This is the homepage for the Synthetic Spectra Reading Seminar for the Spring 2024 and Summer 2024 quarters at Northwestern University.

Co-organizers: Preston Cranford, Lillianna Lipman, Noah Wisdom

Setting: Talks will be 3:00-3:50pm (Central Time) on Fridays in Lunt 101.

Remark: The main reference for this seminar is Piotr Pstrągowski’s Synthetic spectra and the cellular motivic category. Despite not being the dissertation for Piotr’s PhD, we refer to Synthetic spectra and the cellular motivic category as Piotr’s thesis to follow convention.

Please contact the co-organizers if you would like to would like to give a talk, to access the Discord server for this seminar, or to discuss other comments or questions.

Talk 1: Overview

• Date: March 29, 2024
• Speaker: Preston Cranford
• Abstract: Introduce the Adams Spectral Sequence, synthetic spectra, and some of their applications. Give an overview of this seminar.
• Reference: Piotr’s thesis

Talk 2: Filtered Objects of and Deformations of Categories

• Date: April 5, 2024
• Speaker: Noah Wisdom
• Abstract: The goal of this talk is to prepare for defining spherical sheaves on additive sites in Talk 3. Introduce the categorification of the deformation (bigraded homotopy groups) of filtered spectra. Discuss its generic and special fiber.
• References: Talk No. 8 of the 1950-1951 Cartan Seminar, Section 1.2.2 in Lurie’s Higher Algebra, Section 3 in Lurie’s Rotation invariance in algebraic K-theory, Appendix B in Burklund-Hahn-Senger’s Galois reconstruction of Artin-Tate -motivic spectra, Section 2 of Baer-Johnson-Marek’s Stable Comodule Deformations and the Synthetic Adams-Novikov Spectral Sequence

We will skip having a talk on the week containing April 12, 2024.

Talk 3: Adams-type Homology Theories

• Date: April 19, 2024
• Speaker: Lillianna Lipman
• Abstract: The goal of this talk is to cover Section 2 of Piotr’s thesis. Define Adams-type Homology Theories and discuss some of their properties. Define spherical sheaves on additive sites and . Discuss the recognition of spherical sheaves result (Theorem 2.8 in Piotr’s thesis). Discuss Theorem 2.58 and it’s special case Theorem 3.2.
• References: Section 2.2 of Ravenel’s green book and Section 2 of Piotr’s thesis

Talk 4: t-structure on Synthetic Spectra

• Time: Note the irregular time. 4:00-4:50pm (Central Time) on April 25, 2024
• Speaker: Callum Sutton
• Abstract: The goal of this talk is to cover Section 3 and parts of both Sections 4.1 and 4.2 of Piotr’s thesis.. Prove is an additive infinity-site and discuss as done in Section 3.3. Discuss Theorem 3.25/3.26. Define synthetic spectra, state some properties about them, and discuss the t-structure on them.
• Reference: Sections 3 and 4 of Piotr’s thesis

Talk 5: -invertible Synthetic Spectra

• Date: May 3, 2024
• Speaker: Preston Cranford
• Abstract: The goal of this talk is to cover Section 4.1 through 4.4 of Piotr’s thesis. Introduce bigraded spheres and Chow degree. Important results to cover include Corollary 4.12, Theorem 4.18, Proposition 4.21, and Lemma 4.23. Finish by proving that -invertible synthetic spectra are spectra (Theorem 4.37).
• Reference: Section 4 of Piotr’s thesis

Talk 6: The cofiber of

• Date: May 10, 2024
• Speaker: Lillianna Lipman
• Abstract: The goal of this talk is cover Sections 4.5 and 4.6 of Piotr’s thesis. Discuss the cofibers of the maps and show they are commutative algebras in . The main result of Section 4.5 is the equivalence between the categories of -modules and Hovey’s stable category of modules (Theorem 4.54). Show how to recover the -based Adams spectral sequence as in Remark 4.63.
• Reference: Section 4 of Piotr’s thesis

Talk 7: Synthetic Spectra based on

• Date: May 17, 2024
• Speaker: Noah Wisdom
• Abstract: The goal of this talk is cover Sections 5 and 6 of Piotr’s thesis. The important result of Section 5 is about the embedding of even synthetic spectra into synthetic spectra (Theorem 5.13), and the important result of Section 6 is about the structure of the synthetic dual Steenrod algebra (Theorems 6.9 and 6.10). The speaker may choose to discuss Lawson’s result.
• References: Sections 5 and 6 of Piotr’s thesis and Lawson’s Synthetic spectra are (usually) cellular

Talk 8: The Cellular Motivic Category

• Date: May 24, 2024
• Speaker: Callum Sutton
• Abstract: The goal of this talk is cover Sections 7.1, 7.2, and 7.3 of Piotr’s thesis. Introduce cellular motivic spectra and the algebraic cobordism spectrum . The main result is Theorem 7.14. If the speaker has time they should begin discussing spherical sheaves of spectra on the site of finite- projective motivic spectra
• Reference: Section 7 of Piotr’s thesis

Talk 9: The Adjunction between Cellular Motivic Spectra on and Even Synthetic Spectra Based on

• Date: May 31, 2024
• Speaker: TBD
• Abstract: The goal of this talk is cover Sections 7.3, 7.4, and 7.5 of Piotr’s thesis. Details to be filled in later.
• Reference: Section 7 of Piotr’s thesis

The previous nine talks covered Piotr’s thesis, and the remaining talks will introduce applications. We have one talk and speaker with date TBD:

Talk: Haine-Pstrągowski’s Spectral Weight Filtrations

• Date: TBD
• Speaker: Callum Sutton
• Reference: https://arxiv.org/pdf/2309.15072

The following consists of ideas for talk to happen over the summer.

Talk Idea: Pstrągowski-VanKoughnett’s Abstract Goerss-Hopkins theory

• Date: TBD
• Abstract: Introduce Goerss-Hopkins obstruction theory as stated in Theorem 1.2 of Pstrągowski-VanKoughnett as well as section 1.2 of Pstrągowski-VanKoughnett. The main results of sections 3, 4, and 5 of Pstrągowski-VanKoughnett are Theorem 3.8, Theorem 4.9, and Theorem 5.4, and the speaker could go over these for synthetic spectra. The results are then restated in the case of synthetic spectra as in Theorem 6.6.
• References: Pstrągowski-VanKoughnett’s Abstract Goerss-Hopkins theory

Talk Idea: Burklund’s Multiplicative structures on Moore Spectra

• Date: TBD
• Abstract: The goal of this talk is to give an example of the computational potential of synthetic spectra. This is the second of two talks on synthetic spectra. Go over the introduction to explain the problem of equipping Moore Spectra with structure. Discuss how synthetic spectra are used to discover multiplicative structures on Moore spectra that were previously not known as in Theorem 3.1.
• Reference: Burklund’s Multiplicative structures on moore spectra

Other Talk Ideas

• Pstrągowski’s Moduli of spaces with prescribed homotopy groups
• Burklund-Hahn-Senger’s On the boundaries of highly connected, almost closed manifolds
• Gheorghe-Isaksen-Krause-Ricka’s -motivic modular forms
• Burklund’s An extension in the Adams spectral sequence in dimension 54
• Pstrągowski’s Chromatic homotopy theory is algebraic when
• Georghe-Wang-Xu’s The special fiber of the motivic deformation of the stable homotopy category is algebraic
• Patchkoria-Pstrągowski’s Adams spectral sequences and Franke’s algebraicity conjecture
• Balderrama’s Deformations of homotopy theories via algebraic theories
• Burklund-Hahn-Senger’s Galois reconstruction of Artin-Tate -motivic spectra
• Chua’s The page of the Adams spectral sequence
• Burklund-Pstrągowski’s Quivers and the Adams spectral sequence
• Antieau’s Spherical Witt vectors and integral models for spaces
• Lawson’s Synthetic spectra are (usually) cellular
• Marek’s -synthetic homotopy groups of topological modular forms
• Carrick-Davies’s A synthetic approach to detecting -periodic families
• Baer-Johnson-Marek’s Stable Comodule Deformations and the Synthetic Adams-Novikov Spectral Sequence

Acknowledgements: The co-organizers thank the eCHT synthetic spectra reading seminar Winter 2022 organizers William Balderrama, Dan Isaksen, and Piotr Pstrągowski for their syllabus that we’ve largely copied. The co-organizers thank Francis Baer, William Balderrama, and Piotr Pstrągowski for their correspondence on the content of this seminar.