Block seminar: Chow motives and applications to quadratic forms

Summer semester 2026

The algebraic theory of quadratic forms is the study of quadratic forms over arbitrary fields. This theory was founded by Ernst Witt in the 1930s and was further developed in the second half of the last century. Recently, many long-standing problems in this theory have been solved using tools from algebraic geometry. The goal of this seminar is to introduce Chow groups and motives and to demonstrate their applications to algebraic questions about quadratic forms.

Time and place: During the week of 3–7 August. We may also start the week before, depending on the availability of speakers. The exact schedule will be discussed at the first meeting this Friday, 26.06., at 14:15 in B 004.

Information: The seminar is intended for Master students (prerequisites: algebra, commutative algebra and algebraic geometry). If you are interested in attending the seminar and giving a talk, but cannot attend the first meeting, please write me an email.

Preliminary programm:

1. Introduction. Preliminary distribution of the talks. Friday 26.06. at 14:15 in B 004.

2. Overview of Witt theory and Pfister froms. References: [1], Chapter 2, 10.1; [3], Chapter 1, 2.1.

3. Subform theorem, function field of quadrics . References: [1], Chapter 9.1-9.2, 10.3-10.4.

4. Applications: first Witt index, Arason-Pfister Hauptsatz . References: [1], Chapter 10.5.

For Talks 5–8, you can follow my lecture notes (a link will be sent by email). For more details, see references [3] and [4].

5. Chow groups, properties, A^1-invariance. References: for A^1-invariance see Proposition 1.9 in [4].

6. Chow group of cellular varieties. Examples: projective space, slit quadric.

7. Category of Chow motives. Properties, examples.

8. Chow motives of quadrics (split, isotropic cases).

9. Motivic decomposition of Pfister forms .

10. Vishik's theorem on motivic summands of quadrics .

11. Quadratic forms of dimension 10 in I^3 .

12. Overview of recent developments .

References:
1. T. Y. Lam, Introduction to Quadratic forms over fields.
2. B. Kahn, Formes quadratiques sur un corps (available online, in french).
3. R. Elman, N. Karpenko, A. Merkurjev, The algebraic theory of quadratic forms.
4. W. Fulton, Intersection theory.
5. Lectures of Nikita Geldhauser on Youtube.
References for each talk will be updated.