Algebraic and Probabilistic examples in combinatorial geometry

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DMS 0432237Nets KatzIndiana UniversityAlgebraic and Geometric Examples in Combinatorial GeometryABSTRACTWe will study a collection of problems of similar flavor comingfrom different parts of mathematics. The problems are all combinatorialin nature and involve estimates on the overlap between various geometricobjects. We will attempt to produce examples in these problems byboth algebraic and probabilistic methods.A great deal of mathematical analysis involveshandling the infinite. Here however we engage ina deep study of the finite. To be precise, we study the waysin which finite collections of geometric objects like rectangles,patterns of points in a grid, and vectors on a high dimensionalsphere can be arranged to achieve certain unusual effects.We will attempt to discover new and better such arrangements.One way of doing this is by using a known algebraic structureto suggest a construction. Another possibility is to choose the example randomlywithin certain constraints.