Metric Algebraic Geometry by Madeleine Aster Weinstein A dissertation submitted in partial satisfaction of the requirements for the degree of Doctor of Philosophy in Mathematics in the Graduate Division of the University of California, Berkeley Committee in charge: Professor Bernd Sturmfels, Chair Associate Professor Adityanand Guntuboyina Professor David Nadler Spring 2021 Metric Algebraic Geometry Copyright 2021 by Madeleine Aster Weinstein 1 Abstract Metric Algebraic Geometry by Madeleine Aster Weinstein Doctor of Philosophy in Mathematics University of California, Berkeley Professor Bernd Sturmfels, Chair Algebraic geometry is the study of algebraic varieties, zero sets of systems of polynomial equa- tions. Metric algebraic geometry concerns properties of real algebraic varieties that depend on a distance metric. Applications are seen in distance optimization and the geometry of data. Algebraic geometry provides a useful perspective on distance optimization. We study variations n of the nearest point problem, which is stated as follows: given a subset S ⊂ R and point p 2= S, find a point in S of minimal distance to p. An inverse to the nearest point problem can be stated as n follows: Suppose now that p 2 S. Describe the subset of R consisting of points that are closer to p than to any other point of S. This subset is called the Voronoi cell of p with respect to S. We study its algebraic boundary. Voronoi cells enable us to create algorithms to approximate several metric features of varieties S. Bottlenecks are pairs of points on an algebraic variety that are critical points of the distance function between pairs of distinct points on the variety. We study the bottleneck variety consisting of such points and prove a formula for its degree. Algebraic geometry informs the computational study of data. We study the algebraic geometry of the offset hypersurface, the locus of all points at some fixed distance from a given variety. The offset hypersurface allows us to prove the algebraicity of two quantities central to the computation of persistent homology, a method at the heart of topological data analysis. Conversely, numerical and symbolic computational methods yield insight in the analysis of algebraic varieties. We use numerical methods to show that the degree of the Zariski closure of the orbit of a general cubic surface under the action of the projective linear group is 96120. We pair representation theory and numerical algebraic geometry to investigate the real algebraic variety of real symmetric matrices with eigenvalue multiplicities specified by a given partition. Taken together, the results show the power of combining algebraic geometry and numerical meth- ods to produce insights for problems related to distance and metrics. i To Pumpkin p ii Contents Contents ii 1 Introduction1 1.1 Euclidean Distance Degree . .2 1.2 Intersection Theory . .2 1.3 Numerical Algebraic Geometry . .6 1.4 Persistent Homology . .9 1.5 Contributions in this Dissertation . 11 2 Optimizing Distances with Algebraic Varieties 14 2.1 Voronoi Cells . 14 2.2 Bottlenecks . 31 3 Algebraic Geometry of Curvature 56 3.1 Curvature and the Evolute of a Plane Curve . 56 3.2 Algebraic Geometry of Curvature . 59 4 Geometry of Data 68 4.1 Modeling Point Clouds with Varieties . 68 4.2 Persistent Homology with the Offset Filtration . 108 4.3 Voronoi Cells in Metric Algebraic Geometry of Plane Curves . 124 5 Computational Algebraic Geometry 144 5.1 Using Numerical Algebraic Geometry to Compute Degrees . 144 5.2 Real Symmetric Matrices with Partitioned Eigenvalues . 148 Bibliography 155 iii Acknowledgments Many people have supported me throughout my time in graduate school. I would like to thank my academic family for giving me a research community. At every step of graduate school, I have relied on the example and support of my academic sister Maddie Brandt. Thank you to Bernd Sturmfels for inspiring and cultivating a mathematical community that nurtures young researchers. Thank you to my coauthors: Maddie Brandt, Paul Breiding, Laura Brustenga i Moncusí, Diego Ci- fuentes, Sandra Di Rocco, David Eklund, Emil Horobe¸t,Sara Kalisnik, Kristian Ranestad, Bernd Sturmfels, and Sascha Timme. In particular, I am grateful to my senior coauthors Paul Breiding, David Eklund, and Kristian Ranestad for being so generous with their time as they educated me in their areas of expertise. I am grateful to my peer coauthors for bringing fun and energy to our collaborations. Thank you to the members of the Noetherian Ring, who go above and beyond any reasonable requirements of their jobs to improve the climate and culture of the mathematics com- munity and work towards equity and inclusion. I have been particularly inspired and uplifted by the efforts of Maddie Brandt and Mariel Supina. I am also deeply grateful to the woman mathe- maticians who paved the way for our cohort. Thank you to Judie Filomeo, Vicky Lee, and Marsha Snow for enabling the math department at UC Berkeley to function, and Saskia Gutzschebauch for doing the same at MPI Leipzig. Thank you to the math department at Harvey Mudd College, and in particular Dagan Karp and Talithia Williams, for guiding me on my journey from calculus student to graduate student. Thank you to the AAAS Ambassador community for amazing me every day. Thank you to my roommates Esme, Maddie, and Mariel for helping turn Berkeley into my home. Thank you to my family for supporting me in all of my pursuits. Thank you to Hailey, Kathryn, Kayla, and Shifrah for holding the door open for me to the world outside of mathematics. Thank you to my canine companions: Dolce Galileo, Fido, William, Chenille, Bean, and Pumpkin. No matter how short our time together, each of you has a special place in my heart. 1 Chapter 1 Introduction Metric algebraic geometry arose out of a desire to bring the perspective and tools of algebraic geometry to the objects of differential geometry that arise in applications concerning the geometry of data. The necessity of bringing together such tools can be understood through the story of the reach of an algebraic variety. n First introduced by Federer in 1959, the reach is an invariant of a subset S ⊂ R that charac- terizes the difficulty of performing calculations in computational geometry [97]. The reach of S is n the maximum distance from S such that every point of R within this distance has a unique nearest point in S. Sets with nonzero reach are close to being differential. In Section 4.2, we prove that the reach of an algebraic variety is a number algebraic over the field of definition of the variety. With this knowledge comes the hope of finding an algebraic char- acterization of the reach; that is, given an algebraic variety, can we find equations for the subvariety of points critical to the computation of the reach? It can be shown that the reach of a variety is the minimum of two quantities: the minimal radius of curvature of a geodesic of the variety and half of the narrowest bottleneck distance on the variety. Thus, we set out to provide algebraic charac- terizations of these two quantities. Finding an algebraic characterization of curvature is the subject of Chapter3. To obtain an algebraic characterization of bottlenecks, as we do in Section 2.2, we follow the lead of the paper [86], which lays out a framework for finding critical points of distance optimization problems on algebraic varieties by defining the Euclidean distance degree. This is the subject of Section 1.1. Once we have algebraic characterizations of curvature and bottlenecks, we ask for the degree of the associated algebraic varieties. The degree is a measure of algebraic complexity, which is a proxy for computational difficulty. The field of intersection theory provides methods for finding the degree of an algebraic variety. This is the topic of Section 1.2. Having found equations for these varieties and characterized the difficulty of computations, we wish to perform these computations and find specific points on given algebraic varieties. For this, we turn to the field of numerical algebraic geometry, the topic of Section 1.3. Reach exemplifies the spirit of metric algebraic geometry because it plays an important role in the geometry of data analysis. It determines the sampling density required for the method of per- sistent homology to successfully characterize the topology of a manifold. We provide background CHAPTER 1. INTRODUCTION 2 information on persistent homology in Section 1.4. 1.1 Euclidean Distance Degree n n Let X ⊂ R be a variety and u 2 R . The nearest point problem asks what point or points of X are closest to u. Algebraic conditions are unable to distinguish between types of critical points, so we reformulate the problem as follows. Define a function du(x) : X R by 2 2 du(x) = (x1 − u1) + ··· + (xn − un) : Then du(x) is the square of the Euclidean distance from u to x. We seek to find the critical points of the function du. We now describe the construction of the ideal whose variety consists of these critical points. We follow [86]. n Let IX = h f1;:::; fsi ⊂ R[x1;:::;xn] and X = V(IX ) ⊂ C . We denote by J( f ) the s×n Jacobian matrix whose entry in row i and column j is the partial derivative ¶ fi=¶x j. Let c be the codimension of X. The singular locus Xsing of X is defined by IXsing = IX + hc × c − minors of J( f )i: We exclude the singular locus of X from our analysis. To characterize the idea of criticality, we use Lagrange multipliers. Lagrange multipliers is a method for finding optimal values of a function subject to constraints. Here, we wish to optimize du given that x 2 X n Xsing.