Êtes-vous un étudiant de l'EPFL à la recherche d'un projet de semestre?
Travaillez avec nous sur des projets en science des données et en visualisation, et déployez votre projet sous forme d'application sur Graph Search.
Concept
Théorie géométrique des groupes
Résumé
Geometric group theory is an area in mathematics devoted to the study of finitely generated groups via exploring the connections between algebraic properties of such groups and topological and geometric properties of spaces on which these groups act (that is, when the groups in question are realized as geometric symmetries or continuous transformations of some spaces).
Another important idea in geometric group theory is to consider finitely generated groups themselves as geometric objects. This is usually done by studying the Cayley graphs of groups, which, in addition to the graph structure, are endowed with the structure of a metric space, given by the so-called word metric.
Geometric group theory, as a distinct area, is relatively new, and became a clearly identifiable branch of mathematics in the late 1980s and early 1990s. Geometric group theory closely interacts with low-dimensional topology, hyperbolic geometry, algebraic topology, computational group theory and differential geometry. There are also substantial connections with complexity theory, mathematical logic, the study of Lie groups and their discrete subgroups, dynamical systems, probability theory, K-theory, and other areas of mathematics.
In the introduction to his book Topics in Geometric Group Theory, Pierre de la Harpe wrote: "One of my personal beliefs is that fascination with symmetries and groups is one way of coping with frustrations of life's limitations: we like to recognize symmetries which allow us to recognize more than what we can see. In this sense the study of geometric group theory is a part of culture, and reminds me of several things that Georges de Rham practiced on many occasions, such as teaching mathematics, reciting Mallarmé, or greeting a friend".
Geometric group theory grew out of combinatorial group theory that largely studied properties of discrete groups via analyzing group presentations, that describe groups as quotients of free groups; this field was first systematically studied by Walther von Dyck, student of Felix Klein, in the early 1880s, while an early form is found in the 1856 icosian calculus of William Rowan Hamilton, where he studied the icosahedral symmetry group via the edge graph of the dodecahedron.
Source officielle
À propos de ce résultat
Cette page est générée automatiquement et peut contenir des informations qui ne sont pas correctes, complètes, à jour ou pertinentes par rapport à votre recherche. Il en va de même pour toutes les autres pages de ce site. Veillez à vérifier les informations auprès des sources officielles de l'EPFL.
Proximité ontologique
Cours associés (21)
MSE-651: Crystallography of structural phase transformations
The microstructure of many alloys and ceramics are constituted of very fine intricate domains (variants) created by diffusive or displacive phase transformations. The course introduces the crystallogr
MATH-535: Algebraic geometry III - selected topics
This course is aimed to give students an introduction to the theory of algebraic curves, with an emphasis on the interplay between the arithmetic and the geometry of global fields. One of the principl
MATH-731: Topics in geometric analysis I
The subject deals with differential geometry and its relation to global analysis, partial differential equations, geometric measure theory and variational principles to name a few.
Séances de cours associées (223)
Éléments unipotents
Explore les homomorphismes réguliers et les éléments unipotents dans la théorie des groupes.
Construction d'Alpha et de Beta
Couvre la construction de l'alpha et de la bêta en théorie de groupe, mettant l'accent sur la commutativité et les transformations naturelles.
Théorie des groupes combinatoires
Introduit la théorie combinatoire des groupes, en mettant l'accent sur les présentations de groupe, les sous-groupes normaux et les quotients de groupe.
Publications associées (44)