Topics and Lecturers

Lie Algebroids and Characteristic Classes
Paul Bressler (Universidad de los Andes, Bogotá, Colombia)

In my talks I am planning to discuss, time permitting, the following topics:
- 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.


Real Relativistic Fluids in Heavy Ion Collisions
Esteban Calzetta (Universidad de Buenos Aires, Argentina)

The theory of real relativistic fluids is in the rather unique situation that there is a natural relativistic extension of the nonrelativistic theory, but it is physically untenable [1]. On the other hand, mounting evidence that mattter created in relativistic heavy ion collisions behaves as a relativistic fluid with small but finite viscosity has given the quest for an alternative a definite goal [2]. We shall review different approaches to relativistic real fluids, their link to relativistic kinetic theory, and their application to the analysis of heavy ion collisions [3,4].

[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).


Geometrical Aspects of Black Holes
Bruno Carneiro da Cunha (Universidade Federal de Pernambuco, Brazil)

The list of topics to be covered in this course includes:
- 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.


Lie groupoids and Lie algebroids -- connection theory and Poisson geometry
Kirill Mackenzie (University of Sheffield, UK)

First two lectures: The Atiyah sequence of a principal bundle. Connections in transitive Lie algebroids. Cohomological obstruction to integrability. Lie algebroids with given curvature forms.
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


A Mathematical Approach to Some Issues in Quantum Field Theory: A Prolegomenon to Renormalisation Methods
Sylvie Paycha (Potsdam University, Germany)

In this course addressed to master level students, we shall present mathematical aspects of renormalisation methods borrowed from physics, with applications in mathematics. In particular, they will be used to count integer points on cones, to extend the Euler-Maclaurin formula to cones and to renormalise multizeta functions at poles. On the grounds of such examples borrowed from various areas of mathematics, we will get a sense of the manifold of questions related to renormalisation, yet touching on this issue only in its most elementary aspects. This serves as a modest prolegomenon to the otherwise broad and complex renormalisation issue.
Note: This course will continue from August 1st to August 10th at Universidad de los Andes.


C*-algebras and Index Theory of Boundary Value Problems
Elmar Schrohe (Leibniz Universität Hannover, Germany)

Boutet de Monvel's calculus [1,2], provides a pseudodifferential framework which encompasses both the classical differential boundary value problems and their inverses whenever these exist.
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.

- 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

[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.


Operator Algebras and Quantum Physics
Andrés F. Reyes Lega (Universidad de los Andes, Bogotá, Colombia)

Motivated by the sharp contrast between classical and quantum physics as probability theories, in these lectures I will introduce the basic notions of operator algebras that are relevant for the algebraic approach to quantum physics. Aspects of the representation theory of C*-algebras will be motivated and illustrated in physical terms. These will include a discussion of quantum operations, quantum correlations, and symmetry breaking. Particular emphasis will be given to explicit examples from the theory of quantum phase transitions, where concepts coming from strands as diverse as quantum information theory, algebraic quantum physics and statistical mechanics agreeably converge, providing a more complete picture of the physical phenomena involved.


Algebraic Geometry and Conformal Field Theory
Katrin Wendland (Albert-Ludwigs-Universität Freiburg, Germany)

While two-dimensional conformal field theories can be defined abstractly in mathematics without reference to the algebraic geometry of complex varieties of dimension greater than one, some of the most interesting examples are expected to arise as so-called non-linear sigma-models on Calabi-Yau varieties of complex dimension two or higher.
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