Math 720 schedule

Monday Wednesday Friday
Sept. 13

In Lecture (room 4C4, 3 - 4 p.m.):
  • Motivation: Higher dimensional class field theory
  • Definition of Grothendieck groups and K_0(R)

Associated Reading (from gigapedia)
  • Rosenberg, "Algebraic K-theory and its applications," Chapter 1.

Video of class (streaming)
Video of class (downloadable)
Sept. 15

In Lecture (room 4C6, 2:30 - 3 p.m.):
  • Module theoretic approaches to K_0(R)

Associated Reading:
  • Serre, Linear representations of finite groups.

Video of class (streaming)
Video of class (downloadable)
Sept. 17

In Lecture (room 4C6, 1 - 2 p.m.):
  • Cristian Pop will give a lecture in class.
Sept. 20

In Lecture (room 4C4, 3 - 4 p.m.):
  • K_0(R) via infinite matrices: functorial properties and examples

Associated Reading
    links.
  • Rosenberg, "Algebraic K-theory and its applications," Chapter 1.

Video of class (streaming) - not available
Video of class (downloadable) - not available
Sept. 22

In Lecture (room 4C6,10 - 11 a.m..):
  • Applications of K_0(R)

Associated Reading:
  • Rosenberg, "Algebraic K-theory and its applications," Chapter 1.

Video of class (streaming)
Video of class (downloadable)
Sept. 24

In Lecture (room 4C6, 1 - 2 p.m.):
  • Mayer Vietoris construction of projective modules

Associated Reading:
  • TBA

Video of class (streaming)
Video of class (downloadable)
Sept. 27

In Lecture (room 4C4, 3 - 4 p.m.):
  • Mayer Vietoris construction, continued
  • Vector bundles and topological K-theory

Associated Reading
  • Rosenberg, "Algebraic K-theory and its applications," Chapter 1.

Video of class (streaming)
Video of class (downloadable)
Sept. 29

In Lecture (room 4C6,10 - 11 a.m..):
  • Overview of Stieffel Whitney classes and Chern classes
  • Start of the proof of Swan's theorem

Associated Reading:
  • Rosenberg, "Algebraic K-theory and its applications," Chapter 1.

Video of class (streaming)
Video of class (downloadable)
Oct. 1

In Lecture (room 4C6, 1 - 2 p.m.):
  • Completion of the proof of Swan's theorem

Associated Reading:
  • TBA

Video of class (streaming)
Video of class (downloadable)
Oct. 4

In Lecture (room 4C4, 3 - 4 p.m.):
  • Vector bundles and patching
  • Continuous Cech cohomology

Associated Reading
    links.
  • Rosenberg, "Algebraic K-theory and its applications," Chapter 1.
  • J. Milnor and J. Stasheff, "Characteristic Classes."
  • C. Weibel's "K-Book", chapter 1

Video of class (streaming)
Video of class (downloadable)
Oct. 6

In Lecture (room 4C6,10 - 11 a.m..):
  • Stieffel Whitney Classes and Chern Classes

Associated Reading:
  • Rosenberg, "Algebraic K-theory and its applications," Chapter 1.
  • J. Milnor and J. Stasheff, "Characteristic Classes."
  • C. Weibel's "K-Book", chapter 1
  • Lang's algebra on the Clifford Algebras and the Pin group.

Video of class (streaming)
Video of class (downloadable)
Oct. 8

In Lecture (room 4C6, 1 - 2 p.m.):
  • Stieffel Whitney Classes and Chern Classes, continued
  • The Pin group

Associated Reading:
  • Rosenberg, "Algebraic K-theory and its applications," Chapter 1.
  • J. Milnor and J. Stasheff, "Characteristic Classes."
  • C. Weibel's "K-Book", chapter 1
  • Lang's algebra on the Clifford Algebras and the Pin group.

Video of class (streaming)
Video of class (downloadable)
Oct. 11

No class: Fall Break!
Oct. 13

In Lecture (room 4C6,10 - 11 a.m..):
  • The Whitehead group K_1(\Lambda)

Associated Reading:
Video of class (streaming)
Video of class (downloadable)
Oct. 15

No class: postponed.
Oct. 18

In Lecture (room 4C6,10 - 11 a.m..):
  • The Whitehead group K_1(\Lambda), continued
  • Long exact sequences

Associated Reading:
Video of class (streaming)
Video of class (downloadable)
Oct. 20

In Lecture (room 4C4, 12 noon - 1 p.m, note time change):
  • K_2(\Lambda) and the Steinberg group

Associated Reading:
  • Rosenberg, "Algebraic K-theory and its applications," Chapter 4.
  • Milnor, "Algebraic K-theory," Chapter 5.

Video of class (streaming)
Video of class (downloadable)
Oct. 22

No class: postponed.
Oct. 25

In Lecture (room 4C6,10 - 11 a.m..):
  • K-theory and the leading terms in the expansions of L-series at integers

Associated Reading:
  • Be?linson's conjectures on special values of $L$L-functions, 273304, Perspect. Math., 4, Academic Press, Boston, MA, 1988.

Video of class (streaming)
Video of class (downloadable)
Oct. 27

In Lecture
  • Tamagawa numbers

Associated Reading:
  • Kneser, M. Semi-simple algebraic groups. 1967 Algebraic Number. Theory (Proc. Instructional Conf., Brighton, 1965) pp. 250265.

Video of class (streaming)
Video of class (downloadable)
Oct. 29

In Lecture
  • The Steinberg group and K_2, continued

Associated Reading:
  • Rosenberg, "Algebraic K-theory and its applications," Chapter 4.
  • Milnor, "Algebraic K-theory," Chapter 5.

Video of class (streaming) - to be posted
Video of class (downloadable) - to be posted
Nov. 1

In Lecture (room 4C6,10 - 11 a.m..):
  • Universal central extensions and the Steinberg group

Associated Reading:
  • Rosenberg, "Algebraic K-theory and its applications," Chapter 4.
  • Milnor, "Algebraic K-theory," Chapter 5.

Video of class (streaming)
Video of class (downloadable)
Nov. 3

In Lecture
  • Long exact sequences in K-theory
  • K_2 and symbols

Associated Reading:
  • Milnor, "Algebraic K-theory," Chapters 6, 11.
  • Rosenberg, "Algebraic K-theory and its applications," Chapter 4.

Video of class (streaming)
Video of class (downloadable)
Nov. 5

In Lecture
  • K_2 and symbols (continued)

Associated Reading:
  • Rosenberg, "Algebraic K-theory and its applications," Chapter 4.
  • Milnor, "Algebraic K-theory," Chapter 5.

Video of class (streaming)
Video of class (downloadable) -
Nov. 8

In Lecture (room 4C6,10 - 11 a.m..):
  • Quillen's + construction
  • Quillen's Q construction

Associated Reading:
  • Rosenberg, "Algebraic K-theory and its applications," Chapter 5.

Video of class (streaming)
Video of class (downloadable)
Nov. 10

In Lecture
  • Quillen's Q construction, continued

Associated Reading:
  • Rosenberg, "Algebraic K-theory and its applications," Chapter 5.

Video of class (streaming)
Video of class (downloadable)
Nov. 12

In Lecture
  • End of discussion of the Q-construction

Associated Reading:
  • Rosenberg, "Algebraic K-theory and its applications," Chapter 5.

Video of class (streaming)
Video of class (downloadable) -
Nov. 15

In Lecture (room 4C6,10 - 11 a.m..):
  • Overview of some papers on classfield theory to be discussed
  • Recap of Classfield theory for local fields via the Brauer group

Associated Reading:
  • J. P. Serre, Corps Locaux

Video of class (streaming)
Video of class (downloadable)
Nov. 17

In Lecture
  • Recap of classfield theory for local fields: the Artin map
  • Start of classfield theory of curves over local fields

Associated Reading:
  • J. P. Serre, Corps Locaux
  • Algebraic K-Theory and Classfield Theory for Arithmetic Surfaces, by Spencer Bloch Source: The Annals of Mathematics, Second Series, Vol. 114, No. 2 (Sep., 1981), pp. 229-265
  • Class field theory for curves over local fields, by S. Saito, J. Number Theory 21 (1985), no. 1, 4480.

Video of class (streaming)
Video of class (downloadable)
Nov. 19

In Lecture
  • Classfield theory on Curves

Associated Reading:
  • Algebraic K-Theory and Classfield Theory for Arithmetic Surfaces, by Spencer Bloch Source: The Annals of Mathematics, Second Series, Vol. 114, No. 2 (Sep., 1981), pp. 229-265
  • Class field theory for curves over local fields, by S. Saito, J. Number Theory 21 (1985), no. 1, 4480.

Video of class (streaming)
Video of class (downloadable) -
Nov. 22

In Lecture
  • Classfield theory on Curves (continued)

Associated Reading:
  • Algebraic K-Theory and Classfield Theory for Arithmetic Surfaces, by Spencer Bloch Source: The Annals of Mathematics, Second Series, Vol. 114, No. 2 (Sep., 1981), pp. 229-265
  • Class field theory for curves over local fields, by S. Saito, J. Number Theory 21 (1985), no. 1, 4480.

Video of class (streaming)
Video of class (downloadable)
Nov. 24

In Lecture
  • Classfield theory on Curves

Associated Reading:
  • Algebraic K-Theory and Classfield Theory for Arithmetic Surfaces, by Spencer Bloch Source: The Annals of Mathematics, Second Series, Vol. 114, No. 2 (Sep., 1981), pp. 229-265
  • Class field theory for curves over local fields, by S. Saito, J. Number Theory 21 (1985), no. 1, 4480.

Video of class (streaming)
Video of class (downloadable)
Nov. 26

In Lecture
  • No Class - Thanksgiving Break!
Nov. 29

In Lecture
  • Classfield theory on Curves (continued)
  • Tate curves

Associated Reading:
  • Class field theory for curves over local fields, by S. Saito, J. Number Theory 21 (1985), no. 1, 4480.
  • The arithmetic of elliptic curves, J. Silverman
  • Elliptic functions, S. Lang

Video of class (streaming))
Video of class (downloadable)
Dec. 1

In Lecture
  • Tate curves, continued
  • Classfield theory for two dimensional local fields

Associated Reading:
  • Class field theory for curves over local fields, by S. Saito, J. Number Theory 21 (1985), no. 1, 4480.
  • The arithmetic of elliptic curves, J. Silverman
  • Elliptic functions, S. Lang

Video of class (streaming)
Video of class (downloadable)
Dec. 3

In Lecture
  • Classfield theory for two dimensional local fields (continued)

Associated Reading:
  • Class field theory for curves over local fields, by S. Saito, J. Number Theory 21 (1985), no. 1, 4480.

Video of class (streaming)
Video of class (downloadable)
Dec. 6

In Lecture
  • Classfield theory for two dimensional local fields (continued)

Associated Reading:
  • Class field theory for curves over local fields, by S. Saito, J. Number Theory 21 (1985), no. 1, 4480.

Video of class (streaming))
Video of class (downloadable)
Dec. 8

In Lecture
  • Classfield theory for two dimensional complete local rings

Associated Reading:
  • Class field theory for curves over local fields, by S. Saito, J. Number Theory 21 (1985), no. 1, 4480.
  • Invitation to Higher Local Fields, I. Fesenko (see gigapedia).

Video of class (streaming)
Video of class (downloadable)
Dec. 10

In Lecture
  • Classfield theory for two dimensional complete local rings (continued)

Associated Reading:
  • Class field theory for curves over local fields, by S. Saito, J. Number Theory 21 (1985), no. 1, 4480.
  • Invitation to Higher Local Fields, I. Fesenko (see gigapedia).

Video of class (streaming)
Video of class (downloadable)

Last updated: 12/10/10
Send e-mail comments to: ted@math.upenn.edu