ADVANCED GEOMETRY A
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Geometry 1, geometry 2. It may be useful, but not essential, having attended either the course Fundamentals of Advanced Geometry, during the bachelor's degree, or Topics in Advanced Geometry, during the master's degree in mathematics
Oral exam, during which the development of some concrete example will be requested, among those treated during the lectures, and the acquired theoretical skills will be tested. More precisely some topics will be assigned to the students to be exposed in a seminar, which will involve most of the theory developed during the course and also the development of detailed proofs by the students.
EDUCATIONAL OBJECTIVES The course aims to provide students with a suitable introduction to hyperbolic geometry, a significant example of a geometric theory of historical importance and still of great research interest, with numerous connections to other mathematical and physical theories. By the end of the course, students should be able to navigate the existing literature on some developments and applications of hyperbolic geometry.
I. Fundamentals of Hyperbolic Geometry * Models of Hyperbolic Space: * The Hyperbolic Plane (H2): Upper half-plane model, Poincaré disk model, Klein model. Metrics, geodesics, angles, isometries. * Higher-Dimensional Hyperbolic Space (Hn): Introduction to the hyperboloid model and other models. * Geometric Properties: * Triangles in Hyperbolic Space: Angle defect, area formula. * Horocycles and horospheres. * The group of isometries of Hn. * Hyperbolic volume. * Hyperbolic manifolds: * Definition and examples (quotients of Hn by discrete groups of isometries). * Fundamental domains and Dirichlet domains. * Compact and non-compact hyperbolic manifolds. II. Fuchsian Groups and Riemann Surfaces (16 hours) * Discrete Groups of Isometries of H₂ (Fuchsian Groups): * Definition and examples. * Relationship to Riemann surfaces via the uniformization theorem (statement and implications). * Teichmüller space (introduction to the concept of moduli of Riemann surfaces). * The Weil-Petersson metric on Teichmüller space. III. Billiards in Hyperbolic Domains * Basic concepts: trajectories, reflections, periodic orbits. * Comparison with Euclidean billiards. * Billiards in Hyperbolic Polygons: * Geodesic flow on hyperbolic surfaces. * Ergodicity and mixing properties of the geodesic flow on hyperbolic manifolds (brief overview). * Connections to interval exchange transformations and the Teichmüller flow. IV. Connections to the Holographic Principle * Introduction to the Holographic Principle: * AdS/CFT Correspondence (Conceptual Overview): Relevance of Hyperbolic Geometry:
I. Fundamentals of Hyperbolic Geometry * Models of Hyperbolic Space: * The Hyperbolic Plane (H2): Upper half-plane model, Poincaré disk model, Klein model. Metrics, geodesics, angles, isometries. * Higher-Dimensional Hyperbolic Space (Hn): Introduction to the hyperboloid model and other models. * Geometric Properties: * Triangles in Hyperbolic Space: Angle defect, area formula. * Horocycles and horospheres. * The group of isometries of Hn. * Hyperbolic volume. * Hyperbolic manifolds: * Definition and examples (quotients of Hn by discrete groups of isometries). * Fundamental domains and Dirichlet domains. * Compact and non-compact hyperbolic manifolds. II. Fuchsian Groups and Riemann Surfaces (16 hours) * Discrete Groups of Isometries of H₂ (Fuchsian Groups): * Definition and examples. * Relationship to Riemann surfaces via the uniformization theorem (statement and implications). * Teichmüller space (introduction to the concept of moduli of Riemann surfaces). * The Weil-Petersson metric on Teichmüller space. III. Billiards in Hyperbolic Domains * Basic concepts: trajectories, reflections, periodic orbits. * Comparison with Euclidean billiards. * Billiards in Hyperbolic Polygons: * Geodesic flow on hyperbolic surfaces. * Ergodicity and mixing properties of the geodesic flow on hyperbolic manifolds (brief overview). * Connections to interval exchange transformations and the Teichmüller flow. IV. Connections to the Holographic Principle * Introduction to the Holographic Principle: * AdS/CFT Correspondence (Conceptual Overview): Relevance of Hyperbolic Geometry
Lectures, 64 hours, which include sessions dedicated to the explicit calculation of examples. During the course, weekly notes will be made available on e-learning.
email: riccardo.re AT uninsubria.it Contact the teacher via email for any clarification
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Degree course in: MATHEMATICS