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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?

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