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.

Symmetric positive definite (SPD) matrices play important roles in numerous areas of mathematics, statistics, and their applications in machine learning, optimization, computer vision, and related fields. Among the most important topics in the study of SPD matrices are the distances between them that can properly capture the geometry of the set of SPD matrices. Two of the most widely used distances are the affine-invariant distance and the Log-Euclidean distance, which are geodesic distances corresponding to two different Riemannian metrics on this set. In this talk, we present our recently developed concept of Log-Hilbert-Schmidt (Log-HS) distance between positive definite Hilbert-Schmidt operators on a Hilbert space.This is the generalization of the Log-Euclidean distance between SPD matrices to the infinite-dimensional setting. In the case of reproducing kernel Hilbert space (RKHS) covariance operators, we obtain closed form formulas for the Log-HS distance, expressed via Gram matrices. As a practical example, we demonstrate an application of the Log-HS distance to the problem of image classification in computer vision.

Although DNA forensic evidence is widely considered objective and infallible, a great deal of subjectivity and bias can still exist in its interpretation, especially concerning mixtures of DNA. The exact degree of variability across labs, however, is unknown, as DNA forensic examiners are primarily trained in-house, with protocols and quality control up to the discretion of each forensic laboratory. This talk uncovers the current state of forensic DNA mixture interpretation by analyzing the results of a groundbreaking DNA mixture interpretation study initiated by the Department of Defense's Defense Forensic Science Center (DFSC) in the summer of 2014. This talk will be accessible to undergraduates.

Existence of tight contact structures is a fundamental question of contact topology. Etnyre and Honda first gave the example which doesn't admit any tight structure. The existence of fillable tight structures is also a subtle question. Here we give some new examples of hyperbolic 3-manifolds which do not admit any fillable structures.