In this paper we study the properties of the homology of different geometric filtered complexes (such as Vietoris–Rips, Čech and witness complexes) built on top of totally bounded metric spaces. ...Using recent developments in the theory of topological persistence, we provide simple and natural proofs of the stability of the persistent homology of such complexes with respect to the Gromov–Hausdorff distance. We also exhibit a few noteworthy properties of the homology of the Rips and Čech complexes built on top of compact spaces.
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This book provides a comprehensive introduction into the classical topic of projective geometry. It explains how metric concepts may be best understood in projective terms and explores the beauty of ...the interplay of geometry, algebra and combinatorics.
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This volume corresponds to the Banff International Research Station Workshop on Randomization, Relaxation, and Complexity, held from February 28-March 5, 2010 in Banff, Ontario, Canada. This volume ...contains a sample of advanced algorithmic techniques underpinning the solution of systems of polynomial equations. The papers are written by leading experts in algorithmic algebraic geometry and touch upon core topics such as homotopy methods for approximating complex solutions, robust floating point methods for clusters of roots, and speed-ups for counting real solutions. Vital related topics such as circuit complexity, random polynomials over local fields, tropical geometry, and the theory of fewnomials, amoebae, and coamoebae are treated as well. Recent advances on Smale's 17th Problem, which deals with numerical algorithms that approximate a single complex solution in average-case polynomial time, are also surveyed.
This volume contains the proceedings of the Barcelona-Boston-Tokyo Number Theory Seminar, which was held in memory of Fumiyuki Momose, a distinguished number theorist from Chuo University in ...Tokyo.Momose, who was a student of Yasutaka Ihara, made important contributions to the theory of Galois representations attached to modular forms, rational points on elliptic and modular curves, modularity of some families of Abelian varieties, and applications of arithmetic geometry to cryptography. Papers contained in this volume cover these general themes in addition to discussing Momose's contributions as well as recent work and new results.
We introduce the heat method for computing the geodesic distance to a specified subset (e.g., point or curve) of a given domain. The heat method is robust, efficient, and simple to implement since it ...is based on solving a pair of standard linear elliptic problems. The resulting systems can be prefactored once and subsequently solved in near-linear time. In practice, distance is updated an order of magnitude faster than with state-of-the-art methods, while maintaining a comparable level of accuracy. The method requires only standard differential operators and can hence be applied on a wide variety of domains (grids, triangle meshes, point clouds, etc.). We provide numerical evidence that the method converges to the exact distance in the limit of refinement; we also explore smoothed approximations of distance suitable for applications where greater regularity is required.
This volume contains the proceedings of the conference Local and Global Methods in Algebraic Geometry, held from May 12-15, 2016, at the University of Illinois at Chicago, in honor of Lawrence Ein's ...60th birthday.The articles cover a broad range of topics in algebraic geometry and related fields, including birational geometry and moduli theory, analytic and positive characteristic methods, geometry of surfaces, singularity theory, hyper-Kahler geometry, rational points, and rational curves.
This book is a collection of more than 500 attractive open problems in the field. The largely self-contained chapters provide a broad overview of discrete geometry, along with historical details and ...the most important partial results related to these problems.
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We prove a theorem of Hadamard–Stoker type: a connected locally convex complete hypersurface immersed in H n × R ( n ≥ 2 ), where H n is n-dimensional hyperbolic space, is embedded and homeomorphic ...either to the n-sphere or to R n . In the latter case it is either a vertical graph over a convex domain in H n or has what we call a simple end.
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