Georgia Institute of Technology College of Sciences

The asymptotic theory is developed for polynomial sequences that are generated by the three-term higher-order recurrence Q_{n+1} = zQ_n - a_{n-p+1}Q_{n-p}, p \in \mathbb{N}, n\geq p, where z is a complex variable and the coefficients a_k are positive and satisfy the perturbation condition \sum_{n=1}^\infty |a_n-a|<\infty . Our results generalize known results for p = 1, that is, for orthogonal polynomial sequences on the real line that belong to the Blumenthal-Nevai class. As is known, for p\geq 2, the role of the interval is replaced by a starlike set S of p+1 rays emanating from the origin on which the Q_n satisfy a multiple orthogonality condition involving p measures. Here we obtain strong asymptotics for the Q_n in the complex plane outside the common support of these measures as well as on the (finite) open rays of their support. In so doing, we obtain an extension of Weyl's famous theorem dealing with compact perturbations of bounded self-adjoint operators. Furthermore, we derive generalizations of the classical Szeg\"o functions, and we show that there is an underlying Nikishin system hierarchy for the orthogonality measures that is related to the Weyl functions. Our results also have application to Hermite-Pad\'e approximants as well as to vector continued fractions.

The over-abundance of remotely sensed data has resulted inthe realization that we do not have nor could ever acquire asufficient number of highly trained image analysts to parse theavailable data.  Automated techniques are needed to perform low levelfunctions, identifying scenarios of importance from the availabledata, so that analysts may be reserved for higher level interpretativeroles. Data fusion has been an important topic in intelligence sincethe mid-1980s and continues to be a necessary concept in thedevelopment of these automated low-level functions. We propose anapproach to multimodal data fusion to combine images of varyingspatial and spectral resolutions with digital elevation models.Furthermore, our objective is to perform this fusion at the imagefeature level, specifically utilizing Gabor filters because of theirresemblance to the human visual system.

In this talk we will give an introduction of Heegaard-Floer theory through examples. By exploring several explicit examples we hope to show that various aspects of the definitions that seem complicated, really aren't too bad and it really is possible to work with these fairly abstract things. While this is technically a continuation of last weeks talk, we will review enough material so that this talk should be self contained.

Rota asked in the 1960's how one might construct an axiom system for infinite matroids. Among the many suggested answers were the B-matroids of Higgs. In 1978, Oxley proved that any infinite matroid system with the notions of duality and minors must be equivalent to B-matroids. He also provided a simpler mixed basis-independence axiom system for B-matroids, as opposed to the complicated closure system developed by Higgs. In this talk, we examine a recent paper of Bruhn et al that gives independence, basis, circuit, rank, and closure axiom systems for B-matroids. We will also discuss some open problems for infinite matroids.