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).

Random matrix theory is a fast developing topic with connections to so many areas of mathematics: probability, number theory, combinatorics, data analysis, mathematical physics, to mention a few. The determinant is one of the most studied matrix functionals. In our talk, we are going to give a brief survey on the studies of this functional, dated back to Turan in the 1940s. The main focus will be on recent developments that establish the limiting law in various models.

Understanding how the physiology of organisms arises through the dynamic interaction of the molecular constituents of life is an important goal of molecular systems biology, for which mathematical modeling can be very helpful. Different modeling strategies have been used for this purpose. Dynamic mathematical models can be broadly divided into two classes: continuous, such as systems of differential equations and their stochastic variants and discrete, such as Boolean networks and their generalizations. This talk will focus on the discrete modeling approach. Applications will include the study of stochasticity under this setting. No background in mathematical biology is required, and the talk will be accessible to a broad audience.

An isoradial graph is one which can be embedded into the plane such that each face is inscribable in a circle of common radius. We consider the superposition of an isoradial graph, and its interior dual graph, approximating a simply-connected domain, and prove that the height function associated to the dimer configurations is conformally invariant in the scaling limit, and has the same distribution as a Gaussian Free Field.