Metric Algebraic Geometry

Preface
About the Authors
Acknowledgements
Contents
Chapter 1 Historical Snapshot
	1.1 Polars
	1.2 Foci
	1.3 Envelopes
Chapter 2 Critical Equations
	2.1 Euclidean Distance Degree
	2.2 Low-Rank Matrix Approximation
	2.3 Invitation to Polar Degrees
Chapter 3 Computations
	3.1 Gröbner Bases
	3.2 The Parameter Continuation Theorem
	3.3 Polynomial Homotopy Continuation
Chapter 4 Polar Degrees
	4.1 Polar Varieties
	4.2 Projective Duality
	4.3 Chern Classes
Chapter 5 Wasserstein Distance
	5.1 Polyhedral Norms
	5.2 Optimal Transport and Independence Models
	5.3 Wasserstein meets Segre–Veronese
Chapter 6 Curvature
	6.1 Plane Curves
	6.2 Algebraic Varieties
	6.3 Volumes of Tubular Neighborhoods
Chapter 7 Reach and Offset
	7.1 Medial Axis and Bottlenecks
	7.2 Offset Hypersurfaces
	7.3 Offset Discriminant
Chapter 8 Voronoi Cells
	8.1 Voronoi Basics
	8.2 Algebraic Boundaries
	8.3 Degree Formulas
	8.4 Voronoi meets Eckart–Young
Chapter 9 Condition Numbers
	9.1 Errors in Numerical Computations
	9.2 Matrix Inversion and Eckart–Young
	9.3 Condition Number Theorems
	9.4 Distance to the Discriminant
Chapter 10 Machine Learning
	10.1 Neural Networks
	10.2 Convolutional Networks
	10.3 Learning Varieties
Chapter 11 Maximum Likelihood
	11.1 Kullback–Leibler Divergence
	11.2 Maximum Likelihood Degree
	11.3 Scattering Equations
	11.4 Gaussian Models
Chapter 12 Tensors
	12.1 Tensors and their Rank
	12.2 Eigenvectors and Singular Vectors
	12.3 Volumes of Rank-One Varieties
Chapter 13 Computer Vision
	13.1 Multiview Varieties
	13.2 Grassmann Tensors
	13.3 3D Reconstruction from Unknown Cameras
Chapter 14 Volumes of Semialgebraic Sets
	14.1 Calculus and Beyond
	14.2 D-Modules
	14.3 SDP Hierarchies
Chapter 15 Sampling
	15.1 Homology from Finite Samples
	15.2 Sampling with Density Guarantees
	15.3 Markov Chains on Varieties
	15.4 Chow goes to Monte Carlo
References