Spectral Algebraic Geometry 

 In Spring 2024, we are organizing a reading seminar on spectral algebraic geometry at KTH. 

Content: The seminar consists of weekly 60-minute talks given by the participants, followed by questions and discussions. Speakers are also asked to provide notes for their talks, which will be shared on the seminar page.

Topics: We will begin by covering the basics presented in Jacob Lurie's book, after which we will study advanced topics based on participants' interests. Preliminary basic topics include spectral schemes, the classification of étale maps of ring spectra, and spectral stacks, with advanced topics to be decided later. Ideas for advanced topics include, for example, the moduli stack of oriented elliptic curves, ambidexterity, elliptic cohomology, and the Goerss-Hopkins-Miller theorem (Elliptic II, Theorem 5.0.2).

Start: March 5th, 2024 at 2:00 pm

Location: KTH 3418

Timeslot: Tuesdays 2:00 pm (sharp) - 3:00 pm

There are some exceptions to the schedule, which are noted below. March 13th will be a background talk for those who wish to see more details about the topics covered in the first lecture. The titles for talks without an assigned speaker are preliminary.

Time exceptions:

• March 13th, 10:00-11:00 (exception due to division retreat)

• May 8th, 11:00-12:00.

Schedule:

• Preliminary talks:

• March 5th: Introduction + organization (David K.) — Notes

• March 13th: Background on spectra and E_oo-algebras (Elmo) - Notes + see the paper by David Gepner

• March 19th: Spectral schemes (Tilman) - Reading suggestion: SAG chapter 1.1 + HTT 6.2.2 (sheaves)

• March 26th: Spectral DM stacks I: oo-topoi and spectral DM stacks (Max) - Notes

• April 2nd: EASTER BREAK

• April 9th (CANCELLED): Spectral DM stacks II: Functor of points (Elmo) - Notes

• April 16th: Spectral DM stacks III: Quasicoherent sheaves (Max) - Notes

• Advanced topics:

• April 23rd: Deformation theory (Michele)

• April 30th: Comparison between DAG and SAG I (David K.)

Possible references (to be updated as the seminar progresses): 

• Jacob Lurie - Spectral Algebraic Geometry (SAG, the main reference for the seminar)

• Jacob Lurie - Higher Topos Theory (HTT)

• Jacob Lurie - Higher Algebra (HA)

• Jacob Lurie - Elliptic Cohomology (I, II and III)

• Jack Davies - On Lurie's Theorem and Applications

• Papers by Rok Gregoric

• David Gepner - An Introduction to Higher Categorical Algebra

• Charles Rezk - Introduction to SAG

Organizers: David K., Elmo & Michele.