Georgia Institute of Technology College of Sciences

An n-dimensional topological quantum field theory is a functor from the category of closed, oriented (n-1)-manifolds and n-dimensional cobordisms to the category of vector spaces and linear maps. Three and four dimensional TQFTs can be difficult to describe, but provide interesting invariants of n-manifolds and are the subjects of ongoing research. This talk focuses on the simpler case n=2, where TQFTs turn out to be equivalent, as categories, to Frobenius algebras. I'll introduce the two structures -- one topological, one algebraic -- explicitly describe the correspondence, and give some examples.

Some properties of biological memory are briefly described. The examples of short term memory and extra long term memory are drawn from psychological literature and from the personal experience. The short term memory is modeled here with the two types of mathematical models, both models being special cases of the Locally Organized Systems (LOS). The first model belongs to Prof. Mikhail Tsetlin of Moscow State University. His original ?pile of books? model was independently rediscovered a new by a number of scientists throughout the World. Tsetlin?s model demonstrates some very important properties of a natural memory organization. However mathematical study of his model turned out to be rather complicated. The second model belongs to the present author and has somewhat similar properties. However, it is organized in a completely different manner. In particular it contains some parameters, which makes the model rather interesting mathematically and pragmatically. The Stefanuk?s model has many interpretations and will be illustrated here with some biologically inspired examples. Both models founded a number of practical applications. These models demonstrate that the short term memory, which is heavily used by humans and by many biological subsystems is arranged reasonably. For humans it helps to keep the knowledge in the way facilitating its fast extraction. For biological systems the models explain the arrangement of storage of various micro organisms in a cell in an optimal manner to provide for the living.

Let k be a field (not of characteristic 2) and let t be an indeterminate. Legendre's elliptic curve is the elliptic curve over k(t) defined by y^2=x(x-1)(x-t). I will discuss the arithmetic of this curve (group of solutions, heights, Tate-Shafarevich group) over the extension fields k(t^{1/d}). I will also mention several variants and open problems which would make good thesis topics.

There are several definitions of the word shape; of these, the most important to this research is “the external form or appearance of someone or something as produced by its outline.” Shape Analysis in this context focuses specifically on the mathematical study of explicit, parameterized curves in 2D obtained from the boundaries of targets of interest in Synthetic Aperture Sonar (SAS) imagery. We represent these curves with a special “square-root velocity function,” whereby the space of all such functions is a nonlinear Riemannian manifold under the standard L^2 metric. With this curve representation, we form the mathematical space called “shape space” where a shape is considered to be the orbit of an equivalence class under the group actions of scaling, translation, rotation, and re-parameterization. It is in this quotient space that we can quantify the distance between two shapes, cluster similar shapes into classes, and form means and covariances of shape classes for statistical inferences. In this particular research application, I use this shape analysis framework to form probability density functions on sonar target shape classes for use as a shape prior energy term in a Bayesian Active Contour model for boundary extraction in SAS images. Boundary detection algorithms generally perform poorly on sonar imagery due to their typically low signal to noise ratio, high speckle noise, and muddled or occluded target edges; thus, it is necessary that we use prior shape information in the evolution of an active contour to achieve convergence to a meaningful target boundary.