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55469-01 - Vorlesung: Metric geometry: with a view toward Riemannian manifolds 6 KP

Semester Herbstsemester 2019
Angebotsmuster unregelmässig
Dozierende Egor Yasinsky (yahor.yasinsky@unibas.ch, BeurteilerIn)
Inhalt What is geometry? People's understanding of this concept has been changing over centuries. The standard geometry most of you learned in school is the Euclidian Geometry. As you probably remember, Euclid develops plane geometry from some basic definitions and five postulates, one of which (the fifth) states that through a point outside a given line there is one and only one line parallel to the given line.

Although Euclid’s books have been a standard treatise of elementary geometry for more than two thousand years, the fifth postulate was criticized almost from the very beginning. It was believed that this postulate can be deduced from the others (and in fact, most of plane geometry indeed can be proved without it). Only in 19th century it was discovered that the fifth postulate was completely independent of the other ones. Namely, in the works of Gauss, Lobachevsky and Bolyai it was shown that the denial of the fifth postulate leads to a new consistent geometry (called non-Euclidean geometry by Gauss). The first part of our course will be dedicated to some basic properties of such non-Euclidean geometries (we will see that there are infinitely many of them).

But what is the difference between various non-Euclidean geometries, say, spherical and hyperbolic ones? One of possible answers is that the corresponding spaces have different curvature. This is a central notion of differential geometry, whose applications range from Theoretical Physics to Computer Science. Intuitively speaking, you understand that a straight line has a “lower curvature” than an arc (like a “c” letter). However, to formalize this observation (and to generalize it e.g. to surfaces) requires some work and a solid knowledge of analysis.

Luckily, it is now understood that a vast part of differential geometry belongs to the world of metric geometry, i.e. geometry of abstract sets with a distance function. In particular, one can speak about the curvature of such spaces (Alexandrov curvature). This circle of ideas turned out to be so powerful that it basically led to the emergence of new areas of mathematics, e.g. geometric group theory, or to Perelman’s proof of the Poincaré conjecture. On the other hand, methods of metric geometry turned out to be very useful in applied mathematics, Data Science, Statistics and Probability. In this course, we will give an introduction to geometry of metric spaces, and will try to see what happened in Geometry since Euclid.

Tentative list of topics:

1. Basic concepts: metric spaces, geodesic paths, Alexandrov angles.
2. Non-Euclidean geometries: spherical, hyperbolic. Models of hyperbolic space (Poincaré disk and half-space models, hyperboloid model). Alexandrov curvature.
3. Klein's vision of geometry: Das Erlanger Programm. Transformation groups. Isometries in Euclidean, spherical and hyperbolic geometries. Homogeneous spaces.
4. Length spaces, inner metric. Topology of metric spaces. Complete spaces, compactness. The Hopf-Rinow theorem.

If time permits, some of the following topics may be included: projective geometry (homogeneous coordinates, projective transformations, projective duality, Cayley's principle: "projective geometry is all geometry"); more about Spaces of Bounded Curvature; Riemannian length structures (case of 2-dimensional regions); hyperbolic groups in the sense of Gromov.
Literatur 1. Metric Spaces of Non-Positive Curvature, Martin R. Bridson, André Häfliger.
2. A Course in Metric Geometry, Dmitri Burago, Yuri Burago, Sergei Ivanov (available online).
3. Foundations of Hyperbolic Manifolds, John Ratcliffe.
4. Geometries, A. B. Sossinsky.


Teilnahmebedingungen Basic knowledge of analysis and linear algebra. We shall NOT use advanced calculus or algebra, so the second year students are very welcome.
Unterrichtssprache Englisch
Einsatz digitaler Medien kein spezifischer Einsatz


Intervall Wochentag Zeit Raum

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Module Modul: Algebra und Zahlentheorie (Bachelorstudium: Mathematik)
Modul: Vertiefung Mathematik (Bachelorstudium: Computational Sciences (Studienbeginn vor 01.08.2018))
Leistungsüberprüfung Lehrveranst.-begleitend
An-/Abmeldung zur Leistungsüberprüfung Anm.: Belegen Lehrveranstaltung; Abm.: stornieren
Wiederholungsprüfung keine Wiederholungsprüfung
Skala 1-6 0,5
Wiederholtes Belegen beliebig wiederholbar
Zuständige Fakultät Philosophisch-Naturwissenschaftliche Fakultät, studiendekanat-philnat@unibas.ch
Anbietende Organisationseinheit Fachbereich Mathematik