A primer on K-theory and its applications

Lecturers
Ramon Antoine (UAB), Pere Ara (UAB), Natàlia Castellana (UAB), Pere Pascual (UPC) and Francesc Perera (UAB)
Schedule
Mon-Wed 11h-13h, from Feb. 22, 2016
Location
Room C1/028, CRM
Summary

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.

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
  1. B. A. Magurn, An algebraic introduction to K-Theory, Encyclopedia of Mathematics and its Applications 87, Cambridge University Press, 2002.
  2. F. Larsen, N.J. Laustsen, and M. R\ordam, An introduction to K-Theory for C*-algebras, London Mathematical Society Student Texts 49, 2000
  3. J. Rosenberg, Algebraic K-Theory and its applications, Graduate Texts in Mathematics 147, Springer-Verlag 1994.
  4. 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