# MathCS Seminar 2004

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

## Contents |

## Fall 2004

### Thursday, December 2, 2004, 3 pm

*Speaker:* Dr. Mohamed Allali

*Title:* Interpolation and Equidistribution on the unit sphere

*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. This
talk will introduce different methods of equidistributing points on the
sphere. Moreover, of practical importance is the problem of
interpolating scattered data on the sphere. Strictly positive definite
functions shall be introduced and connected to the interpolation
problem.

### Thursday, November 18, 2004, 3 pm

*Speaker:* Prof. Mihaela Vajiac

*Title:* Harmonic Maps on SU(n) and Virasoro actions, part II

Joint work with Professor Karen Uhlenbech, UT Austin

*Abstract:* This introductive talk covers the interaction between Harmonic
Maps, Loop Groups and Integrable Systems.

We continue with the construction of the Virasoro algebra in the SU(n) case.

### Thursday, October 28, 2004, 3 pm

*Speaker:* Prof. Mihaela Vajiac

*Title:* Harmonic Maps on SU(n) and Virasoro actions

Joint work with Professor Karen Uhlenbech, UT Austin

*Abstract:* This introductive talk covers the interaction between Harmonic Maps, Loop Groups and Integrable Systems.

We will continue next thursday with the construction of the Virasoro algebra in the SU(n) case.

## Spring 2004

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

### Thursday, April 29, 2004, 4 pm

*Speaker:* Dr. Joanne Walters-Wayland

*Title:* $G_{\delta}$-density in a pointfree setting

*Abstract:* I shall endeavour to give a short history of, and motivation for
studying, "generalized topoplogy". I shall also attempt to highlight the
beauty of this theory by considering $G_{\delta}$-density in this setting. In
particular, I will introduce the cozero part of a frame - the cozeros are
essential in understanding many concepts especially when dealing with
complete regularity and related notions.

### Thursday, April 8, 2004, 1:30 pm

*Speaker:* Dr. Catalin Zara (Penn State)

*Title:* Hamiltonian GKM Spaces: From Geometry To Combinatorics, And Back.

### Thursday, April 1, 2004, 2:45 pm

*Speakers:* Catherine Parsons, Vicki Shultz and Zack Wheatly (Chapman University)

*Title:* Morphing, Warping and SVD Through Linear Algebra.

*Abstract:* Linear Algebra is used throughout Digital Image Processing (DIP) for both effects and compression. Morphing and warping, two techniques in DIP used to make one image "transform" into another, and Singular Value Decomposition (SVD) are examples of this which we will present in this talk. Morphing fades one image into another, while warping stretches and skews the images based off chosen lines. SVD takes an original image, breaks it into three distinct matrices, and extracts enough pertinent information to still
recreate the image while using less memory. Through our presentation we
will show how these three techniques in DIP can be achieved with the basic tools of Linear Algebra.

### Thursday, March 18, 2004, 3 pm

*Speaker:* Noah Salvaterra (Indiana University, Bloomington)

*Title:* A Brief Introduction to Khovanov Homology

*Abstract:* By a modification of Kauffman's bracket for a link diagram one can construct a chain complex of graded vector spaces. Just as a renormalization of the Kauffman bracket leads to a link invariant, the Jones polynomial, a similar renormalization of this complex gives rise to interesting homologies which are actually link invariants. While in fact the Jones polynomial arises as the graded Euler characteristic in this construction, the Poincare polynomial of these homologies is a strictly stronger invariant.

I hope to give a brief overview of this construction with lots of pictures, filling in the necessary knot theory along the way.

### Thursday, March 11, 2004, 3 pm

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

*Title:* Quantum Cohomology and Quantum Products.

*Abstract:* The theory of quantum cohomology and Gromov-Witten invariants was
first developed by Witten and has been the subject of
active research in algebraic and symplectic geometry.
In my talk, gauge theory techniques and the theory of flat
connections are used to prove that the small quantum product is a
gauge theoretic deformation of the cup product on a symplectic
manifold M and to construct a moduli space of products on a
vector space which are associative, commutative, Frobenius, and have
unit (we will call these quantum-type products).

### Thursday, February 25, 2004, 3 pm

Meeting on assessment of GE Calculus

### Thursday, February 19, 2004, 3 pm

Meeting to discuss GE, precalculus textbook adoption.

### Thursday, February 12, 2004, 3 pm

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

*Title:* Representable l-groups are algebras of binary relations, and representable finite GBL-algebras are commutative.

*Abstract:* It is shown that any representable l-group G is isomorphic to a set of binary relations $R(G)$ closed under union, intersection, composition and residuals. The identity element of the l-group corresponds to the partial order relation rather than the diagonal relation, and the collection of binary relations is not closed under relation converse. However this construction never-the-less shows that the variety of representable l-groups is contained in the class of all binary relations closed under union, intersection, composition and residuals. It is interesting to note that, in contrast to the variety of representable l-groups, the latter class is not finitely axiomatizable. The construction shows that representable l-groups are subreducts of relation algebras with an additional constant, and leads to a number of interesting problems regarding the algebras of binary relations generated by $R(G)$ if we also include relation converse and/or complementation as basic operations.

In the second part of the talk we will show that representable generalized basic logic algebras are commutative, hence simply BL-hoops. Some possible generalizations of this result will also be discussed.