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 |