Aim and description: The aim of the seminar is to study some Derived Algebraic Geometry (DAG) mostly in the form of Jacob Lurie's thesis. We will go through the foundations of the subject and then we will cover some additional advanced topics, depending on the taste of the participants. Some general references we will follow can be found below. More specialized papers will be listed while we proceed.

This seminar is a follow up seminar of our Infinity categories seminar. But we will review prerequisites.

References: see below.

Schedule:

Week	Speaker	Topic
Week 1	Marco Schlichting	Motivation and overview.
Week 2	Marco Schlichting	Infinity categories for DAG
Week 3	Dylan Madden	Stable infinity categories.
Week 4	Dylan Madden	Stable infinity categories II
Week 5	Parvez Sarwar	Derived rings and their modules
Week 6	Ferdinando Zanchetta	Derived schemes I
Week 7	Ferdinando Zanchetta	Derived schemes II
Week 8	Ferdinando Zanchetta	Derived schemes III
Week 9	No seminar
Week 10	No seminar. Topology seminar instead

Some References:

• Derived algebraic geometry
  • We will mainly follow Jacob Lurie's thesis DAG
• Infinity categories
  • UIUC Higher Category Theory Seminar
  • Rezk: Stuff about quasicategories
  • Lurie: Higher Topos Theory
  • Cisinski's Bourbaki talk Catégories supérieures et théorie des topos and video
• Stable infinity categories
  • Chapter 1 of Lurie's Higher Algebra
• Simplicial rings
  • Quillen's unpublished notes Homology of commutative rings
  • The relevant chapter in Quillen's Homotopical algebra. Lecture Notes in Mathematics, No. 43
  • Section 3 in DAG
• Other
  • Spectral algebraic geometry SAG
  • D. Gaitsgory and N. Rozenblyum, "A study in derived algebraic geometry". Link.
  • Toen's overview "Derived Algebraic Geometry" .
  • Regensburg modules Descent in algebraic K-theory and The Grothendieck–Riemann–Roch theorem .