Georgia Institute of Technology College of Sciences

Many real-world problems require an approximation to an algebraic variety (e.g., determination of the roots of a polynomial). To solve such problems, the standard techniques are either symbolic or numeric. Symbolic techniques are globally correct, but they are often time consuming to compute. Numerical techniques are typically fast, but include more limited correctness statements. Recently, attention has shifted to hybrid techniques that combine symbolic and numerical techniques. In this talk, I will discuss hybrid subdivision algorithms for approximating a variety. These methods recursively subdivide an initial region into smaller and simpler domains which are easier to characterize. These algorithms are typically recursive, making them both easy to implement (in practice) and adaptive (performing more work near difficult features). There are two challenges: to develop algorithms with global correctness guarantees and to determine the efficiency of such algorithms. I will discuss solutions to these challenges by presenting two hybrid subdivision algorithms. The first algorithm computes a piecewise-linear approximation to a real planar curve. This is one of the first numerical algorithms whose output is guaranteed to be topologically correct, even in the presence of singularities. The primitives in this algorithm are numerical (i.e., they evaluate a polynomial and its derivatives), but its correctness is justified with algebraic geometry and symbolic algebra. The second algorithm isolates the real roots of a univariate polynomial. I will analyze the number of subdivisions performed by this algorithm using a new technique called continuous amortization. I will show that the number of subdivisions performed by this algorithm is nearly optimal and is comparable with standard symbolic techniques for solving this problem (e.g., Descartes' rule of signs or Sturm sequences).

We introduce a new invariant of an abelian variety defined over a number field, and study its arithmetic properties. We then show how an extended version of Mazur's visibility theorem yields non-trivial elements in this invariant and explain how such a construction provides theoretical evidence for the Birch and Swinnerton-Dyer Conjecture.

The talk will discuss the concept of test ideal for rings of positive characteristic. In some cases test ideals enjoy remarkable algebraic properties related to the integral closure of ideals. We will present this connection in some detail.