Mini-workshop in Mathematical Physics Noncommutative Topics

02 de agosto de 2016

August 9-18 & September 6-16 Noncommutative geometry, in the sense of this workshop, is the exploration of geometric concepts through operator-algebraic methods, as initiated and outlined by Alain Connes. Many of the topics central to non-commutative geometry (index theory and K-theory, spectral geometry, the measure-theoretic and topological analysis of operator algebra) arose from classical problems in analysis, topology, representation theory and dynamical systems. Collectively they have become powerful and effective tools in various areas of pure mathematics and mathematical physics such as non-commutative algebraic geometry, foliations, groupoids, stacks, gerbes, deformation-quantization and number theory. Noncommutative geometry has also found applications to several topics in physics such as quantum field theory, renormalization, gauge theory, string theory, cosmology, gravity, mirror symmetry, condensed matter physics and statistical mechanics.

In order to stimulate the interest towards these modern topics the Pontificia Universidad Católica de Chile (PUC) in collaboration with the Núcleo Milenio Física Matemática has decided to organize a mini-workshop designed to treat some aspects of current interest in noncommutative geometry. The workshop is structured in five mini-courses of three classes (of 90 minutes) each. The first three courses will be taught in the period August 9-18 while the remaining two courses will be taught in the period September 6-16. The workshop relies on the collaboration of three internationational lecturers, K. Gomi (Shinshu University), G. Landi (University of Trieste) and M. Mantoiu (University of Chile) and with the support of two local lecturers G. De Nittis and R. Rebolledo (PUC). The workshop is aimed to postgraduate students and academics from the departments of mathematics and physics which are interested in expand their knowledge and become familiar with some concepts of noncommutative geometry.

The mini-courses will be self contained but with high and demanding teaching standards. The interested post-graduated students have the opportunity to register the workshop as Seminario I with the code MPG3125, NRC16097. In this case, the achievement of the final grade (15 credits) will be subject to the presentation of an investigation by the student about one topic related to the content of the workshop and coordinated with the supervision of G. De Nittis. Basics of analysis and linear algebra are required as minimum knowledge to attend the workshop. The course of differential geometry (which will be delivered in parallel in the second semester) is strongly recommended. For more details the interested students are encouraged to contact directly the respective directors of the post-graduate program: O. Bourget (bourget@mat.puc.cl) for mathematics and J. Maze (jmaze@fis.puc.cl) for physics.

The program of the proposed mini-courses is listed below:

First Part (August 9-18)

1-Brownian motions on noncommutative manifolds (R. Rebolledo, PUC)
•    Heat semigroup and Brownian Motion on classical manifolds.
• Heat semigroup and Laplacians on noncommutative manifolds (Examples).
•    Examples of quantum Brownian motions on noncommutative manifolds.

2. Elements of Noncommutative Geometry (G. Landi, U. Trieste)
•    Spectral triples.
•    Solitons on noncommutative spaces.
•    Non commutative line bundles.
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3.Operator algebras associated to some mathematical structures (M. Mantoiu, U. Chile)
•    Algebras associated to topological spaces and groups.
•    Algebras associated to dynamical systems.
•    Quantization and pseudodifferential calculus.

Second Part (September 6-16)

1. Introduction to algebraic K-theory (G. De Nittis, PUC)
•    K0-group of rings.
•    K1-group of rings.
•    Cyclic homology and K-theory.

2. Introduction to topological K-theory (K. Gomi, U. Shinshu)
•   Basics of topological K-theory.
•   Calculations of K-theory.
•   Variants of K-theory.

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For more informations the interested may contact G. De Nittis (gidenittis@mat.uc.cl) or visit the web-page https://gdenittis.wordpress.com/courses/ncg-workshop/ .For practical reasons (reservation of rooms, etc.) we invite people interested in attending the workshop to send a confirmation email to gidenittis@mat.uc.cl or bourget@mat.puc.cl.