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A primer on K-theory and its applications

A primer on K-theory and its applications

Date
Mon-Wed 11h-13h, from Feb. 22, 2016
Registration
Location
Room C1/028, CRM
K-Theory has an interdisciplinary flavour within modern mathematics, being present in diverse subjects such as algebraic topology, number theory, operator theory and dynamical systems, to mention but a few. In this course we will present the basic tools of the subject, with a stress on applications in various areas by means of examples. We also plan on offering special sessions related to three areas in which K-Theory plays a significant role: Topology, Analysis and Algebraic Geometry. These will be conducted by an expert on the area.
Date
Mon-Wed 11h-13h, from Feb. 22, 2016
Registration
Location
Room C1/028, CRM
K-Theory has an interdisciplinary flavour within modern mathematics, being present in diverse subjects such as algebraic topology, number theory, operator theory and dynamical systems, to mention but a few. In this course we will present the basic tools of the subject, with a stress on applications in various areas by means of examples. We also plan on offering special sessions related to three areas in which K-Theory plays a significant role: Topology, Analysis and Algebraic Geometry. These will be conducted by an expert on the area.
Lecturers
Ramon Antoine (UAB)

Pere Ara (UAB)

Natàlia Castellana (UAB)

Pere Pascual (UPC)

Francesc Perera (UAB)

Contents
  • Modules over a ring. Free and projective modules
  • Construction of the Grothendieck group K0, using finitely generated projective modules and idempotents
  • Computation of K0 for a number of classes of rings (including PIDs and local rings)
  • The Whitehead Lemma and the Construction of the group K1, and computation for various classes of rings
  • Relative K1. The relative Whitehead Lemma. The exact sequence associated to an extension
  • The Theorem of Bass-Heller-Swan. Negative K-Theory
  • Topological K-Theory as a motivation for the work of Quillen
  • An overview on Quillen’s +-construction

 

Special Sessions

  • K-Theory and Topology: Vector bundles
  • K-Theory and Analysis: C*-algebras. Bott’s periodicity and the six-terms exact sequence
  • K-Theory and Geometry

 

Additional activity

Barcelona Spring 2016 workshop on Number Theory and K-theory.

Bibliography
  • B. A. Magurn, An algebraic introduction to K-Theory, Encyclopedia of Mathematics and its Applications 87, Cambridge University Press, 2002.
  • F. Larsen, N.J. Laustsen, and M. R\ordam, An introduction to K-Theory for C*-algebras, London Mathematical Society Student Texts 49, 2000
  • J. Rosenberg, Algebraic K-Theory and its applications, Graduate Texts in Mathematics 147, Springer-Verlag 1994.
  • C. A. Weibel, The K-book: an introduction to algebraic K-theory, Graduate Studies in Mathematics 145, AMS, 2013.
List of Participants
First Name Last Name Email address Affiliation Degree Area of interest
Xavier Soria Poma xsoria@cvc.uab.es Yes Master Computational Mathematics
Louis Carlier louiscarlier@mat.uab.cat UAB Doctorate algebraic topology, category theory
Teresa Gálvez mtgcia@gmail.com UAB Doctorate
David Bachiller dbachiller@mat.uab.cat UAB PhD student Group theory
Carlos Calvo charlie1988@gmail.com CRM Master in Advanced Mathematics Symplectic geometry and group actions
Mikel Lluvia lluviamikel@gmail.com UB Doctorate
Joan Claramunt joan.claramunt1@gmail.com UAB Master Algebra
Ricard Garcí­a riba@mat.uab.cat UAB Doctorate Topology
Daniel Torres Moral dani10sa2@hotmail.com UPC Doctorate