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# 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 • Motivation: if a formal group were for sale, why might you buy it?     • Why is the group law called “formal”? • How “should” one think of a formal group? • What is a formal exponential and formal logarithm?     • What are examples of formal groups? What is a morphism? When are they isomorphic? • What are the additive and multiplicative formal groups? • How are the additive and multiplicative formal groups related? • What is the height of a formal group?  • How can one extend formal groups over rings R to algebras over R? • What is the formal scheme interpretation of formal groups? • How does iterated K-theory show up in the theory of formal groups? ## Other Connections to Elliptic Curves

• What is the invariant differential of an elliptic curve? • How does one extract the formal group of an elliptic curve using differential forms? • How does the formal group of a curve relate to torsion and the Mordell-Weil theorem?    ## Lazard Ring • What does it mean to be a “universal” coefficient ring? • How exactly is the Lazard ring universal? • How are formal groups related to spectra?

• What is MU?

• How are formal groups related to cobordism?  • What is a complex oriented cohomology theory? • How is the Lazard ring directly related to cobordism classification? • What is the moduli stack of formal groups? ## Stable Homotopy

### tmf

• How is tmf related to modular forms?   ### K-Theories

• How is the algebraic K-theory of a commutative ring defined/constructed in terms of monoidal categories?  #projects/notes/reading #projects/my-talks #projects/notes/seminars #todo #homotopy/stable-homotopy #web/blog #resources/notes #resources/summaries