Vladimir Boltyansky est un mathématicien soviétique et russe reconnu pour ses contributions à la vulgarisation des mathématiques. Son domaine d'expertise couvre la topologie et la géométrie combinatoire, avec un intérêt marqué pour le troisième problème de Hilbert. Il est l'auteur de livres et d'articles mathématiques populaires qui rendent les concepts complexes accessibles à un large public. L'œuvre de Boltyansky se distingue par son approche de l'éducation mathématique et de la communication scientifique.
Topology is a relatively young and very important branch of mathematics, which studies the properties of objects that are preserved through deformations, twistings, and stretchings. This book deals with the topology of curves and surfaces as well as with the fundamental concepts of homotopy and homology, and does this in a lively and well-motivated way. This book is well suited for readers who are interested in finding out what topology is all about.
The book covers a comprehensive exploration of convexity and its various aspects across multiple dimensions. It begins with an in-depth analysis of convex sets, faces, supporting hyperplanes, and the concept of polarity, leading to discussions on the lower semicontinuity of the exponential operator and convex cones. The Farkas Lemma and its generalizations are examined, along with separable systems of convex cones.
The focus then shifts to d-convexity in normed spaces, defining d-convex sets and their support properties, and exploring the separability and Helly dimension of these sets. The section on H-convexity introduces the functional md for vector systems, the ?-displacement Theorem, and the continuity properties of H-convex sets, along with applications and connections to d-convexity.
The Szökefalvi-Nagy Problem is addressed, detailing the theorem and its generalizations, as well as descriptions of vector systems and compact convex bodies. Borsuk’s partition problem is formulated and surveyed, particularly in relation to bodies of constant width in various spaces.
The book also delves into homothetic covering and illumination, discussing key problems and results, including inner illumination of convex bodies. It concludes with a focus on the combinatorial geometry of belt bodies, presenting integral representations, definitions, and solutions to relevant problems. The work is rounded off with an author index and a list
Inhaltsverzeichnis1. Topologie der Kurven.1.1. Der Begriff der Stetigkeit.1.2. Womit beschäftigt sich die Topologie?.1.3. Einfachste topologische Invarianten.1.4. Die Eulersche Charakteristik eines Graphen.1.5. Schnittindex.1.6. Der Jordansche Kurvensatz.1.7. Was ist eine Kurve?.1.8. Peanokurven.2. Die Topologie der Flächen.2.1. Der Satz von Euler.2.2. Flächen.2.3. Die Eulersche Charakteristik der Fläche.2.4. Klassifizierung der geschlossenen orientierbaren Flächen.2.5. Klassifizierung der geschlossenen nichtorientierbaren Flächen.2.6. Vektorfelder auf Flächen.2.7. Das Vierfarbenproblem.2.8. Färbung von Karten auf Flächen.2.9. Wilde Sphären.2.10. Knoten.2.11. Verschlingungszahlen.3. Homotopie und Homologie.3.1. Perioden mehrdeutiger Funktionen.3.2. Die Fundamentalgruppe.3.3. Zellenzerlegungen und Polyeder.3.4. Überlagerungen.3.5. Der Abbildungsgrad und der Fundamentalsatz der Algebra.3.6. Knotengruppen.3.7. Zyklen und Homologie.3.8. Topologische Produkte.3.9. Faserbündel.3.10. Morse-Theorie.Anhang. Topologische Objekte in nematischen Flüssigkristallen.1. Nematik.2. Disklination in der Nematik.3. Disklination und Topologie.4. Singuläre Punkte.5. Was gibt es noch?.Literatur.Namen- und Sachverzeichnis.
