Georgia Institute of Technology College of Sciences

The Macaulay dual space offers information about a polynomial ideal localized at a point such as initial ideal and values of the Hilbertfunction, and can be computed with linear algebra. Unlike Gr\"obner basis methods, it is compatible with floating point arithmetic making it anatural fit for the toolbox of numerical algebraic geometry. I willpresent an algorithm using the Macaulay dual space for computing theregularity index of the local Hilbert function.

Systems biology focuses on modeling complex biological systems, such as metabolic and cell signaling networks. These biological networks are modeled with polynomial dynamical systems. Analyzing these systems at steady-state results in algebraic varieties that live in high-dimensional spaces. By understanding these varieties, we can provide insight into the behavior of the models. Furthermore, this algebro-geometric framework yields techniques for model selection and parameter estimation that can circumvent challenges such as limited or noisy data. In this talk, we will introduce biochemical reaction networks and their resulting steady-state varieties. In addition, we will discuss the questions asked by modelers and their corresponding geometric interpretation, particularly in regards to model selection and parameter estimation.

We define a variant of tropical varieties for exponential sums. These polyhedral complexes can be used to approximate, within an explicit distance bound, the real parts of complex zeroes of exponential sums. We also discuss the algorithmic efficiency of tropical varieties in relation to the computational hardness of algebraic sets. This is joint work with Maurice Rojas and Grigoris Paouris.

Hurwitz correspondences are certain multivalued self-maps of the moduli space M0,N parametrizing marked genus zero curves. We study the dynamics of these correspondences via numerical invariants called dynamical degrees. We compare a given Hurwitz correspondence H on various compactifications of M0,N to show that, for k ≥ ( dim M0,N )/2, the k-th dynamical degree of H is the largest eigenvalue of the pushforward map induced by H on a comparatively small quotient of H2k(M0,N). We also show that this is the optimal result of this form.