The sixth meeting of the Carolina Mathematics Seminar took place at the University of South Carolina Salkehatchie, Salkehatchie, SC. This is a list of our speakers, titles and abstracts.
Fidele Ngwane (USC Salkehatchie)
Title: Computing Integral Closures.
Abstract: Monomial orderings will be discussed. They are vital in our polynomial computations. Integral extensions will be analyzed, in particular, type I integral extensions. Integral closures of type I integral extensions have great applications. We will present a method for computing integral closures that is different from others.
Balaji Iyangar (Benedict College)
Title: Multigrid Methods.
Abstract: Iterative processes for solving the algebraic equations arising from discretizing partial differential equations are stalling numerical processes, in which the error has relatively small changes from one iteration to the next. The computer grinds very hard for very small or slow real physical effect with the use of too-fine discretization grids. In this case, in large parts of the computational domain the meshsize is much smaller than the real scale of solution changes. Such problems can be overcome by the multigrid method, or more generally, the Multi-Level Adaptive Technique (MLAT). Stalling numerical processes are usually related to the existence of several solution components with different scales, which conflict with each other. By using interactively several scales of discretization, multigrid techniques resolve such conflicts, avoiding stalling and also being computationally efficient.
Antara Mukherjee (The Citadel)
Title: Isoperimetric Inequalities Using Varopoulos.
Abstract: I will start by introducing Dehn functions and then show the compu tation of upper bounds of the second order Dehn functions for lattices of three-dimensional geometries, namely Nil and Sol. These upper bounds are obtained by using the Varopoulos transport argument on dual graphs. The idea is to reduce the original isoperimetric problem involving volume of three-dimensional balls and areas of their boundary spheres to a problem involving Varopoulos' notion of volume and boundary of nite domains in dual graphs.
Upasana Kashyap (The Citadel)
Title: A Morita theorem for dual operator algebras.
Abstract: We consider some new variants of the notion of Morita equivalence appropriate to algebras of Hilbert space operators which are closed in the `weak* -topology' (or equivalently, which are dual spaces and known as dual operator algebras), and we will describe how the earlier theory of strong Morita equivalence due to Blecher, Muhly, and Paulsen, transfers to this `weak*-topology setting'. We will present our main theorem, that two dual operator algebras are weak*-Morita equivalent in our sense if and only if they have equivalent categories of dual operator modules. A key ingredient in the proof of our main theorem is W*-dilation, which connects the non-selfadjoint dual operator algebra with the W*-algebraic.
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