Lecture notes: https://drive.google.com/drive/folders/1GnlwXn0fyBswuMq_FfuTo40WifUBKE2n?usp=sharing
Submit homeworks on Canvas.
8/31
Moduli problems, Yoneda lemma, fine moduli space as a representing object, universal family.
9/7
Examples: families of endomorphisms.
9/9
Local universal families, categorical quotients, orbit spaces. Algebraic groups and actions.
9/14
Basic facts. Affine algebraic groups are linear. Quotients of linear algebraic groups are qproj varieties (without proof).
9/16
Reductive groups. Quotients of affine varieties.
9/21
Orbits are open in their closure. Good and geometric quotients. Good quotients are categorical quotients.
9/23
Linearizations and GIT quotients.
9/28
Groupoids, pseudofunctors, fibered categories.
9/28
CANCELED.
10/05
Examples of fibered categories: QCoh, reverse (fiberwise oppposite). 2-category of fibered categories, 2-Yoneda lemma. Categories fibered in groupoids (prestacks).
10/07
STUDENT PRESENTATION (Emma, Andrew): following King's paper "Moduli of representations of finite-dimensional algebras".
10/12
STUDENT PRESENTATION (Elijah): Straightening and unstraightening.
10/14
FALL BREAK.
10/19
Topologies.
10/21
Algebraic spaces and stacks.
10/26
Presentations of algebraic spaces and algebraic stacks.
10/28
STUDENT PRESENTATION (Yidi): introduction to group schemes.
11/02
STUDENT PRESENTATION (Ming, Deependra): examples of stacks.
11/04
Sheaves on stacks.
11/09
Beyond GIT. Moduli problems as pseudofunctors. Coarse moduli spaces, Keel-Mori theorem. Good moduli spaces.
11/11
STUDENT PRESENTATION (Ben, Marc): properties of good moduli spaces.
11/16
Universality of good moduli spaces.
11/18
STUDENT PRESENTATION (Ming, Emma): criterion for existence of good moduli spaces.
11/23
Grothendieck abelian categories, base change and tensor product.
11/30
STUDENT PRESENTATION (Deependra): numerical invariants and stability on algebraic stacks.
Moduli of objects in abelian categories.
12/2
12/7
STUDENT PRESENTATION (Ben, Marc): derived group schemes.
12/9
STUDENT PRESENTATION (Andrew): moduli of stable bundles on elliptic curves.