The K-group at Reed
See the poster!
The K-group is a small summer research team at Reed College focused on Milnor-style K-theories and their applications in number theory, algebraic geometry, and motivic homotopy theory. Under the supervision of Kyle Ormsby, paricipants will undertake a two-week mini-course in K-theory and then pursue various research projects related to Milnor and Milnor-Witt K-theory.
Dates: July 6 - August 28, 2015
Support: Stipends available to qualified applicants.
Requirements: Participants must be students at Reed College who have completed Math 332 or the equivalent by the end of spring term.
July 6: Milnor K-theory, symbols, and Hilbert reciprocity. Notes [pdf].
July 7: Quadratic forms and Witt rings (selections from Chapters I and II of Lam, Introduction to quadratic forms over fields).
July 8: The discriminant, presentation of the Grothendieck-Witt ring, Pfister forms, and the Milnor conjecture (Lam II.2, II.4, and X.6).
July 10: Milnor-Witt K-theory: basic properties. Notes [pdf].
July 13: Stiefel-Whitney invariants (Nico).
- J. Milnor, Algebraic K-theory and quadratic forms
- The original source for Milnor K-theory.
- J. Milnor, Gauss and quadratic reciprocity
- A selection from Chapter 11 of Milnor's Introduction to algebraic K-theory.
- P. Clark, Quadratic forms Chapter I: Witt's theory, Chapter II: Structure of the Witt ring
- Notes on quadratic forms and the Witt ring.
- T.Y. Lam, Introduction to quadratic forms over fields
- Even better than Scharlau's classic, this is a complete, accessible, beautifully written treatment of quadratic forms over fields, including connections with Milnor K-theory.
- F. Morel, Unramified Milnor-Witt K-theories
- This is Chapter 3 of Morel's A1-algebraic topology over a field.
- F. Morel, Sur les puissances de l'ideal fondamental de l'anneau de Witt
- Morel proves that Milnor-Witt K-theory and J*(F) are isomorphic in this paper.
- D. Dugger, Notes on the Milnor conjectures
Kyle M. Ormsby