Georgia Institute of Technology College of Sciences

Granville and Soundararajan have recently introduced thenotion of pretentiousness in the study of multiplicative functions ofmodulus bounded by 1, essentially the idea that two functions whichare similar in a precise sense should exhibit similar behavior. Itturns out, somewhat surprisingly, that this does not directly extendto detecting power cancellation - there are multiplicative functionswhich exhibit as much cancellation as possible in their partial sumsthat, modified slightly, give rise to functions which exhibit almostas little as possible. We develop two new notions of pretentiousnessunder which power cancellation can be detected, one of which appliesto a much broader class of multiplicative functions. This work isjoint with Junehyuk Jung.

Geometric modeling builds computer models for industrial design and manufacture from basic units, called patches, such as, Bézier curves and surfaces. The control polygon of a Bézier curve is well-defined and has geometric significance—there is a sequence of weights under which the limiting position of the curve is the control polygon. For a Bezier surface patch, there are many possible polyhedral control structures, and none are canonical. In this talk, I will present a not necessarily polyhedral control structure for surface patches, regular control surfaces, which are certain C^0 spline surfaces. While not unique, regular control surfaces are exactly the possible limiting positions of a Bezier patch when the weights are allowed to vary. While our primary interest is to explain the meaning of control nets for the classical rational Bezier patches, we work in the generality of Krasauskas’ toric Bezier patches. Toric Bezier patches are multi-sided parametric patches based on the geometry of toric varieties and depend on a polytope and some weights. Our results rely upon a construction in computational algebraic geometry called a toric degeneration.

The talk will discuss the notion of Hilbert-Kunz multiplicity, presenting its general theory and listing some of the outstanding open problems together with recent progress on them.

In the last decades, path following methods have become a very popular strategy to solve systems of polynomial equations. Many of the advances are due to the correct understanding of the geometrical properties of an algebraic object, the so-called solution variety for polynomial system solving. I summarize here some of the most recent advances in the understanding of this object, focusing also on the certifcation and complexity of the numerical procedures involved in path following methods.