Formal Groups Reading Notes

Last modified date: <%+ tp.file.last_modified_date() %>



Formal Groups and Spectra

Formal Groups

Basics

  • Motivation: why should one care about formal groups?

    • Idea: substitute for Lie algebras in positive characteristic
    • Formal group and lie algebra carry the same info in char 0
    • Functor: LieGroups (or AlgGrp) → LieAlgebras factors through FormalGroups

    Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled.png

  • Motivation: if a formal group were for sale, why might you buy it?

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%201.png

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%202.png

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%203.png

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%204.png

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%205.png

  • Why is the group law called “formal”?

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%206.png

  • How “should” one think of a formal group?

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%207.png

  • What is a formal exponential and formal logarithm?

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%208.png

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%209.png

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%2010.png

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%2011.png

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%2012.png

  • What are examples of formal groups? What is a morphism? When are they isomorphic?

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%2013.png

  • What are the additive and multiplicative formal groups?

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%2014.png

  • How are the additive and multiplicative formal groups related?

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%2015.png

  • What is the height of a formal group?

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%2016.png

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%2017.png

  • How can one extend formal groups over rings R to algebras over R?

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%2018.png

  • What is the formal scheme interpretation of formal groups?

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%2019.png

  • How does iterated K-theory show up in the theory of formal groups?

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%2020.png

Other Connections to Elliptic Curves

  • What is the invariant differential of an elliptic curve?

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%2021.png

  • How does one extract the formal group of an elliptic curve using differential forms?

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%2022.png

  • How does the formal group of a curve relate to torsion and the Mordell-Weil theorem?

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%2023.png

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%2024.png

  • What can a formal group tell you about your curve?

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%2025.png

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%2026.png

Lazard Ring

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%2027.png

  • What does it mean to be a “universal” coefficient ring?

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%2028.png

  • How exactly is the Lazard ring universal?

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%2029.png

  • How are formal groups related to spectra?

  • What is MU?

  • How are formal groups related to cobordism?

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%2030.png

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%2031.png

  • What is a complex oriented cohomology theory?

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%2032.png

  • How is the Lazard ring directly related to cobordism classification?

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%2033.png

  • What is the moduli stack of formal groups?

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%2034.png

Stable Homotopy

tmf

  • How is tmf related to modular forms?

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%2035.png

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%2036.png

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%2037.png

K-Theories

  • How is the algebraic K-theory of a commutative ring defined/constructed in terms of monoidal categories?

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%2038.png

Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%2039.png

#projects/notes/reading #projects/my-talks #projects/notes/seminars #todo #homotopy/stable-homotopy #web/blog #resources/notes #resources/summaries