Topics and Lecturers
- Lie algebroids: basic definitions and examples of Lie algebroids. - Basic operations on Lie algebroids. - Modules over Lie algebroids, universal enveloping algebra of a Lie algebroid. - Lie algebroid cohomology, the de Rham complex. - Transitive Lie algebroids, connections, curvature. - Picard-Lie Algebroids and the first Chern class. - Chern-Weil theory for Lie algebroids and characteristic classes.
References:
[1] W. Hiscock and L. Lindblom, Ann. Phys. 151, 466 (1983); Phys. Rev. D 31, 725 (1985). [2] D. Rischke, Fluid dynamics for relativistic nuclear collisions, Proceedings of the 11th Chris Engelbrecht Summer School in Theoretical Physics, Cape Town, Feb. 4-13, 1998. [3] E. Calzetta and B-L Hu, Nonequilibrium quantum field theory (Cambridge University Press, Cambridge (England), 2008). [4] J. Peralta Ramos and E. Calzetta, Phys. Rev. D 82, 106003 (2010); Phys. Rev. C 82, 054905 (2010); JHEP 02, 085 (2012); Phys. Rev. D 86, 125024 (2012).
- Classical Solutions: Schwarzschild, Reissner-Nordström, Kerr, Kerr-Newman - SUGRA Solutions: Black-branes and extended charges. - Thermodynamics of Black Holes - The near horizon limit and partial recovering of (super)symmetries - Quantum fields on classical black-holes: the meaning of entropy.
Lecture 3,4,5: Lie groupoids and Lie algebroids. The gauge groupoid of a principal bundle. Symplectic groupoids. Induced Poisson structure on the base. Poisson Lie groups and Lie bialgebras. Poisson groupoids and Lie bialgebroids. Lecture 6: Introduction to double Lie groupoids and associated Lie algebroids
Note: This course will continue from August 1st to August 10th at Universidad de los Andes.
It associates to each operator two symbols: a pseudodifferential principal symbol and an operator-valued boundary symbol. Ellipticity requires the invertibility of both. If the underlying manifold is compact, elliptic elements define Fredholm operators. Boutet de Monvel showed how then the index can be computed in topological terms, reducing the problem to the Atiyah-Singer index theorem for closed manifolds [3]. In the lectures I want to give a short introduction to the basics of index theory and Boutet de Monvel's calculus and then show how C*-algebra K-theory can be used to prove Boutet de Monvel's index theorem for boundary value problems. Topics:
- Fredholm operators and their index- Pseudodifferential operators and the parametrix construction - The topological index and the Atiyah-Singer index theorem - The idea of Boutet de Monvel's calculus: Relation to classical boundary value problems, operator classes, symbols and ellipticity - Basics on K-theory for C*-algebras - The C*-closure of the algebra of operators of order and class zero - The kernel and the range of the boundary symbol map - The index theorem for boundary value problems References:
[1] L. Boutet de Monvel. Boundary problems for pseudo-differential operators.
Acta Math. 126 (1971), no. 1-2, 11-51.[2] E.Schrohe. A short introduction to Boutet de Monvel's calculus. In Approaches to Singular Analysis (Berlin, 1999), Oper. Theory Adv. Appl. 125, 85-116, Birkhäuser, Basel, 2001. [3] M. F. Atiyah & I. M. Singer. The index of elliptic operators I. Ann. of Math. (2) 87 (1968), 484-530.
In these lectures, we shall present some of the basic constructions in conformal field theory, as well as some of the relevant background in Calabi-Yau geometry. As a key example, we will study K3 surfaces, where algebraic geometry and conformal field theory can be linked in a non-trivial manner. Time permitting, the topics will include: - The complex and Kähler geometry of Calabi-Yau varieties - Vertex algebras - Chiral de Rham complex and elliptic genus - Symmetries of K3 surfaces and of conformal field theories on K3 |