A primer on K-theory and its applications
Ramon Antoine (UAB), Pere Ara (UAB), Natàlia Castellana (UAB), Pere Pascual (UPC) and Francesc Perera (UAB)
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.
- 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
- 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
- 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 Pomafirstname.lastname@example.org||Yes||Master||Computational Mathematics|
|Louis||Carlieremail@example.com||UAB||Doctorate||algebraic topology, category theory|
|David||Bachillerfirstname.lastname@example.org||UAB||PhD student||Group theory|
|Carlos||Calvoemail@example.com||CRM||Master in Advanced Mathematics||Symplectic geometry and group actions|