Synthetic spectra

During the autumn semester of 2022, we are organizing a reading group on synthetic spectra at Stockholm University. We will meet once every week, when someone will give a 2x45 minute talk.

Topic: Synthetic spectra, originally developed by Pstrągowski, have led to new computational techniques in stable homotopy theory that have found many applications. In this reading group, we will first treat the construction and basic properties of the category of synthetic spectra and then study some of these applications.

Location: Cramèr room, Hus 1, Albano, Department of Mathematics, Stockholm University.

Time slot: Until November 10: Thursday morning 10:15 AM - 12:00 PM.
From November 24 onwards: Thursday afternoon 1:15 PM - 3:00 PM

Start: October 6, 10:15 AM.

Schedule:

• September 15, Thomas: Organizational meeting; Introduction to synthetic spectra (notes).

• October 6, Josefien: Sheaves on ∞-sites (Part I).

• October 13, Josefien: Sheaves on ∞-sites (Part II).

• October 20, Louis: Synthetic spectra and their basic properties.

• October 27, Tilman: An extension in the Adams spectral sequence in dimension 54.

• November 3, Robin: On the boundaries of highly connected, almost closed manifolds (Part I).

• November 10 (Kovalevsky room), Jan: On the boundaries of highly connected, almost closed manifolds (Part II).

• November 17: No talk.

• November 24 (Mittag-Leffler room), Thomas: On the boundaries of highly connected, almost closed manifolds (Part III).

• TBD

Possible further topics and references (this list is not exhaustive):

• The main reference for the talks of October 6 & 13 is Synthetic spectra and the cellular motivic category by Pstrągowski.

• An extension in the Adams spectral sequence in dimension 54 by Burklund.

• On the boundaries of highly connected, almost closed manifolds by Burklund-Hahn-Senger.

• Chromatic homotopy is algebraic when p > n² + n + 1 by Pstrągowski.

• Adams spectral sequences and Franke's algebraicity conjecture by Patchkoria-Pstrągowski.

• Abstract Goerss-Hopkins theory by Pstrągowski-Vankoughnett.

• HF₂-synthetic homotopy groups of topological modular forms by Marek

• The relation to motivic homotopy theory (I'm not sure what the best papers would be here, Synthetic spectra and the cellular motivic category by Pstrągowski would be a good place to start)

• Multiplicative structures on Moore spectra by Burklund.