Conformal geometry and related topics
Informacje ogólne
Kod przedmiotu: | 1100-CGRT |
Kod Erasmus / ISCED: | (brak danych) / (brak danych) |
Nazwa przedmiotu: | Conformal geometry and related topics |
Jednostka: | Wydział Fizyki |
Grupy: |
Physics (Studies in English), 2nd cycle; courses from list "Topics in Contemporary Physics" Physics (Studies in English); 2nd cycle |
Punkty ECTS i inne: |
(brak)
|
Język prowadzenia: | angielski |
Kierunek podstawowy MISMaP: | fizyka |
Założenia (opisowo): | (tylko po angielsku) This course focuses on relatively advanced topics in differential geometry, and students are expected to have taken either a first course in differential geometry or a first course in general relativity. |
Tryb prowadzenia: | w sali |
Pełny opis: |
(tylko po angielsku) Conformal geometry is concerned with the study of spaces or spacetimes equipped with a structure that allows one to measure angles, but not arc lengths. Many differential equations in mathematical relativity, notably those connected to massless particles, are conformally invariant. In fact, a number of powerful theorems in mathematical relativity, such as the Robinson theorem, the Goldberg-Sachs theorem and the Kerr theorem, find a more natural formulation in conformal geometry. From a purely mathematical perspective, a number of important geometric structures on Riemannian and pseudo-Riemannian manifolds, such as Hermitian and Robinson manifolds, are conformally invariant. Conformal geometry also arises from other types of related geometries, such as Cauchy--Riemann (CR) geometry, projective geometry and generic vector distributions. This course is intended to Master-level and PhD-level students. The topics covered include the basics of conformal geometry, its associated calculus, known as tractor calculus, and the Fefferman-Graham ambient metric. In addition, we shall discuss its relation to twistor geometry and underlying geometric structures, such as totally null complex vector distributions and CR geometry. |
Literatura: |
(tylko po angielsku) Bailey, T. N., Eastwood, M. G. and Gover, A. R., "Thomas's structure bundle for conformal, projective and related structures." Rocky Mountain J. Math. 24 (1994), no. 4, 1191-1217. Curry S. N., Gover A. R., "An introduction to conformal geometry and tractor calculus, with a view to applications in general relativity.", Asymptotic analysis in general relativity, 86--170, London Math. Soc. Lecture Note Ser., 443, Cambridge Univ. Press, Cambridge, 2018. Fefferman, C. and Graham, C. R., "The ambient metric". Annals of Mathematics Studies, 178. Princeton University Press, Princeton, NJ, 2012. x+113 pp. ISBN: 978-0-691-15313-1 Penrose, R. and Rindler, W., "Spinors and space-time. Vol. 2. Spinor and twistor methods in space-time geometry." Cambridge Monographs on Mathematical Physics. Cambridge University Press, Cambridge, 1986. x+501 pp. ISBN: 0-521-25267-9 |
Efekty uczenia się: |
(tylko po angielsku) By the end of the course, the students will demonstrate a conceptual understanding of conformal geometry and of how it relates to other geometric structures. Besides this, they will be able to work with the tools of conformal geometry, such as tractor calculus and the Fefferman-Graham ambient metric, and apply these techniques to problems of mathematical physics. |
Metody i kryteria oceniania: |
(tylko po angielsku) Participation is recommended. Principles for the class and course credit award (including re-sit credit award): The students must meet the requirements specified below. Methods and criteria of assessment: Either an oral exam during the exam period, or an essay due by 30.06.2022. |
Właścicielem praw autorskich jest Uniwersytet Warszawski, Wydział Fizyki.