# A primer on K-theory and its applications

# A primer on K-theory and its applications

Mon-Wed 11h-13h, from Feb. 22, 2016

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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.

Mon-Wed 11h-13h, from Feb. 22, 2016

–

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.

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 |