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This book gives an overview of research in the topology and geometry of intersections of quadrics in $mathbb{R}^n$, with a focus on intersections of concentric ellipsoids and related spaces. Unifying and organizing material previously spread over many articles, it also contains new results.
The first part provides very detailed foundations of a wide-ranging theory that could be useful for future developments. It includes chapters on general intersections of quadrics, operations on them, and intersections of concentric and coaxial quadrics. Moving from the general to the specific, the second part focuses on a topological description of transverse intersections of concentric ellipsoids, including a complete description of the case of three ellipsoids, and of some large families of more than three of them. The third part looks at relations to other areas of mathematics such as dynamical systems, complex geometry, contact and symplectic geometry, andother applications. An appendix gathers some technical items and also gives an account of the origins, motivations and progression of the subject, including historical recollections of the author, who has been central to its development.
List of contents
1 Introduction.- PART I: General Results.- 2 General intersections of quadrics.- 3 Intersections of coaxial quadrics.- 4 Intersections of coaxial ellipsoids.- PART II: Topological description of transverse intersections of concentric ellipsoids.- 5 Characterization of connected sums.- 6 Three coaxial ellipsoids.- 7 Three concentric ellipsoids.- 8 More than three coaxial ellipsoids.- 9 A family of surfaces that are intersections of concentric, non-coaxial, ellipsoid.- PART III: Relations with other areas of Mathematics.- 10 Dynamical systems.- 11 Complex Geometry.- 12 Contact and symplectic Geometry.- 13 Intersections with dihedral symmetry.- 14 Toric Topology and polyhedral products.- PART IV: Appendices.- 15 Appendix 1. Proof of Theorem 2.1.- 16 Appendix 2. Origins.- 17 Appendix 3. Diagonalizability of matrices.- 18 Appendix 4: Complements of products of spheres in spheres.
About the author
¿Santiago López de Medrano is a Research Professor in Mathematics at the Universidad Nacional Autónoma de México (UNAM). After completing his PhD at Princeton University in 1968, his thesis was published as the highly influential book Involutions on Manifolds (Springer-Verlag, 1971). After Princeton, he returned to UNAM, where he has been ever since. He was President of the Mexican Mathematical Society (1969-1973) and has been an invited researcher and speaker at numerous international institutes and conferences. His research is primarily in differential topology, singularity theory, dynamical systems, mathematical biology and their interaction. Recently his focus has been on the topology of intersections of ellipsoids in $\mathbb{R}^n$, and its applications to dynamical systems and geometry. He is the author of over 60 publications. In addition to mathematical research, he has also been interested in the improvement of the teaching of mathematics from high schoolto graduate university level.
Report
"The author often invites us to share his process of discovery or sometimes rediscovery of this very concrete material which reaches so widely into different areas of mathematics. The book is full of detail which a short summary ... . There are many illustrations and a short index." (Peter Giblin, Mathematical Reviews, November, 2024)
"The book is structured in three parts. ... All three parts are of equal importance for understanding the subject. The book is nicely illustrated by figures which help the understanding." (Ivailo M. Mladenov, zbMATH 1534.53001, 2024)
