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Formal Groups and Spectra

• Notes

• Survey Paper

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?

• What is a formal group law?

• 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 R to R-algebras?

• 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?

• How do formal groups arise from Elliptic Curves?

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

Lazard Ring

• What is the 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 Motivation?

• 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?

• How is tmf defined in terms of the moduli stack of elliptic curves?

K-Theories

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