We seek to determine a real algebraic variety from a fixed finite subset of points. Existing
methods are studied and new methods are developed. Our focus lies on aspects of topology …

Metric algebraic geometry combines concepts from algebraic geometry and differential
geometry. Building on classical foundations, it offers practical tools for the 21st century …

A bottleneck of a smooth algebraic variety X⊂C^n is a pair (x,y) of distinct points x,y∈X such
that the Euclidean normal spaces at x and y contain the line spanned by x and y. The …

The configuration spaces of parallel manipulators exhibit more nonlinearity than serial
manipulators. Qualitatively, they can be seen to possess extra folds. Projection onto smaller …

We introduce numerical algebraic geometry methods for computing lower bounds on the
reach, local feature size, and weak feature size of the real part of an equidimensional and …

Consider the set of solutions to a system of polynomial equations in many variables. An
algebraic manifold is an open submanifold of such a set. We introduce a new method for …

In this paper we present an efficient algorithm to produce a provably dense sample of a
smooth compact affine variety. The procedure is partly based on computing bottlenecks of …

We review supervised learning and deep neural network design for learning membership on
algebraic varieties. We demonstrate that these trained artificial neural networks can predict …

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In recent years, persistent homology (PH) has been successfully applied to real-world data
in many different settings. Despite significant computational advances, PH algorithms do not …