclude the talk with tangible and concrete steps for improving the overall teaching results in remedial and MATH 206 class.

### Thursday, November 2, 2006, 12:30-1:50 pm

*Speaker:* Dr. Peter Jipsen

*Title:* Involutive residuated frames with applications to decidability

(joint work with Dr. Nikolaos Galatos, Japan Advanced Institute for Science and Technology)

*Abstract:* We consider frames for involutive residuated lattices. We use them to prove a general cut-elimination result for this variety and derive the finite model property for classical substructural logics (without any assumption of commutativity or cyclicity). Connections with relation algebras are highlighted, and we examine several constructions on frames (various unions and products) that illustrate how involutive frames provide a unifying perspective for involutive lattices.

### Thursday, October 26, 2006, 12:30-1:50 pm

*Speaker:* Dr. Peter Jipsen

*Title:* Residuated (Kripke) frames

(joint work with Dr. Nikolaos Galatos, Japan Advanced Institute for Science and Technology)

*Abstract:* The notion of frames from modal logic has been generalized in a number of different ways. We present some background on residuated lattice ordered groupoids,
FL-algebras and several important subvarieties (involutive, cyclic, classical, commutative, distributive, Boolean). The represention of lattices via Galois correspondences and the concept of nucleus are shown to lead naturally to the notion of nuclear Galois relation. This results in the definition of residuated frames as a suitable generalization of Kripke semantics for FL-algebras.

### Thursday, October 19, 2006, 12:30-1:50 pm

*Speaker:* Dr. M. Andrew Moshier

*Title:* Hofmann-Mislove Theorems in Bitopological Settings

*Abstract:* The Hofmann-Mislove Theorem for sober topological spaces establishes a valuable connection between compact subsets and certain filters of opens, roughly speaking, by extending the bijection between points and completely prime filters that characterizes sober spaces. The theorem has proved to be very useful in domain theory and other areas of topology.

In the first part of this talk, we will consider ways to interpret the classical result as well as its frame-theoretic analogue. Following that, we will discuss new results on bitopological spaces and their Stone duals.

*N.B*. This talk is self-contained. Although last seminar talk "On the Bitopological Nature of Stone Duality" provides some motivation, I will not use any technical ideas from that talk.

### Thursday, September 28, 2006, 12:30-1:30 pm

*Speaker:* Dr. M. Andrew Moshier

*Title:* On the Bitopological Nature of Stone Duality

*Abstract:*
The relationship between classical Stone Duality and the theory of frames is much more subtle than one is often lead to believe. In particular, Stone Duality has to do with bounded distributive lattices (and more specially, Boolean rings) that need not be complete, whereas a frame is a special sort of complete lattice. So Stone's original results, as well as several more recent Stone-type dualities, do not actually generalize, nor specialize, the duality between spaces and frames (here refered to as "Papert/Isbell duality").

We provide a uniform treatment of Stone-type dualities as well as Papert/Isbell duality, both as specializations of a general concrete adjunction. Toward this end, we introduce the category of "d-frames" as the duals of bi topological spaces. In frames, two distinct notions of order are conflated: a frame is a certain kind of dcpo (information ordered) and is a certain kind of distributive lattice (logically ordered). In d-frames, the symmetry between information and logic breaks naturally. It is this broken symmetry on the "frame-theoretic" side along with the asymmetry of bispaces on the "topological side" that allows for a uniform treatment of Stone and Isbell duality.

### Thursday, September 21, 2006, 12:30-1:30 pm

*Speaker:* Dr. Raymundo Marcial Romero

*Title:* A computational framework for exact real number arithmetic.

*Abstract:* Almost every mechanical device to do real number calculations uses the well-known floating-point arithmetic. However floating-point arithmetic cannot represent every real number in an exact form. Hence, in a calculation a rounding is done to the closest floating-point number. Although this kind of representation is useful for a large number of calculations, there are other frameworks in which more accurate solutions are required. Exact real number arithmetic is a different paradigm to do real number computation in an "exact" form. In this talk we sketch exact real number arithmetic and present a programming language to work with it.

### Thursday, September 14, 2006, 12:30-1:30 pm

*Speaker:* Professor Daniele Struppa

*Title:* Algebraic Analysis of Dirac Systems