Enumerative Geometry of Algebraic Data

Solving systems of polynomial equations is central for many scientific fields and everyday life applications.

The study of the geometry of the solution sets of polynomial systems (algebraic varieties) is the core of Algebraic Geometry.

This project proposes to develop an efficient theory for sampling (finding real points on) algebraic varieties in order to recover geometrical properties such as the number of connected components or their singularities. The primary aim of the planned project is to develop new techniques for studying the varieties’ geometry.

One way to understand the shape of a variety is to find its algebraic invariants, i.e, the properties that remain the same even if the variety is deformed.

We propose to recover these invariants via a cloud of points on the variety, which we call a sample.The main point of this proposal is to develop an algebraic theory to produce an efficient sample, with the right density in order to recover the shape of the algebraic object.

This is in general a deep and difficult problem, at the center of research in Algebraic Geometry and analysis of algebraic data.We introduce and propose to study  alternative density invariants for sampling algebraic varieties, specifically k-bottlenecks and the  weak feature size. These are invariants  reflect  underline algebraic  structures.

Sampling-algorithms using these new invariants have the advantage of being computationally more accessible as they require considerably less sample points