2025/2026
Геометрия и динамика гомеоморфизмов поверхностей
Статус:
Дисциплина общефакультетского пула
Кто читает:
Факультет математики
Где читается:
Факультет математики
Когда читается:
1, 2 модуль
Охват аудитории:
для всех кампусов НИУ ВШЭ
Язык:
английский
Контактные часы:
60
Course Syllabus
Abstract
A self-homeomorphism of a surface is an object of TOPOLOGY. Considered up to homotopy, it gives rise to an element of the mapping class group, an important and powerful ALGEBRAIC invariant of a 2-manifold. It is determined by its action on a suitably chosen graph; hence it an object of COMBINATORICS. On the other hand, it can be iterated, and, for this reason, it is an object of DYNAMICAL SYSTEMS theory. Highly nontrivial and chaotic dynamics can arise from surface homeomorphisms. Finally, and most importantly, William Thurston in 1976 realized that a surface diffeomorphism endows a surface with an essentially unique GEOMETRY. This idea of Thurston has led to a classification of surface homeomorphisms that is satisfactory from both algebraic and dynamical viewpoints. Using the Nielsen-Thurston classification of surface homeomorphisms as a guiding line, we discuss relevant questions of surface topology, geometry, and dynamics.
Learning Objectives
- Knowing fundamental concepts and results in topology/geometry/dynamics of surfaces such as uniformization, hyperbolic metrics, change of coordinates principle, geometric intersection numbers, Dehn twists, mapping class groups, measured foliations, pseudo-Anosov maps, topological entropy, Teichmueller spaces, train tracks.
- Understanding principles of geometrization program as applied to classification of surface homomorphisms
- Being able to perform specific computations with the surface gluing schemes and the mapping class groups of surfaces.
Bibliography
Recommended Core Bibliography
- Itiro Tamura. (2018). Topology of Foliations: An Introduction. AMS.
- Scárdua, B., & Rojas, C. A. M. (2017). Geometry, Dynamics And Topology Of Foliations: A First Course. World Scientific Publishing Company.
Recommended Additional Bibliography
- Boissonnat, J.-D., & Teillaud, M. (2006). Effective Computational Geometry for Curves and Surfaces. Berlin: Springer. Retrieved from http://search.ebscohost.com/login.aspx?direct=true&site=eds-live&db=edsebk&AN=176637