Metric Algebraic Geometry Maddie Weinstein University of California Berkeley [email protected] 1 / 24 What is metric algebraic geometry? This is the title of my PhD dissertation. Metric algebraic geometry is concerned with properties of real algebraic varieties that depend on a distance metric. The distance metric can be the Euclidean metric in the ambient space or a metric intrinsic to the variety. Results can be applied in the computational study of the geometry of data with nonlinear models. 2 / 24 Application: Computational Study of Geometry of Data with Nonlinear Models n Suppose we are given of sample of points in R and we choose to model them with an algebraic variety V . What can we learn about the variety? In Learning Algebraic Varieties from Samples, joint with Paul Breiding, Sara Kalisnik, Bernd Sturmfels, we survey, develop, and implement methods to study the following properties: What is the dimension of V ? What equations vanish on V ? What is the degree of V ? What are the irreducible components of V ? What are the homology groups of V ? 3 / 24 Topological Data Analysis Figure: Persistent homology barcodes for the quartic Trott curve in R2. See Topology and Data by Gunnar Carlsson for more information. 4 / 24 Algebraicity of Persistent Homology In Offset Hypersurfaces and Persistent Homology of Algebraic Varieties, joint with Emil Horobet, we study the persistent homology of the offset filtration of an algebraic variety, bringing the perspective of real algebraic geometry to the study of persistent homology and showing that barcodes can be computed exactly. Figure: The quartic Viviani curve (left) and its = 1 offset surface (right), which is of degree 10. Theorem (Horobet-W. '18) Let f1;:::; fs be polynomials in Q[x1;:::; xn] with XR = VR(f1;:::; fs ). Then the values of the persistence parameter at which a bar in the offset filtration barcode appears or disappears are real numbers algebraic over Q. 5 / 24 Guaranteeing Persistent Homology: The Reach of an Algebraic Variety Theorem (Niyogi, Smale, Weinberger 2006) N Let M be a compact submanifold of R of dimension k with reach τ. Let x¯ = fx1;:::; xng be a set of n points drawn in i.i.d fashion according to 1 [ the uniform probability measure on M. Let 0 < < 2τ . Let U = B(x) x2x¯ N vol(M) be a corresponding random open subset of R . Let β1 = k k (cos (θ1))vol(B/4) vol(M) τ τ and β2 = k k where θ1 = arcsin( 8 ) and θ2 = arcsin( 16 ). (cos (θ2))vol(B/8) Then for all 1 n > β log(β ) + log 1 2 δ the homology of U equals the homology of M with high confidence (probability > 1 − δ). 6 / 24 The Reach of an Algebraic Variety Definition n The medial axis of a variety V ⊂ R is the set Med(V ) of all points n u 2 R such that the minimum distance from V to u is attained by two distinct points. The reach τV is the infimum of all distances from points on the variety V to points in its medial axis Med(V ). Figure: The medial axis of the quartic butterfly curve can be seen in its Voronoi approximation. 7 / 24 Algebraicity of Reach Proposition (Horobet-W. '18) n Let V be a smooth algebraic variety in R . Let f1;:::; fs 2 Q[x1;:::; xn] with V = VR(f1;:::; fs ). Then the reach of V is an algebraic number over Q. 8 / 24 Reach, Bottlenecks, and Curvature Figure: The reach of a manifold is attained by a bottleneck, two points on a circular arc, or a point of maximal curvature. Figure and Theorem due to Aamari-Kim-Chazal-Michel-Rinaldo-Wasserman '17. 9 / 24 Bottlenecks Definition n Let V ⊂ R be a smooth variety. A line is orthogonal to V if it is n orthogonal to the tangent space Tx V ⊂ R at x. The bottlenecks of V are pairs (x; y) of distinct points x; y 2 V such that the line spanned by x and y is orthogonal to V at both points. Figure: The real bottleneck pairs of the quartic butterfly curve. 10 / 24 Bottlenecks as Distance Optimization Remark Bottlenecks are the critical points of the squared distance function n n 2 R × R :(x; y) 7! jjx − yjj ; subject to the constraints x; y 2 V as well as the non-triviality condition x 6= y. 11 / 24 Bottleneck Degree n Denote by BND(V ) the bottleneck degree of V ⊂ C . Under certain conditions, this coincides with twice the number of bottleneck pairs. Theorem (Di Rocco-Eklund-W. '19) 2 Let V ⊂ C be a \general" curve of degree d. Then BND(V ) = d4 − 5d2 + 4d. 3 Let V ⊂ C be a \general" surface of degree d. Then BND(V ) = d6 − 2d5 + 3d4 − 15d3 + 26d2 − 13d. For any smooth variety V ⊂ n in \general position," we have an PC algorithm to express the bottleneck degree in terms of the polar classes of V . 12 / 24 Polar Classes An m-dimensional variety has m + 1 polar varieties defined by exceptional tangent loci. x V Tx V p Ty V y Figure: Polar locus of a point p with respect to an ellipse V : P1(V ; p) = fx; yg. 13 / 24 Polar Classes Example 3 3 For a smooth surface V ⊂ P we have two polar varieties. Let p 2 P be a 3 general point and l ⊂ P a general line. Then P1(V ; p) is the set of points 3 x 2 V such that the projective tangent plane Tx V ⊂ P contains p. This is a curve on V . Similarly, P2(V ; l) = fx 2 V : l ⊆ Tx V g, which is finite. Definition n Let V ⊂ P be a smooth variety of dimension m. For j = 0;:::; m and a n general linear space L ⊆ P of dimension n − m − 2 + j the polar variety is given by Pj (V ; L) = fx 2 V : dim Tx V \ L ≥ j − 1g: For each polar variety Pj (V ; L), there is a corresponding polar class [Pj (V ; L)] = pj which represents Pj (V ; L) up to rational equivalence. 14 / 24 Voronoi Diagrams of Point Clouds Figure: A Voronoi diagram of the bookstores in Berkeley. Created using program by Rodion Chachura. 15 / 24 Voronoi Cells of Varieties Definition n Let V be a real algebraic variety of codimension c in R and y a smooth point on V . Its Voronoi cell consists of all points whose closest point in V is y, i.e. n 2 VorV (y) := u 2 R : y 2 arg min kx − uk : x2V The Voronoi cell VorV (y) is a convex semialgebraic set of dimension c, living in n the normal space NV (y) to V at y. Its boundary consists of the points in R that have at least two closest points in V , including y. Figure: A quartic space curve and the Voronoi cell in one of its normal planes. 16 / 24 Voronoi Degrees Remark When discussing degree, we identify the variety V and δalg VorV (y) with their Zariski closures in n . PC Definition The algebraic boundary of the Voronoi cell VorV (y) is a hypersurface in the normal space to V at y. Its degree δV (y) is called the Voronoi degree of V at y. 17 / 24 Voronoi Degrees Theorem (Cifuentes-Ranestad-Sturmfels-W. '18) n Let V ⊂ P be a curve of degree d and geometric genus g with at most ordinary multiple points as singularities. The Voronoi degree at a general point y 2 V equals δV (y) = 4d + 2g − 6; n provided V is in general position in P . n Let V ⊂ P be a smooth surface of degree d. Then its Voronoi degree equals δV (y) = 3d + χ(V ) + 4g(V ) − 11; n provided the surface V is in general position in P and y is a general point on V , where χ(V ) := c2(V ) is the topological Euler characteristic and g(V ) is the genus of the curve obtained by n intersecting V with a general smooth quadratic hypersurface in P . 18 / 24 Application: Low Rank Approximation on the Space of Symmetric Matrices Remark The Frobenius and Euclidean norms have dramatically different properties with respect to low rank approximation of symmetric matrices. The (n+1) Eckart-Young Theorem is valid for the Frobenius norm on R 2 but not for the Euclidean norm. Figure: The Voronoi cell of a symmetric 3×3 matrix of rank 1 is a convex body of dimension 3. It is shown for the Frobenius norm (left) and for the Euclidean norm (right). 19 / 24 Voronoi Convergence Theorem (Brandt-W. '19) As sampling density increases, the Voronoi diagrams of a point sample of a variety \converge" to the Voronoi decomposition of the variety. Figure: Voronoi cells of 101, 441, and 1179 points sampled from the quartic butterfly curve: x 4 − x 2y 2 + y 4 − 4x 2 − 2y 2 − x − 4y + 1 = 0. 20 / 24 Curvature and the Evolute Figure: The eleven real points of critical curvature on the butterfly curve (purple) joined by green line segments to their centers of curvature. These give cusps on the evolute (light blue). 21 / 24 Degree of Critical Curvature Theorem (Brandt-W. '19) 2 Let V ⊂ R be a smooth, irreducible curve of degree d ≥ 3. Then the degree of critical curvature of V is 6d2 − 10d. 22 / 24 Current and Future Work Algebraic study of curvature Degree of critical curvature for surfaces and beyond Degree of the reach of an algebraic variety Algorithms to compute the reach Applying the perspective of metric algebraic geometry to differential geometry 23 / 24 Thank you! 24 / 24 Polar Classes and Chern Classes Pj (V ; L) is either empty or of pure codimension j and j X m − i + 1 p = (−1)i hj−i c (T ); j j − i i X i=0 where h 2 An−1(X ) is the hyperplane class.