Math 8815-1

Ulam Seminar on K-theory

Professor:   

Guillermo Cortiñas

office: Math 253

phone: (303) 492–2312

e-mail: gcorti at dm.uba.ar

web: http://mate.dm.uba.ar/~gcorti

Office hours: By appointment.

Lectures

1. 1/ 22/13. Definition of K_0 and K_1.  Ref: [1, Sec. 2.1], [11, 1.1, 1.2, 2.1] More info: [13, Ch 1, 2, 3.1-3.3].
   
2. 1/29/13. Excision, stability, nilinvariance. Ref: [1, Sec. 2.1, 2.4], [11, Ch 1.5, 2.5]
   
3. 2/5/13. Matrix stability and negative K-theory. Ref: [1,2.2, 2.3, 2.5], [10, Sec. 7] More info: [11, 3.3]. [13, 3.4]
   

4. 2/12/13. Topological K-theory, Karoubi-Villamayor K-theory. Ref:[1,  Sec. 3.1, 3.2, 4] [10, 1.2], [8]

5. 2/19/13. Homotopy algebraic K-theory. Ref: [1, Sec. 5.1-5.2], [10]. 

6. 2/26/13. The fundamental theorem for KH and Bott periodicity. Notes: part 1, part 2. Ref:[2, 7.3],[4].

7. 3/5/13. The homotopy invariance theorem, part 1. Ref: [8, 3.1.1 and 3.2.1].

9. 3/12/13. The homotopy invariance theorem, part 2. Notes.

10. 3/19/13. Homotopy algebraic K-theory of stable algebras and topological K-theory. Ref:  [3] (comparison theorems), [6] (Cuntz' theorem in the C*-setting), [7] (Cuntz' theorem the locally convex setting), [9] (Karoubi's density theorem, Karoubi's conjecture).

11. 4/2/13. Quillen's K-theory. Ref: [1, Sec. 6.1-6.6] (Plus construction and Suslin-Wodzicki excision theorems) [1,Sec. 7.1] (Karoubi's conjecture).

12. 4/9/13. K-theory spectra. Ref: [14, Sec. 10.9] (Generalities about spectra), [1, Sec. 10] (K-theory spectra). 

13. 4/16/13. Chern Characters. Ref: [1, Sec. 11], [3, Sec. 2, 4, 6.3], [13]

14. 4/23/13. K-theory of singular algebraic varieties. Ref: [2].

15. 4/30/13. K-theory of algebras of continuous functions. Ref: [3].

References

1. G. Cortiñas. Algebraic v. topological K-theory: a friendly match. In: Topics in algebraic and topological K-theory. Springer Lecture Notes in Mathematics 2008. 
   
2. G. Cortiñas, C. Haesemeyer, M. Schlichting, C. Weibel. Cyclic homology, cdh-cohomology and negative K-theory. Annals of Mathematics. 167, (2008) 549-573.
3. G. Cortiñas, A. Thom. Bivariant algebraic K-theory. J. reine angew. Math. 510 (2007) 71-124.
4. G. Cortiñas, A. Thom. Comparison between algebraic and topological K-theory of locally convex algebras.
   Adv. Math. 218 (2008), no. 1, 266–307.
5. G. Cortiñas, A. Thom. Algebraic geometry of topological spaces I.  Acta Mathematica 209 (2012) 83-131.
6.   J. Cuntz. K-theory and C∗-algebras. Algebraic K-theory, number theory, geometry and analysis (Bielefeld, 1982), 55–79, Lecture Notes in Math., 1046, Springer, Berlin, 1984.
7. J. Cuntz. Bivariante K-Theorie für lokalkonvexe Algebren und der Chern-Connes-Charakter.  Doc. Math. J. DMV. 2 (1997) 139-182. 
8. J. Cuntz, R. Meyer, J. Rosenberg. Top.ological and bivariant K-theory.
   Oberwolfach Seminars, 36. Birkhäuser Verlag, Basel, 2007.

9. M. Karoubi. K-théorie algébrique de certaines algèbres de'operateurs. Springer Lecture Notes in Mathematics 725, 254-290 (1979).

10. M. Karoubi, O. E. Villamayor. K-théorie algébrique et K-théorie topologique. I. (French) Math. Scand. 28 (1971), 265–307 (1972).
11.  J. Rosenberg. Algebraic K-theory and its applications.
    Graduate Texts in Mathematics, 147. Springer-Verlag, New York, 1994.
12.  C. Weibel. Homotopy algebraic K-theory. Algebraic K-theory and algebraic number theory (Honolulu, HI, 1987), 461–488, Contemp. Math., 83, Amer. Math. Soc., Providence, RI, 1989.
    
13.  C. Weibel. The K-book: an introduction to algebraic K-theory. Available at http://www.math.rutgers.edu/~weibel/Kbook.html
14. C. Weibel. An introduction to homological algebra. Cambridge Studies in Advanced Mathematics, 38. Cambridge University Press, 1994.
15. M. Wodzicki. Algebraic K-theory and functional analysis, in: First European Congress of Mathematics, Vol. II, Paris, 1992, in: Progr. Math., vol. 120, Birkhäuser, Basel, 1994, pp. 485–496.