--- date: 2021-10-11 19:35 modification date: Thursday 12th May 2022 16:54:15 title: "Formal Groups Reading Notes" aliases: [Formal Groups Reading Notes] --- Last modified date: <%+ tp.file.last_modified_date() %> --- - Tags: - #projects/notes/reading #projects/my-talks #projects/notes/seminars #todo #homotopy/stable-homotopy #web/blog - Refs: - [2020 Juvitop Notes](https://math.mit.edu/juvitop/pastseminars/notes_2020_Spring/Levy.pdf) #resources/notes - [Survey Paper](http://people.math.harvard.edu/~lurie/papers/survey.pdf) #resources/summaries - Links: - [Formal group](Unsorted/Formal%20group.md) - [stable homotopy](Unsorted/stable%20homotopy.md) --- # Formal Groups and Spectra ## Formal Groups ### Basics - Motivation: why should one care about [formal groups](Formal%20group.md)? - 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) - What is a [formal group law](Unsorted/Formal%20group.md)? ![](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](Mordell-Weil%20theorem)? ![](Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%2023.png) - How do formal groups arise from [Projects/2022 Advanced Qual Projects/Elliptic Curves/Elliptic Curves](Projects/2022%20Advanced%20Qual%20Projects/Elliptic%20Curves/Elliptic%20Curves.md)? ![](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 - What is the [Lazard ring](Unsorted/Lazard%20ring.md)? ![](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](spectra.md)? - What is [MU](MU)? - How are formal groups related to [cobordism](cobordism.md)? ![](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](complex%20oriented%20cohomology%20theory.md)? ![](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](moduli%20stack%20of%20formal%20groups)? ![](Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%2034.png) ## Stable Homotopy ### tmf - How is [tmf](tmf.md) related to [modular forms](modular%20forms)? ![](Projects/2020%20Formal%20Groups%20and%20Spectra/attachments/Untitled%2035.png) - How is tmf defined in terms of the [moduli stack of elliptic curves](Unsorted/moduli%20stack%20of%20elliptic%20curves.md)? ![](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](algebraic%20K-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)