## 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.

### Schedule

**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).

### Resources

- 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 A
^{1}-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