A real algebraic variety is a set of points in real Euclidean space that satisfy a system of polynomial equations. Metric algebraic geometry is the study of properties of real algebraic varieties that depend on a metric. In this talk, we introduce metric algebraic geometry through a discussion of Voronoi cells, bottlenecks, and the reach of an algebraic variety. We describe applications to the study of the geometry of data with nonlinear models and to low-rank matrix approximation.

We introduce two novel applied topology techniques to address real-world problems in different fields. The first, used in information theory for the study of complex systems, is a principled compression method of higher order signals obtained from a combinatorial version of the Laplace-Beltrami operator. The second, used for the study of conformation spaces of molecules, is an enhancement of the traditional persistent homology pipeline obtained from recent developments in the effective construction of cohomology operations.