## Math 790, Section 01, Tannakian Categories

## Spring, 2016

**Schedule:** 10:05 to 11:20, Tuesdays and Thursdays, February 13 to March 20; Physics 205

**Instructor:** Richard Hain

**References:**

- Pierre Deligne:
*Catégories tannakiennes*, The Grothendieck Festschrift, Vol. II, 111-195,
Progr. Math., 87, Birkhaüser, Boston, MA, 1990. (pdf)
- P. Deligne and J. Milne:
*Tannakian Categories*. (pdf) --- Just sections 1 and 2.
- S. Mac Lane:
*Categories for the Working Mathematician*, Springer GTM no. 5, 1971.

**Topics:**

- Motivation: the category of representations of a group
- Tannakian categories: definition and examples
- Affine group schemes
- Reductive representations and reductive groups
- Tannaka duality
- Example: unipotent and relative unipotent completion
- Example: graded polarizable mixed Hodge structures; Mumford-Tate groups
- Example: Brown's enriched mixed Hodge structures; periods
- Example: variations of MHS and enriched MHS; families of periods
- Extensions and cohomology (as time permits)

**Notes:**

Return to: Richard Hain *
Duke Mathematics Department *
Duke University