Quillen's Work on Formal Group Laws and Complex Cobordism Theory

Home , Daniel Quillen, Formal group law, Jack Morava, Michel Lazard

Quillen’s work on formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s work on Quillen’s 6 page paper Enter the formal group law formal group laws Show it is universal The Brown-Peterson and complex theorem p-typicality cobordism theory Operations in BP-theory

Complex cobordism theory after Quillen Conference Morava’s work honoring the legacy Chromatic homotopy theory Elliptic cohomology and of Daniel Quillen topological modular forms October 6-8, 2012 Conclusion

Doug Ravenel University of Rochester

1.1 Quillen’s work on The six author paper formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.2 Quillen’s work on Quillen’s cryptic and insightful masterpiece: formal group laws and complex Six pages that rocked our world cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.3 Table of contents

1. Formal group laws. 2. The formal group law of complex cobordism. 3. The universal nature of cobordism group laws. 4. Typical group laws (after Cartier). ∗ 5. Decomposition of Ω(p). (The p-local splitting of Ω = MU.) ∗ 6. Operations in ΩT . (The structure of BP∗(BP).)

Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.4 Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and Table of contents topological modular forms

Conclusion 1. Formal group laws. 2. The formal group law of complex cobordism. 3. The universal nature of cobordism group laws. 4. Typical group laws (after Cartier). ∗ 5. Decomposition of Ω(p). (The p-local splitting of Ω = MU.) ∗ 6. Operations in ΩT . (The structure of BP∗(BP).)

1.4 just as we deﬁne them today.

Deﬁnition A formal group law over a ring R is a power series F(X, Y ) ∈ R[[X, Y ]] with

F(X, 0) = F(0, X) = X F(Y , X) = F(X, Y ) F(X, F(Y , Z )) = F(F(X, Y ), Z).

Examples: • X + Y , the additive formal group law. • X + Y + XY , the multiplicative formal group law. • Euler’s addition formula for a certain elliptic integral, √ √ X 1 − Y 4 + Y 1 − X 4 ∈ Z[1/2][[X, Y ]]. 1 − X 2Y 2

Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex cobordism theory

Quillen begins by deﬁning formal group laws Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.5 Deﬁnition A formal group law over a ring R is a power series F(X, Y ) ∈ R[[X, Y ]] with

F(X, 0) = F(0, X) = X F(Y , X) = F(X, Y ) F(X, F(Y , Z )) = F(F(X, Y ), Z).

Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex cobordism theory

Quillen begins by deﬁning formal group laws just as we deﬁne Doug Ravenel

them today. Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.5 Examples: • X + Y , the additive formal group law. • X + Y + XY , the multiplicative formal group law. • Euler’s addition formula for a certain elliptic integral, √ √ X 1 − Y 4 + Y 1 − X 4 ∈ Z[1/2][[X, Y ]]. 1 − X 2Y 2

Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex cobordism theory

Quillen begins by deﬁning formal group laws just as we deﬁne Doug Ravenel

them today. Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson Deﬁnition theorem p-typicality A formal group law over a ring R is a power series Operations in BP-theory Complex cobordism F(X, Y ) ∈ R[[X, Y ]] with theory after Quillen Morava’s work Chromatic homotopy theory F(X, 0) = F(0, X) = X Elliptic cohomology and topological modular forms

F(Y , X) = F(X, Y ) Conclusion F(X, F(Y , Z )) = F(F(X, Y ), Z).

1.5 • X + Y + XY , the multiplicative formal group law. • Euler’s addition formula for a certain elliptic integral, √ √ X 1 − Y 4 + Y 1 − X 4 ∈ Z[1/2][[X, Y ]]. 1 − X 2Y 2

Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex cobordism theory

Quillen begins by deﬁning formal group laws just as we deﬁne Doug Ravenel

F(Y , X) = F(X, Y ) Conclusion F(X, F(Y , Z )) = F(F(X, Y ), Z).

Examples: • X + Y , the additive formal group law.

1.5 • Euler’s addition formula for a certain elliptic integral, √ √ X 1 − Y 4 + Y 1 − X 4 ∈ Z[1/2][[X, Y ]]. 1 − X 2Y 2

Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex cobordism theory

Quillen begins by deﬁning formal group laws just as we deﬁne Doug Ravenel

F(Y , X) = F(X, Y ) Conclusion F(X, F(Y , Z )) = F(F(X, Y ), Z).

Examples: • X + Y , the additive formal group law. • X + Y + XY , the multiplicative formal group law.

1.5 Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex cobordism theory

Quillen begins by deﬁning formal group laws just as we deﬁne Doug Ravenel

F(Y , X) = F(X, Y ) Conclusion F(X, F(Y , Z )) = F(F(X, Y ), Z).

Deﬁnition

Let L1 and L2 be complex line bundles over a space X with Conner-Floyd Chern classes

2 2 c1(L1), c1(L2) ∈ MU (X) = Ω (X)

Then the formal group law over the complex cobordism ring is

Ω F (c1(L1), c1(L2)) = c1(L1 ⊗ L2).

Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper He then deﬁnes the formal group law of complex cobordism in Enter the formal group law terms of the ﬁrst Chern class of the tensor product of two Show it is universal The Brown-Peterson complex line bundles, theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.6 Then the formal group law over the complex cobordism ring is

Ω F (c1(L1), c1(L2)) = c1(L1 ⊗ L2).

Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex cobordism theory

Doug Ravenel

Deﬁnition Complex cobordism theory after Quillen Let L1 and L2 be complex line bundles over a space X with Morava’s work Chromatic homotopy theory Conner-Floyd Chern classes Elliptic cohomology and topological modular forms 2 2 Conclusion c1(L1), c1(L2) ∈ MU (X) = Ω (X)

1.6 Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex cobordism theory

Doug Ravenel

Then the formal group law over the complex cobordism ring is

Ω F (c1(L1), c1(L2)) = c1(L1 ⊗ L2).

1.6 This mysterious statement leads to a new determination of the logarithm of the formal group law.

Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex Then he states his ﬁrst theorem. cobordism theory Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.7 This mysterious statement leads to a new determination of the logarithm of the formal group law.

Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex Then he states his ﬁrst theorem. cobordism theory Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.7 Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex Then he states his ﬁrst theorem. cobordism theory Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

This mysterious statement leads to a new determination of the logarithm of the formal group law.

1.7 Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex Then he states his ﬁrst theorem. cobordism theory Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

This mysterious statement leads to a new determination of the logarithm of the formal group law.

1.7 The logarithm `(X) of a formal group law is a power series deﬁning an isomorphism (after tensoring with the rationals) with the additive formal group law, so we have

`(F(X, Y )) = `(X) + `(Y ).

The above Corollary identiﬁes the logarithm for the formal group law associated with complex cobordism theory.

Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.8 so we have

`(F(X, Y )) = `(X) + `(Y ).

The above Corollary identiﬁes the logarithm for the formal group law associated with complex cobordism theory.

Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and The logarithm `(X) of a formal group law is a power series topological modular forms Conclusion deﬁning an isomorphism (after tensoring with the rationals) with the additive formal group law,

1.8 The above Corollary identiﬁes the logarithm for the formal group law associated with complex cobordism theory.

Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

`(F(X, Y )) = `(X) + `(Y ).

1.8 Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

`(F(X, Y )) = `(X) + `(Y ).

The above Corollary identiﬁes the logarithm for the formal group law associated with complex cobordism theory.

1.8 His proof uses two previously known facts: • Michel Lazard had determined the ring L over which the universal formal group law F L is deﬁned. The previous corollary im- plies that the map L → Ωev (pt) carrying F L to F Ω is a rational isomorphism. The target was known to be torsion free, so it sufﬁces to show the map is onto.

Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex cobordism theory

Doug Ravenel

Then he shows that the formal groups law for complex Quillen’s 6 page paper Enter the formal group law cobordism is universal. Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.9 His proof uses two previously known facts: • Michel Lazard had determined the ring L over which the universal formal group law F L is deﬁned. The previous corollary im- plies that the map L → Ωev (pt) carrying F L to F Ω is a rational isomorphism. The target was known to be torsion free, so it sufﬁces to show the map is onto.

Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex cobordism theory

Doug Ravenel

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.9 • Michel Lazard had determined the ring L over which the universal formal group law F L is deﬁned. The previous corollary im- plies that the map L → Ωev (pt) carrying F L to F Ω is a rational isomorphism. The target was known to be torsion free, so it sufﬁces to show the map is onto.

Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex cobordism theory

Doug Ravenel

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory His proof uses two previously known facts: Elliptic cohomology and topological modular forms

Conclusion

1.9 The previous corollary im- plies that the map L → Ωev (pt) carrying F L to F Ω is a rational isomorphism. The target was known to be torsion free, so it sufﬁces to show the map is onto.

Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex cobordism theory

Doug Ravenel

1.9 The target was known to be torsion free, so it sufﬁces to show the map is onto.

Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex cobordism theory

Doug Ravenel

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory His proof uses two previously known facts: Elliptic cohomology and topological modular forms • Conclusion Michel Lazard had determined the ring L over which the universal formal group law F L is deﬁned. The previous corollary im- plies that the map L → Ωev (pt) carrying F L to F Ω is a rational isomorphism.

1.9 so it sufﬁces to show the map is onto.

Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex cobordism theory

Doug Ravenel

1.9 Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex cobordism theory

Doug Ravenel

1.9 It is torsion free and generated by as a ring by the cobordism classes of the Milnor hypersurfaces, Hm,n ⊂ CPm × CPn. Hm,n is the zero locus of a bilinear function on CPm × CPn.

Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex cobordism theory

Doug Ravenel

Conclusion

1.10 Hm,n is the zero locus of a bilinear function on CPm × CPn.

Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality John Milnor Sergei Novikov Operations in BP-theory Complex cobordism • Milnor and Novikov had determined the structure of the theory after Quillen Morava’s work ring MU∗. It is torsion free and generated by as a ring by Chromatic homotopy theory Elliptic cohomology and the cobordism classes of the Milnor hypersurfaces, topological modular forms Conclusion Hm,n ⊂ CPm × CPn.

1.10 Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex cobordism theory

Doug Ravenel

1.10 Denote the latter as usual by

X i j F(X, Y ) = ai,j X Y where ai,j ∈ MU2(i+j−1) i,j≥0 X n with P(X) = PnX n≥0 = `0(X) where `(X) is the logarithm X and H(X, Y ) = [Hm,n]X mY n. m,n≥0

These are related by the formula

H(X, Y ) = P(X)P(Y )F(X, Y ),

so the cobordism class each Milnor hypersurface is deﬁned in terms of the formal group law. This is Quillen’s proof of Theorem 2.

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.11 X n with P(X) = PnX n≥0 = `0(X) where `(X) is the logarithm X and H(X, Y ) = [Hm,n]X mY n. m,n≥0

These are related by the formula

H(X, Y ) = P(X)P(Y )F(X, Y ),

so the cobordism class each Milnor hypersurface is deﬁned in terms of the formal group law. This is Quillen’s proof of Theorem 2.

Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex These imply that it sufﬁces to show that the cobordism classes cobordism theory of the Hm,n can be deﬁned in terms of the formal group law. Doug Ravenel Quillen’s 6 page paper Enter the formal group law Denote the latter as usual by Show it is universal The Brown-Peterson theorem X i j p-typicality F(X, Y ) = ai,j X Y where ai,j ∈ MU2(i+j−1) Operations in BP-theory i,j≥0 Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.11 X and H(X, Y ) = [Hm,n]X mY n. m,n≥0

These are related by the formula

H(X, Y ) = P(X)P(Y )F(X, Y ),

so the cobordism class each Milnor hypersurface is deﬁned in terms of the formal group law. This is Quillen’s proof of Theorem 2.

Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex These imply that it sufﬁces to show that the cobordism classes cobordism theory of the Hm,n can be deﬁned in terms of the formal group law. Doug Ravenel Quillen’s 6 page paper Enter the formal group law Denote the latter as usual by Show it is universal The Brown-Peterson theorem X i j p-typicality F(X, Y ) = ai,j X Y where ai,j ∈ MU2(i+j−1) Operations in BP-theory i,j≥0 Complex cobordism theory after Quillen X n Morava’s work with P(X) = PnX Chromatic homotopy theory Elliptic cohomology and n≥0 topological modular forms 0 = ` (X) where `(X) is the logarithm Conclusion

1.11 These are related by the formula

H(X, Y ) = P(X)P(Y )F(X, Y ),

so the cobordism class each Milnor hypersurface is deﬁned in terms of the formal group law. This is Quillen’s proof of Theorem 2.

Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex These imply that it sufﬁces to show that the cobordism classes cobordism theory of the Hm,n can be deﬁned in terms of the formal group law. Doug Ravenel Quillen’s 6 page paper Enter the formal group law Denote the latter as usual by Show it is universal The Brown-Peterson theorem X i j p-typicality F(X, Y ) = ai,j X Y where ai,j ∈ MU2(i+j−1) Operations in BP-theory i,j≥0 Complex cobordism theory after Quillen X n Morava’s work with P(X) = PnX Chromatic homotopy theory Elliptic cohomology and n≥0 topological modular forms 0 = ` (X) where `(X) is the logarithm Conclusion X and H(X, Y ) = [Hm,n]X mY n. m,n≥0

1.11 so the cobordism class each Milnor hypersurface is deﬁned in terms of the formal group law. This is Quillen’s proof of Theorem 2.

Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex These imply that it sufﬁces to show that the cobordism classes cobordism theory of the Hm,n can be deﬁned in terms of the formal group law. Doug Ravenel Quillen’s 6 page paper Enter the formal group law Denote the latter as usual by Show it is universal The Brown-Peterson theorem X i j p-typicality F(X, Y ) = ai,j X Y where ai,j ∈ MU2(i+j−1) Operations in BP-theory i,j≥0 Complex cobordism theory after Quillen X n Morava’s work with P(X) = PnX Chromatic homotopy theory Elliptic cohomology and n≥0 topological modular forms 0 = ` (X) where `(X) is the logarithm Conclusion X and H(X, Y ) = [Hm,n]X mY n. m,n≥0

These are related by the formula

H(X, Y ) = P(X)P(Y )F(X, Y ),

1.11 This is Quillen’s proof of Theorem 2.

Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex These imply that it sufﬁces to show that the cobordism classes cobordism theory of the Hm,n can be deﬁned in terms of the formal group law. Doug Ravenel Quillen’s 6 page paper Enter the formal group law Denote the latter as usual by Show it is universal The Brown-Peterson theorem X i j p-typicality F(X, Y ) = ai,j X Y where ai,j ∈ MU2(i+j−1) Operations in BP-theory i,j≥0 Complex cobordism theory after Quillen X n Morava’s work with P(X) = PnX Chromatic homotopy theory Elliptic cohomology and n≥0 topological modular forms 0 = ` (X) where `(X) is the logarithm Conclusion X and H(X, Y ) = [Hm,n]X mY n. m,n≥0

These are related by the formula

H(X, Y ) = P(X)P(Y )F(X, Y ),

so the cobordism class each Milnor hypersurface is deﬁned in terms of the formal group law.

1.11 Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex These imply that it sufﬁces to show that the cobordism classes cobordism theory of the Hm,n can be deﬁned in terms of the formal group law. Doug Ravenel Quillen’s 6 page paper Enter the formal group law Denote the latter as usual by Show it is universal The Brown-Peterson theorem X i j p-typicality F(X, Y ) = ai,j X Y where ai,j ∈ MU2(i+j−1) Operations in BP-theory i,j≥0 Complex cobordism theory after Quillen X n Morava’s work with P(X) = PnX Chromatic homotopy theory Elliptic cohomology and n≥0 topological modular forms 0 = ` (X) where `(X) is the logarithm Conclusion X and H(X, Y ) = [Hm,n]X mY n. m,n≥0

These are related by the formula

H(X, Y ) = P(X)P(Y )F(X, Y ),

so the cobordism class each Milnor hypersurface is deﬁned in terms of the formal group law. This is Quillen’s proof of Theorem 2. 1.11 Here there is a formal group law deﬁned in terms of Stiefel-Whitney classes instead of Chern classes.

As in the complex case, the cobordism ring is generated by real analogs of the Milnor hypersurfaces. Unlike the complex case, the tensor product square of any real line bundle is trivial.

This forces the formal group law to have characteristic 2 and satisfy the relation F(X, X) = 0.

Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex cobordism theory He proved a similar result about unoriented cobordism. Doug Ravenel Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.12 As in the complex case, the cobordism ring is generated by real analogs of the Milnor hypersurfaces. Unlike the complex case, the tensor product square of any real line bundle is trivial.

This forces the formal group law to have characteristic 2 and satisfy the relation F(X, X) = 0.

Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex cobordism theory He proved a similar result about unoriented cobordism. Here Doug Ravenel there is a formal group law deﬁned in terms of Stiefel-Whitney Quillen’s 6 page paper Enter the formal group law classes instead of Chern classes. Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.12 Unlike the complex case, the tensor product square of any real line bundle is trivial.

This forces the formal group law to have characteristic 2 and satisfy the relation F(X, X) = 0.

real analogs of the Milnor hypersurfaces. Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.12 This forces the formal group law to have characteristic 2 and satisfy the relation F(X, X) = 0.

real analogs of the Milnor hypersurfaces. Unlike the complex Complex cobordism theory after Quillen case, the tensor product square of any real line bundle is trivial. Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.12 Quillen’s work on Six pages that rocked our world (continued) formal group laws and complex cobordism theory He proved a similar result about unoriented cobordism. Here Doug Ravenel there is a formal group law deﬁned in terms of Stiefel-Whitney Quillen’s 6 page paper Enter the formal group law classes instead of Chern classes. Show it is universal The Brown-Peterson theorem p-typicality As in the complex case, the cobordism ring is generated by Operations in BP-theory

Conclusion

This forces the formal group law to have characteristic 2 and satisfy the relation F(X, X) = 0.

1.12 Quillen’s work on formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Table of contents Chromatic homotopy theory Elliptic cohomology and topological modular forms 1. Formal group laws. Conclusion 2. The formal group law of complex cobordism. 3. The universal nature of cobordism group laws. 4. Typical group laws (after Cartier). ∗ 5. Decomposition of Ω(p). (The p-local splitting of MU.) ∗ 6. Operations in ΩT . (The structure of BP∗(BP).)

1.13 In 1966 Brown and Peterson showed that after localization at a prime p, Ω (or MU) splits into a wedge of smaller spectra now known as BP and denoted by Quillen as ΩT . This splitting is sug- gested by a corresponding decomposition of H∗MU as a module over the mod p Steenrod algebra. Their theorem was a diamond in the rough.

Quillen’s work on The Brown-Peterson theorem formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.14 This splitting is sug- gested by a corresponding decomposition of H∗MU as a module over the mod p Steenrod algebra. Their theorem was a diamond in the rough.

Quillen’s work on The Brown-Peterson theorem formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory In 1966 Brown and Peterson showed Elliptic cohomology and topological modular forms

that after localization at a prime p, Ω Conclusion (or MU) splits into a wedge of smaller spectra now known as BP and denoted by Quillen as ΩT .

1.14 Their theorem was a diamond in the rough.

Quillen’s work on The Brown-Peterson theorem formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory In 1966 Brown and Peterson showed Elliptic cohomology and topological modular forms

that after localization at a prime p, Ω Conclusion (or MU) splits into a wedge of smaller spectra now known as BP and denoted by Quillen as ΩT . This splitting is sug- gested by a corresponding decomposition of H∗MU as a module over the mod p Steenrod algebra.

1.14 Quillen’s work on The Brown-Peterson theorem formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory In 1966 Brown and Peterson showed Elliptic cohomology and topological modular forms

1.14 By using some algebra developed by Pierre Cartier, he gave a much cleaner form of the splitting, thereby showing that BP is a ring spectrum.

A formal group law F over a ring R deﬁnes a group structure on the set of curves over R, meaning power series with trivial constant term. Given a curve f (X) and a positive integer n, let

n X F 1/n (Fnf )(X) = f (ζi X ), i=1

where the ζi are the nth roots of unity, and the addition on the right is deﬁned by the formal group law F.

Quillen’s work on p-typicality formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson Quillen made it sparkle! theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.15 thereby showing that BP is a ring spectrum.

A formal group law F over a ring R deﬁnes a group structure on the set of curves over R, meaning power series with trivial constant term. Given a curve f (X) and a positive integer n, let

n X F 1/n (Fnf )(X) = f (ζi X ), i=1

where the ζi are the nth roots of unity, and the addition on the right is deﬁned by the formal group law F.

Quillen’s work on p-typicality formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson Quillen made it sparkle! theorem p-typicality Operations in BP-theory

1.15 A formal group law F over a ring R deﬁnes a group structure on the set of curves over R, meaning power series with trivial constant term. Given a curve f (X) and a positive integer n, let

n X F 1/n (Fnf )(X) = f (ζi X ), i=1

where the ζi are the nth roots of unity, and the addition on the right is deﬁned by the formal group law F.

Quillen’s work on p-typicality formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson Quillen made it sparkle! theorem p-typicality Operations in BP-theory

1.15 Given a curve f (X) and a positive integer n, let

n X F 1/n (Fnf )(X) = f (ζi X ), i=1

where the ζi are the nth roots of unity, and the addition on the right is deﬁned by the formal group law F.

Quillen’s work on p-typicality formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson Quillen made it sparkle! theorem p-typicality Operations in BP-theory

Complex cobordism By using some algebra developed by Pierre theory after Quillen Morava’s work Cartier, he gave a much cleaner form of the split- Chromatic homotopy theory Elliptic cohomology and ting, thereby showing that BP is a ring spectrum. topological modular forms Conclusion A formal group law F over a ring R deﬁnes a group structure on the set of curves over R, meaning power series with trivial constant term.

1.15 where the ζi are the nth roots of unity, and the addition on the right is deﬁned by the formal group law F.

Quillen’s work on p-typicality formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson Quillen made it sparkle! theorem p-typicality Operations in BP-theory

n X F 1/n (Fnf )(X) = f (ζi X ), i=1

1.15 and the addition on the right is deﬁned by the formal group law F.

Quillen’s work on p-typicality formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson Quillen made it sparkle! theorem p-typicality Operations in BP-theory

n X F 1/n (Fnf )(X) = f (ζi X ), i=1

where the ζi are the nth roots of unity,

1.15 Quillen’s work on p-typicality formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson Quillen made it sparkle! theorem p-typicality Operations in BP-theory

n X F 1/n (Fnf )(X) = f (ζi X ), i=1

where the ζi are the nth roots of unity, and the addition on the right is deﬁned by the formal group law F.

1.15 Note that if we replace the formal sum by an ordinary one and

X j X j f (X) = fj X , then (Fnf )(X) = n fnj X . j>0 j>0

The curve f is said to be p-typical if Fqf = 0 for each prime q 6= p. In the case of ordinary summation this means that f has the form X pk f (X) = f(k)X . k≥0 The formal group law itself is said to be p-typical if the curve X is p-typical with respect to it. Over a torsion free ring, this is equivalent to the logarithm having the form above.

Quillen’s work on p-typicality (continued) formal group laws and complex cobordism theory

Doug Ravenel

n Quillen’s 6 page paper X F 1/n Enter the formal group law (Fnf )(X) = f (ζi X ). Show it is universal The Brown-Peterson i=1 theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.16 X j then (Fnf )(X) = n fnj X . j>0

Quillen’s work on p-typicality (continued) formal group laws and complex cobordism theory

Doug Ravenel

n Quillen’s 6 page paper X F 1/n Enter the formal group law (Fnf )(X) = f (ζi X ). Show it is universal The Brown-Peterson i=1 theorem p-typicality Note that if we replace the formal sum by an ordinary one and Operations in BP-theory

Complex cobordism X j theory after Quillen f (X) = fj X , Morava’s work Chromatic homotopy theory j>0 Elliptic cohomology and topological modular forms

Conclusion

1.16 The curve f is said to be p-typical if Fqf = 0 for each prime q 6= p. In the case of ordinary summation this means that f has the form X pk f (X) = f(k)X . k≥0 The formal group law itself is said to be p-typical if the curve X is p-typical with respect to it. Over a torsion free ring, this is equivalent to the logarithm having the form above.

Quillen’s work on p-typicality (continued) formal group laws and complex cobordism theory

Doug Ravenel

Complex cobordism X j X j theory after Quillen f (X) = fj X , then (Fnf )(X) = n fnj X . Morava’s work Chromatic homotopy theory j>0 j>0 Elliptic cohomology and topological modular forms

Conclusion

1.16 In the case of ordinary summation this means that f has the form X pk f (X) = f(k)X . k≥0 The formal group law itself is said to be p-typical if the curve X is p-typical with respect to it. Over a torsion free ring, this is equivalent to the logarithm having the form above.

Quillen’s work on p-typicality (continued) formal group laws and complex cobordism theory

Doug Ravenel

Complex cobordism X j X j theory after Quillen f (X) = fj X , then (Fnf )(X) = n fnj X . Morava’s work Chromatic homotopy theory j>0 j>0 Elliptic cohomology and topological modular forms

Conclusion The curve f is said to be p-typical if Fqf = 0 for each prime q 6= p.

1.16 The formal group law itself is said to be p-typical if the curve X is p-typical with respect to it. Over a torsion free ring, this is equivalent to the logarithm having the form above.

Quillen’s work on p-typicality (continued) formal group laws and complex cobordism theory

Doug Ravenel

Complex cobordism X j X j theory after Quillen f (X) = fj X , then (Fnf )(X) = n fnj X . Morava’s work Chromatic homotopy theory j>0 j>0 Elliptic cohomology and topological modular forms

Conclusion The curve f is said to be p-typical if Fqf = 0 for each prime q 6= p. In the case of ordinary summation this means that f has the form X pk f (X) = f(k)X . k≥0

1.16 Over a torsion free ring, this is equivalent to the logarithm having the form above.

Quillen’s work on p-typicality (continued) formal group laws and complex cobordism theory

Doug Ravenel

Complex cobordism X j X j theory after Quillen f (X) = fj X , then (Fnf )(X) = n fnj X . Morava’s work Chromatic homotopy theory j>0 j>0 Elliptic cohomology and topological modular forms

Conclusion The curve f is said to be p-typical if Fqf = 0 for each prime q 6= p. In the case of ordinary summation this means that f has the form X pk f (X) = f(k)X . k≥0 The formal group law itself is said to be p-typical if the curve X is p-typical with respect to it.

1.16 Quillen’s work on p-typicality (continued) formal group laws and complex cobordism theory

Doug Ravenel

Complex cobordism X j X j theory after Quillen f (X) = fj X , then (Fnf )(X) = n fnj X . Morava’s work Chromatic homotopy theory j>0 j>0 Elliptic cohomology and topological modular forms

1.16 there is a canonical coordinate change that converts any formal group law into a p-typical one.

Quillen used this to deﬁne an idempotent map ξˆ on Ω(p) = MU(p) whose telescope is ΩT = BP. This construction is much more convenient than that of Brown and Peterson.

This process changes the logarithm from

k k X [CPn]X n+1 X [CPp −1]X p to . n + 1 pk n≥0 k≥0

This new logarithm is much simpler than the old one.

An analogous construction converts the unoriented cobordism spectrum MO to the mod 2 Eilenberg-Mac Lane spectrum HZ/2.

Quillen’s work on p-typicality (continued) formal group laws and complex cobordism theory Cartier showed that when R is a Z(p)-algebra, Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.17 Quillen used this to deﬁne an idempotent map ξˆ on Ω(p) = MU(p) whose telescope is ΩT = BP. This construction is much more convenient than that of Brown and Peterson.

This process changes the logarithm from

k k X [CPn]X n+1 X [CPp −1]X p to . n + 1 pk n≥0 k≥0

This new logarithm is much simpler than the old one.

An analogous construction converts the unoriented cobordism spectrum MO to the mod 2 Eilenberg-Mac Lane spectrum HZ/2.

Quillen’s work on p-typicality (continued) formal group laws and complex cobordism theory Cartier showed that when R is a Z(p)-algebra, there is a Doug Ravenel

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.17 This construction is much more convenient than that of Brown and Peterson.

This process changes the logarithm from

k k X [CPn]X n+1 X [CPp −1]X p to . n + 1 pk n≥0 k≥0

This new logarithm is much simpler than the old one.

An analogous construction converts the unoriented cobordism spectrum MO to the mod 2 Eilenberg-Mac Lane spectrum HZ/2.

Quillen’s work on p-typicality (continued) formal group laws and complex cobordism theory Cartier showed that when R is a Z(p)-algebra, there is a Doug Ravenel

canonical coordinate change that converts any formal group Quillen’s 6 page paper law into a p-typical one. Enter the formal group law Show it is universal The Brown-Peterson theorem Quillen used this to deﬁne an idempotent map ξˆ on p-typicality Operations in BP-theory Ω = Ω = (p) MU(p) whose telescope is T BP. Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.17 This process changes the logarithm from

k k X [CPn]X n+1 X [CPp −1]X p to . n + 1 pk n≥0 k≥0

This new logarithm is much simpler than the old one.

An analogous construction converts the unoriented cobordism spectrum MO to the mod 2 Eilenberg-Mac Lane spectrum HZ/2.

Quillen’s work on p-typicality (continued) formal group laws and complex cobordism theory Cartier showed that when R is a Z(p)-algebra, there is a Doug Ravenel

canonical coordinate change that converts any formal group Quillen’s 6 page paper law into a p-typical one. Enter the formal group law Show it is universal The Brown-Peterson theorem Quillen used this to deﬁne an idempotent map ξˆ on p-typicality Operations in BP-theory Ω = Ω = (p) MU(p) whose telescope is T BP. This construction Complex cobordism is much more convenient than that of Brown and Peterson. theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.17 k k X [CPp −1]X p to . pk k≥0

This new logarithm is much simpler than the old one.

An analogous construction converts the unoriented cobordism spectrum MO to the mod 2 Eilenberg-Mac Lane spectrum HZ/2.

Quillen’s work on p-typicality (continued) formal group laws and complex cobordism theory Cartier showed that when R is a Z(p)-algebra, there is a Doug Ravenel

canonical coordinate change that converts any formal group Quillen’s 6 page paper law into a p-typical one. Enter the formal group law Show it is universal The Brown-Peterson theorem Quillen used this to deﬁne an idempotent map ξˆ on p-typicality Operations in BP-theory Ω = Ω = (p) MU(p) whose telescope is T BP. This construction Complex cobordism is much more convenient than that of Brown and Peterson. theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and This process changes the logarithm from topological modular forms Conclusion X [CPn]X n+1 n + 1 n≥0

1.17 This new logarithm is much simpler than the old one.

An analogous construction converts the unoriented cobordism spectrum MO to the mod 2 Eilenberg-Mac Lane spectrum HZ/2.

Quillen’s work on p-typicality (continued) formal group laws and complex cobordism theory Cartier showed that when R is a Z(p)-algebra, there is a Doug Ravenel

canonical coordinate change that converts any formal group Quillen’s 6 page paper law into a p-typical one. Enter the formal group law Show it is universal The Brown-Peterson theorem Quillen used this to deﬁne an idempotent map ξˆ on p-typicality Operations in BP-theory Ω = Ω = (p) MU(p) whose telescope is T BP. This construction Complex cobordism is much more convenient than that of Brown and Peterson. theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and This process changes the logarithm from topological modular forms Conclusion k k X [CPn]X n+1 X [CPp −1]X p to . n + 1 pk n≥0 k≥0

1.17 An analogous construction converts the unoriented cobordism spectrum MO to the mod 2 Eilenberg-Mac Lane spectrum HZ/2.

Quillen’s work on p-typicality (continued) formal group laws and complex cobordism theory Cartier showed that when R is a Z(p)-algebra, there is a Doug Ravenel

canonical coordinate change that converts any formal group Quillen’s 6 page paper law into a p-typical one. Enter the formal group law Show it is universal The Brown-Peterson theorem Quillen used this to deﬁne an idempotent map ξˆ on p-typicality Operations in BP-theory Ω = Ω = (p) MU(p) whose telescope is T BP. This construction Complex cobordism is much more convenient than that of Brown and Peterson. theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and This process changes the logarithm from topological modular forms Conclusion k k X [CPn]X n+1 X [CPp −1]X p to . n + 1 pk n≥0 k≥0

This new logarithm is much simpler than the old one.

1.17 Quillen’s work on p-typicality (continued) formal group laws and complex cobordism theory Cartier showed that when R is a Z(p)-algebra, there is a Doug Ravenel

canonical coordinate change that converts any formal group Quillen’s 6 page paper law into a p-typical one. Enter the formal group law Show it is universal The Brown-Peterson theorem Quillen used this to deﬁne an idempotent map ξˆ on p-typicality Operations in BP-theory Ω = Ω = (p) MU(p) whose telescope is T BP. This construction Complex cobordism is much more convenient than that of Brown and Peterson. theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and This process changes the logarithm from topological modular forms Conclusion k k X [CPn]X n+1 X [CPp −1]X p to . n + 1 pk n≥0 k≥0

This new logarithm is much simpler than the old one.

An analogous construction converts the unoriented cobordism spectrum MO to the mod 2 Eilenberg-Mac Lane spectrum HZ/2.

1.17 one needs to understand the graded algebra AE of maps from E to itself.

In favorable cases one can set up an E-based Adams spectral sequence and wonder about its E2-term. This is usually some Ext group deﬁned in terms of the algebra of operations AE .

Finding it explicitly is a daunting task.

Quillen’s work on Operations in BP-theory formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson ∗ theorem In order to get the most use out of a cohomology theory E p-typicality represented by a spectrum E, Operations in BP-theory Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.18 In favorable cases one can set up an E-based Adams spectral sequence and wonder about its E2-term. This is usually some Ext group deﬁned in terms of the algebra of operations AE .

Finding it explicitly is a daunting task.

Quillen’s work on Operations in BP-theory formal group laws and complex cobordism theory

Doug Ravenel

Conclusion

1.18 This is usually some Ext group deﬁned in terms of the algebra of operations AE .

Finding it explicitly is a daunting task.

Quillen’s work on Operations in BP-theory formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson ∗ theorem In order to get the most use out of a cohomology theory E p-typicality represented by a spectrum E, one needs to understand the Operations in BP-theory E Complex cobordism graded algebra A of maps from E to itself. theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and In favorable cases one can set up an E-based Adams spectral topological modular forms sequence and wonder about its E2-term. Conclusion

1.18 Finding it explicitly is a daunting task.

Quillen’s work on Operations in BP-theory formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson ∗ theorem In order to get the most use out of a cohomology theory E p-typicality represented by a spectrum E, one needs to understand the Operations in BP-theory E Complex cobordism graded algebra A of maps from E to itself. theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and In favorable cases one can set up an E-based Adams spectral topological modular forms sequence and wonder about its E2-term. This is usually some Conclusion Ext group deﬁned in terms of the algebra of operations AE .

1.18 Quillen’s work on Operations in BP-theory formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson ∗ theorem In order to get the most use out of a cohomology theory E p-typicality represented by a spectrum E, one needs to understand the Operations in BP-theory E Complex cobordism graded algebra A of maps from E to itself. theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and In favorable cases one can set up an E-based Adams spectral topological modular forms sequence and wonder about its E2-term. This is usually some Conclusion Ext group deﬁned in terms of the algebra of operations AE .

Finding it explicitly is a daunting task.

1.18 • For E = HZ/2, the algebra AE is the mod 2 Steenrod algebra, which has proven to be a fertile source of theorems in algebraic topology. The corresponding Ext group has been extensively studied.

210 dimensions worth of Adams Ext groups, as computed by Christian Nassau in 1999.

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.19 The corresponding Ext group has been extensively studied.

210 dimensions worth of Adams Ext groups, as computed by Christian Nassau in 1999.

Quillen’s work on Operations in BP-theory (continued) formal group laws ∗ and complex In order to get the most use out of a cohomology theory E cobordism theory represented by a spectrum E, one needs to understand the Doug Ravenel E graded algebra A of maps from E to itself. Quillen’s 6 page paper E Enter the formal group law • For E = HZ/2, the algebra A is the mod 2 Steenrod Show it is universal The Brown-Peterson algebra, which has proven to be a fertile source of theorem p-typicality theorems in algebraic topology. Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.19 210 dimensions worth of Adams Ext groups, as computed by Christian Nassau in 1999.

Conclusion

1.19 Quillen’s work on Operations in BP-theory (continued) formal group laws ∗ and complex In order to get the most use out of a cohomology theory E cobordism theory represented by a spectrum E, one needs to understand the Doug Ravenel E graded algebra A of maps from E to itself. Quillen’s 6 page paper E Enter the formal group law • For E = HZ/2, the algebra A is the mod 2 Steenrod Show it is universal The Brown-Peterson algebra, which has proven to be a fertile source of theorem p-typicality theorems in algebraic topology. The corresponding Ext Operations in BP-theory group has been extensively studied. Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

210 dimensions worth of Adams Ext groups, as computed by Christian Nassau in 1999.

1.19 • For E = MU, AE was determined in 1967 by Novikov. He found a small but extremely rich portion of its Ext group, rich enough to include the denominator of the value of the Riemann zeta function at each negative odd integer!

The color coded argument of ζ(x + iy) for −20 ≤ x ≤ 2 and −5 ≤ y ≤ 5.

Quillen’s work on Operations in BP-theory (continued) formal group laws and complex In order to get the most use out of a cohomology theory E ∗ cobordism theory Doug Ravenel represented by a spectrum E, one needs to understand the E Quillen’s 6 page paper graded algebra A of maps from E to itself. Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.20 He found a small but extremely rich portion of its Ext group, rich enough to include the denominator of the value of the Riemann zeta function at each negative odd integer!

The color coded argument of ζ(x + iy) for −20 ≤ x ≤ 2 and −5 ≤ y ≤ 5.

Quillen’s work on Operations in BP-theory (continued) formal group laws and complex In order to get the most use out of a cohomology theory E ∗ cobordism theory Doug Ravenel represented by a spectrum E, one needs to understand the E Quillen’s 6 page paper graded algebra A of maps from E to itself. Enter the formal group law Show it is universal E The Brown-Peterson • For E = MU, A was determined in 1967 by Novikov. theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.20 rich enough to include the denominator of the value of the Riemann zeta function at each negative odd integer!

The color coded argument of ζ(x + iy) for −20 ≤ x ≤ 2 and −5 ≤ y ≤ 5.

Quillen’s work on Operations in BP-theory (continued) formal group laws and complex In order to get the most use out of a cohomology theory E ∗ cobordism theory Doug Ravenel represented by a spectrum E, one needs to understand the E Quillen’s 6 page paper graded algebra A of maps from E to itself. Enter the formal group law Show it is universal E The Brown-Peterson • For E = MU, A was determined in 1967 by Novikov. He theorem found a small but extremely rich portion of its Ext group, p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.20 The color coded argument of ζ(x + iy) for −20 ≤ x ≤ 2 and −5 ≤ y ≤ 5.

Quillen’s work on Operations in BP-theory (continued) formal group laws and complex In order to get the most use out of a cohomology theory E ∗ cobordism theory Doug Ravenel represented by a spectrum E, one needs to understand the E Quillen’s 6 page paper graded algebra A of maps from E to itself. Enter the formal group law Show it is universal E The Brown-Peterson • For E = MU, A was determined in 1967 by Novikov. He theorem found a small but extremely rich portion of its Ext group, p-typicality Operations in BP-theory

rich enough to include the denominator of the value of the Complex cobordism Riemann zeta function at each negative odd integer! theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.20 Quillen’s work on Operations in BP-theory (continued) formal group laws and complex In order to get the most use out of a cohomology theory E ∗ cobordism theory Doug Ravenel represented by a spectrum E, one needs to understand the E Quillen’s 6 page paper graded algebra A of maps from E to itself. Enter the formal group law Show it is universal E The Brown-Peterson • For E = MU, A was determined in 1967 by Novikov. He theorem found a small but extremely rich portion of its Ext group, p-typicality Operations in BP-theory

Conclusion

The color coded argument of ζ(x + iy) for −20 ≤ x ≤ 2 and −5 ≤ y ≤ 5.

1.20 • For E = BP, AE was determined by Quillen. The details are too technical for this talk. He gave a precise description in less than two pages, with little indication of proof.

The resulting Ext group is the same as Novikov’s localized at the prime p. The underlying formulas are easier to use than Novikov’s, once one knows how to use them.

It took the rest of us about 5 years to ﬁgure that out.

Quillen’s work on Operations in BP-theory (continued) formal group laws and complex cobordism theory

Doug Ravenel

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.21 The details are too technical for this talk. He gave a precise description in less than two pages, with little indication of proof.

The resulting Ext group is the same as Novikov’s localized at the prime p. The underlying formulas are easier to use than Novikov’s, once one knows how to use them.

It took the rest of us about 5 years to ﬁgure that out.

Quillen’s work on Operations in BP-theory (continued) formal group laws and complex cobordism theory

Doug Ravenel

Conclusion

1.21 He gave a precise description in less than two pages, with little indication of proof.

The resulting Ext group is the same as Novikov’s localized at the prime p. The underlying formulas are easier to use than Novikov’s, once one knows how to use them.

It took the rest of us about 5 years to ﬁgure that out.

Quillen’s work on Operations in BP-theory (continued) formal group laws and complex cobordism theory

Doug Ravenel

Conclusion

1.21 with little indication of proof.

The resulting Ext group is the same as Novikov’s localized at the prime p. The underlying formulas are easier to use than Novikov’s, once one knows how to use them.

It took the rest of us about 5 years to ﬁgure that out.

Quillen’s work on Operations in BP-theory (continued) formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper ∗ In order to get the most use out of a cohomology theory E Enter the formal group law Show it is universal represented by a spectrum E, one needs to understand the The Brown-Peterson E theorem graded algebra A of maps from E to itself. p-typicality Operations in BP-theory E • For E = BP, A was determined by Quillen. The details Complex cobordism theory after Quillen are too technical for this talk. He gave a precise Morava’s work Chromatic homotopy theory description in less than two pages, Elliptic cohomology and topological modular forms

Conclusion

1.21 The resulting Ext group is the same as Novikov’s localized at the prime p. The underlying formulas are easier to use than Novikov’s, once one knows how to use them.

It took the rest of us about 5 years to ﬁgure that out.

Quillen’s work on Operations in BP-theory (continued) formal group laws and complex cobordism theory

Doug Ravenel

1.21 The underlying formulas are easier to use than Novikov’s, once one knows how to use them.

It took the rest of us about 5 years to ﬁgure that out.

Quillen’s work on Operations in BP-theory (continued) formal group laws and complex cobordism theory

Doug Ravenel

1.21 once one knows how to use them.

It took the rest of us about 5 years to ﬁgure that out.

Quillen’s work on Operations in BP-theory (continued) formal group laws and complex cobordism theory

Doug Ravenel

1.21 It took the rest of us about 5 years to ﬁgure that out.

Quillen’s work on Operations in BP-theory (continued) formal group laws and complex cobordism theory

Doug Ravenel

1.21 Quillen’s work on Operations in BP-theory (continued) formal group laws and complex cobordism theory

Doug Ravenel

It took the rest of us about 5 years to ﬁgure that out.

1.21 Frank Adams did it for him.

J.F. Adams 1930-1989 Adams explained all the proofs with great care. His book be- came the deﬁnitive reference for Quillen’s results.

He also introduced a very helpful but counterintuitive point of view. Instead of studying the algebra ABP , one should com- pute in terms of its suitably deﬁned linear dual.

Quillen’s formal variables ti are crying out to be lo- cated in BP∗(BP). This proved to be a huge technical simpliﬁcation.

Quillen’s work on Complex cobordism theory after Quillen formal group laws and complex cobordism theory

Quillen never wrote a detailed account of his Doug Ravenel

6 page 1969 Bulletin announcement. Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.22 J.F. Adams 1930-1989 Adams explained all the proofs with great care. His book be- came the deﬁnitive reference for Quillen’s results.

He also introduced a very helpful but counterintuitive point of view. Instead of studying the algebra ABP , one should com- pute in terms of its suitably deﬁned linear dual.

Quillen’s formal variables ti are crying out to be lo- cated in BP∗(BP). This proved to be a huge technical simpliﬁcation.

Quillen’s work on Complex cobordism theory after Quillen formal group laws and complex cobordism theory

Quillen never wrote a detailed account of his Doug Ravenel

6 page 1969 Bulletin announcement. Frank Quillen’s 6 page paper Adams did it for him. Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.22 Adams explained all the proofs with great care. His book be- came the deﬁnitive reference for Quillen’s results.

He also introduced a very helpful but counterintuitive point of view. Instead of studying the algebra ABP , one should com- pute in terms of its suitably deﬁned linear dual.

Quillen’s formal variables ti are crying out to be lo- cated in BP∗(BP). This proved to be a huge technical simpliﬁcation.

Quillen’s work on Complex cobordism theory after Quillen formal group laws and complex cobordism theory

Quillen never wrote a detailed account of his Doug Ravenel

6 page 1969 Bulletin announcement. Frank Quillen’s 6 page paper Adams did it for him. Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen J.F. Adams Morava’s work 1930-1989 Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.22 His book be- came the deﬁnitive reference for Quillen’s results.

He also introduced a very helpful but counterintuitive point of view. Instead of studying the algebra ABP , one should com- pute in terms of its suitably deﬁned linear dual.

Quillen’s formal variables ti are crying out to be lo- cated in BP∗(BP). This proved to be a huge technical simpliﬁcation.

Quillen’s work on Complex cobordism theory after Quillen formal group laws and complex cobordism theory

Quillen never wrote a detailed account of his Doug Ravenel

6 page 1969 Bulletin announcement. Frank Quillen’s 6 page paper Adams did it for him. Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen J.F. Adams Morava’s work 1930-1989 Chromatic homotopy theory Elliptic cohomology and topological modular forms Adams explained all the proofs with great care. Conclusion

1.22 He also introduced a very helpful but counterintuitive point of view. Instead of studying the algebra ABP , one should com- pute in terms of its suitably deﬁned linear dual.

Quillen’s formal variables ti are crying out to be lo- cated in BP∗(BP). This proved to be a huge technical simpliﬁcation.

Quillen’s work on Complex cobordism theory after Quillen formal group laws and complex cobordism theory

Quillen never wrote a detailed account of his Doug Ravenel

6 page 1969 Bulletin announcement. Frank Quillen’s 6 page paper Adams did it for him. Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

1.22 Instead of studying the algebra ABP , one should com- pute in terms of its suitably deﬁned linear dual.

Quillen’s formal variables ti are crying out to be lo- cated in BP∗(BP). This proved to be a huge technical simpliﬁcation.

Quillen’s work on Complex cobordism theory after Quillen formal group laws and complex cobordism theory

Quillen never wrote a detailed account of his Doug Ravenel

6 page 1969 Bulletin announcement. Frank Quillen’s 6 page paper Adams did it for him. Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

He also introduced a very helpful but counterintuitive point of view.

1.22 one should com- pute in terms of its suitably deﬁned linear dual.

Quillen’s formal variables ti are crying out to be lo- cated in BP∗(BP). This proved to be a huge technical simpliﬁcation.

Quillen’s work on Complex cobordism theory after Quillen formal group laws and complex cobordism theory

Quillen never wrote a detailed account of his Doug Ravenel

6 page 1969 Bulletin announcement. Frank Quillen’s 6 page paper Adams did it for him. Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

He also introduced a very helpful but counterintuitive point of view. Instead of studying the algebra ABP ,

1.22 Quillen’s formal variables ti are crying out to be lo- cated in BP∗(BP). This proved to be a huge technical simpliﬁcation.

Quillen’s work on Complex cobordism theory after Quillen formal group laws and complex cobordism theory

Quillen never wrote a detailed account of his Doug Ravenel

6 page 1969 Bulletin announcement. Frank Quillen’s 6 page paper Adams did it for him. Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

He also introduced a very helpful but counterintuitive point of view. Instead of studying the algebra ABP , one should com- pute in terms of its suitably deﬁned linear dual.

1.22 This proved to be a huge technical simpliﬁcation.

Quillen’s work on Complex cobordism theory after Quillen formal group laws and complex cobordism theory

Quillen never wrote a detailed account of his Doug Ravenel

6 page 1969 Bulletin announcement. Frank Quillen’s 6 page paper Adams did it for him. Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

He also introduced a very helpful but counterintuitive point of view. Instead of studying the algebra ABP , one should com- pute in terms of its suitably deﬁned linear dual.

Quillen’s formal variables ti are crying out to be lo- cated in BP∗(BP).

1.22 Quillen’s work on Complex cobordism theory after Quillen formal group laws and complex cobordism theory

Quillen never wrote a detailed account of his Doug Ravenel

6 page 1969 Bulletin announcement. Frank Quillen’s 6 page paper Adams did it for him. Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

He also introduced a very helpful but counterintuitive point of view. Instead of studying the algebra ABP , one should com- pute in terms of its suitably deﬁned linear dual.

Quillen’s formal variables ti are crying out to be lo- cated in BP∗(BP). This proved to be a huge technical simpliﬁcation.

1.22 Homotopy theorists have been expanding that bridge ever since.

The BP Bridge in Chicago, where Adams lectured on Quillen’s work in 1971.

Quillen’s work on Complex cobordism theory after Quillen (continued) formal group laws and complex cobordism theory

Quillen’s work was a bridge connecting algebraic topology with Doug Ravenel

algebraic geometry and number theory. Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.23 The BP Bridge in Chicago, where Adams lectured on Quillen’s work in 1971.

Quillen’s work on Complex cobordism theory after Quillen (continued) formal group laws and complex cobordism theory

Quillen’s work was a bridge connecting algebraic topology with Doug Ravenel

algebraic geometry and number theory. Quillen’s 6 page paper Enter the formal group law Show it is universal Homotopy theorists have been expanding that bridge ever The Brown-Peterson theorem since. p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.23 Quillen’s work on Complex cobordism theory after Quillen (continued) formal group laws and complex cobordism theory

Quillen’s work was a bridge connecting algebraic topology with Doug Ravenel

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

The BP Bridge in Chicago, where Adams lectured on Quillen’s work in 1971.

1.23 In the early 1970s Morava applied deeper results (due mostly to Lazard) from the theory of formal group laws to algebraic topology. These included • A classification of formal group laws over the algebraic closure of the field Fp. There is a complete isomorphism invariant called the height, which can be any positive integer or infinity. • The automorphism group of a height n formal group law is a certain p-adic Lie group. It is now known as the Morava stabilizer group Sn. • He defined a cohomology theory associated with height n formal group laws now known as Morava K-theory K (n).

Quillen’s work on Expanding the bridge: the work of Jack Morava formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.24 These included • A classification of formal group laws over the algebraic closure of the field Fp. There is a complete isomorphism invariant called the height, which can be any positive integer or infinity. • The automorphism group of a height n formal group law is a certain p-adic Lie group. It is now known as the Morava stabilizer group Sn. • He defined a cohomology theory associated with height n formal group laws now known as Morava K-theory K (n).

Quillen’s work on Expanding the bridge: the work of Jack Morava formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen In the early 1970s Morava applied deeper results (due mostly Morava’s work Chromatic homotopy theory to Lazard) from the theory of formal group laws to algebraic Elliptic cohomology and topological modular forms

topology. Conclusion

1.24 There is a complete isomorphism invariant called the height, which can be any positive integer or inﬁnity. • The automorphism group of a height n formal group law is a certain p-adic Lie group. It is now known as the Morava stabilizer group Sn. • He deﬁned a cohomology theory associated with height n formal group laws now known as Morava K-theory K (n).

Quillen’s work on Expanding the bridge: the work of Jack Morava formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

topology. These included Conclusion • A classiﬁcation of formal group laws over the algebraic closure of the ﬁeld Fp.

1.24 which can be any positive integer or inﬁnity. • The automorphism group of a height n formal group law is a certain p-adic Lie group. It is now known as the Morava stabilizer group Sn. • He deﬁned a cohomology theory associated with height n formal group laws now known as Morava K-theory K (n).

Quillen’s work on Expanding the bridge: the work of Jack Morava formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

topology. These included Conclusion • A classiﬁcation of formal group laws over the algebraic closure of the ﬁeld Fp. There is a complete isomorphism invariant called the height,

1.24 • The automorphism group of a height n formal group law is a certain p-adic Lie group. It is now known as the Morava stabilizer group Sn. • He deﬁned a cohomology theory associated with height n formal group laws now known as Morava K-theory K (n).

Quillen’s work on Expanding the bridge: the work of Jack Morava formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

1.24 It is now known as the Morava stabilizer group Sn. • He deﬁned a cohomology theory associated with height n formal group laws now known as Morava K-theory K (n).

Quillen’s work on Expanding the bridge: the work of Jack Morava formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

1.24 • He deﬁned a cohomology theory associated with height n formal group laws now known as Morava K-theory K (n).

Quillen’s work on Expanding the bridge: the work of Jack Morava formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

1.24 Quillen’s work on Expanding the bridge: the work of Jack Morava formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

topology. These included Conclusion • A classification of formal group laws over the algebraic closure of the field Fp. There is a complete isomorphism invariant called the height, which can be any positive integer or infinity. • The automorphism group of a height n formal group law is a certain p-adic Lie group. It is now known as the Morava stabilizer group Sn. • He defined a cohomology theory associated with height n formal group laws now known as Morava K-theory K (n). 1.24 It is now known as the moduli stack of formal group laws.

After passage to characteristic p, the orbits under this action are isomorphism classes of formal group laws as classiﬁed by Lazard.

The Zariski closures of these orbits form a nested sequence of afﬁne subspaces of the afﬁne variety.

The isotropy group of the height n orbit is the height n automorphism group Sn, hence the name stabilizer group.

Quillen’s work on The work of Jack Morava (continued) formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen He also studied the afﬁne variety Spec(MU ) and deﬁned an Morava’s work ∗ Chromatic homotopy theory action on it by a group of power series substitutions. Elliptic cohomology and topological modular forms

Conclusion

1.25 After passage to characteristic p, the orbits under this action are isomorphism classes of formal group laws as classiﬁed by Lazard.

The Zariski closures of these orbits form a nested sequence of afﬁne subspaces of the afﬁne variety.

The isotropy group of the height n orbit is the height n automorphism group Sn, hence the name stabilizer group.

Quillen’s work on The work of Jack Morava (continued) formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

known as the moduli stack of formal group laws. Conclusion

1.25 The Zariski closures of these orbits form a nested sequence of afﬁne subspaces of the afﬁne variety.

The isotropy group of the height n orbit is the height n automorphism group Sn, hence the name stabilizer group.

Quillen’s work on The work of Jack Morava (continued) formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

known as the moduli stack of formal group laws. Conclusion

After passage to characteristic p, the orbits under this action are isomorphism classes of formal group laws as classiﬁed by Lazard.

1.25 The isotropy group of the height n orbit is the height n automorphism group Sn, hence the name stabilizer group.

Quillen’s work on The work of Jack Morava (continued) formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

known as the moduli stack of formal group laws. Conclusion

After passage to characteristic p, the orbits under this action are isomorphism classes of formal group laws as classiﬁed by Lazard.

The Zariski closures of these orbits form a nested sequence of afﬁne subspaces of the afﬁne variety.

1.25 hence the name stabilizer group.

Quillen’s work on The work of Jack Morava (continued) formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

known as the moduli stack of formal group laws. Conclusion

After passage to characteristic p, the orbits under this action are isomorphism classes of formal group laws as classiﬁed by Lazard.

The Zariski closures of these orbits form a nested sequence of afﬁne subspaces of the afﬁne variety.

The isotropy group of the height n orbit is the height n automorphism group Sn, 1.25 Quillen’s work on The work of Jack Morava (continued) formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

known as the moduli stack of formal group laws. Conclusion

After passage to characteristic p, the orbits under this action are isomorphism classes of formal group laws as classiﬁed by Lazard.

The Zariski closures of these orbits form a nested sequence of afﬁne subspaces of the afﬁne variety.

The isotropy group of the height n orbit is the height n automorphism group Sn, hence the name stabilizer group. 1.25 In the late 1970s Haynes Miller, Steve Wilson and I showed that Morava’s stratification leads to a nice filtration of the Adams-Novikov E2- term. In the early 80s we learned that the stable homotopy category itself possesses a filtration similar to the one found by Morava in Spec(MU∗). A key technical tool in defining it is Bousfield localization. Pete Bousfield As in the algebraic case, for each positive integer n, there is a layer of the stable homotopy category (localized at a prime p) related to height n formal group laws.

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.26 In the early 80s we learned that the stable homotopy category itself possesses a filtration similar to the one found by Morava in Spec(MU∗). A key technical tool in defining it is Bousfield localization. Pete Bousfield As in the algebraic case, for each positive integer n, there is a layer of the stable homotopy category (localized at a prime p) related to height n formal group laws.

Conclusion

1.26 A key technical tool in defining it is Bousfield localization. Pete Bousfield As in the algebraic case, for each positive integer n, there is a layer of the stable homotopy category (localized at a prime p) related to height n formal group laws.

Quillen’s work on Chromatic homotopy theory formal group laws and complex Morava’s insights led to the for- cobordism theory Doug Ravenel mulation of the chromatic point of Quillen’s 6 page paper view in stable homotopy theory. In Enter the formal group law Show it is universal the late 1970s Haynes Miller, Steve The Brown-Peterson theorem Wilson and I showed that Morava’s p-typicality stratification leads to a nice fil- Operations in BP-theory Complex cobordism tration of the Adams-Novikov E2- theory after Quillen term. Morava’s work Chromatic homotopy theory Elliptic cohomology and In the early 80s we learned that topological modular forms the stable homotopy category it- Conclusion self possesses a filtration similar to the one found by Morava in Spec(MU∗).

1.26 As in the algebraic case, for each positive integer n, there is a layer of the stable homotopy category (localized at a prime p) related to height n formal group laws.

1.26 there is a layer of the stable homotopy category (localized at a prime p) related to height n formal group laws.

1.26 related to height n formal group laws.

1.26 Quillen’s work on Chromatic homotopy theory formal group laws and complex Morava’s insights led to the for- cobordism theory Doug Ravenel mulation of the chromatic point of Quillen’s 6 page paper view in stable homotopy theory. In Enter the formal group law Show it is universal the late 1970s Haynes Miller, Steve The Brown-Peterson theorem Wilson and I showed that Morava’s p-typicality stratification leads to a nice fil- Operations in BP-theory Complex cobordism tration of the Adams-Novikov E2- theory after Quillen term. Morava’s work Chromatic homotopy theory Elliptic cohomology and In the early 80s we learned that topological modular forms the stable homotopy category it- Conclusion self possesses a filtration similar to the one found by Morava in Spec(MU∗). A key technical tool in defining it is Bousfield localization. Pete Bousfield As in the algebraic case, for each positive integer n, there is a layer of the stable homotopy category (localized at a prime p) related to height n formal group laws.

1.26 Homotopy groups of objects in the nth layer tend to repeat themselves every 2pk (pn − 1) dimensions for various k. This is known as vn-periodicity. The term chromatic refers to this separation into varying fre- quencies.

The ﬁrst known example of this phenomenon was the Bott Periodicity Theorem of 1956, describing the homotopy of the stable uni- tary and orthogonal groups. It is an example of v1-periodicity. Raoul Bott 1923-2005

Quillen’s work on Chromatic homotopy theory (continued) formal group laws and complex cobordism theory

Doug Ravenel

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.27 This is known as vn-periodicity. The term chromatic refers to this separation into varying fre- quencies.

Quillen’s work on Chromatic homotopy theory (continued) formal group laws and complex cobordism theory

Doug Ravenel

Conclusion

1.27 The term chromatic refers to this separation into varying fre- quencies.

Quillen’s work on Chromatic homotopy theory (continued) formal group laws and complex cobordism theory

Doug Ravenel

Conclusion

1.27 The ﬁrst known example of this phenomenon was the Bott Periodicity Theorem of 1956, describing the homotopy of the stable uni- tary and orthogonal groups. It is an example of v1-periodicity. Raoul Bott 1923-2005

Quillen’s work on Chromatic homotopy theory (continued) formal group laws and complex cobordism theory

Doug Ravenel

Roughly speaking, the structure of the nth layer is controlled Quillen’s 6 page paper by the cohomology of the nth Morava stabilizer group Sn. Enter the formal group law Show it is universal The Brown-Peterson Homotopy groups of objects in the nth layer tend to repeat theorem k n p-typicality themselves every 2p (p − 1) dimensions for various k. This Operations in BP-theory is known as v -periodicity. Complex cobordism n theory after Quillen Morava’s work The term chromatic refers to this separation into varying fre- Chromatic homotopy theory Elliptic cohomology and quencies. topological modular forms Conclusion

1.27 describing the homotopy of the stable uni- tary and orthogonal groups. It is an example of v1-periodicity. Raoul Bott 1923-2005

Quillen’s work on Chromatic homotopy theory (continued) formal group laws and complex cobordism theory

Doug Ravenel

1.27 It is an example of v1-periodicity. Raoul Bott 1923-2005

Quillen’s work on Chromatic homotopy theory (continued) formal group laws and complex cobordism theory

Doug Ravenel

1.27 Quillen’s work on Chromatic homotopy theory (continued) formal group laws and complex cobordism theory

Doug Ravenel

1.27 Research on them in the 1950s and 60s indicated a very disorganized picture, a zoo of erratic phenomena. It was seen then to contain one systematic pattern related to Bott periodicity. The known homotopy groups of the stable orthogonal group mapped to the unknown stable homotopy groups of spheres by the Hopf-Whitehead J-homomorphism.

Its image was determined by Adams. It contained the rich arithmetic structure detected by the Novikov calculation referred to earlier.

Quillen’s work on Chromatic homotopy theory (continued) formal group laws and complex cobordism theory

Doug Ravenel

The motivating problem behind this work was understanding Quillen’s 6 page paper the stable homotopy groups of spheres. Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.28 It was seen then to contain one systematic pattern related to Bott periodicity. The known homotopy groups of the stable orthogonal group mapped to the unknown stable homotopy groups of spheres by the Hopf-Whitehead J-homomorphism.

Its image was determined by Adams. It contained the rich arithmetic structure detected by the Novikov calculation referred to earlier.

Quillen’s work on Chromatic homotopy theory (continued) formal group laws and complex cobordism theory

Doug Ravenel

The motivating problem behind this work was understanding Quillen’s 6 page paper the stable homotopy groups of spheres. Research on them Enter the formal group law Show it is universal in the 1950s and 60s indicated a very disorganized picture, a The Brown-Peterson theorem zoo of erratic phenomena. p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.28 The known homotopy groups of the stable orthogonal group mapped to the unknown stable homotopy groups of spheres by the Hopf-Whitehead J-homomorphism.

Its image was determined by Adams. It contained the rich arithmetic structure detected by the Novikov calculation referred to earlier.

Quillen’s work on Chromatic homotopy theory (continued) formal group laws and complex cobordism theory

Doug Ravenel

Complex cobordism It was seen then to contain one systematic pattern related to theory after Quillen Bott periodicity. Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.28 Its image was determined by Adams. It contained the rich arithmetic structure detected by the Novikov calculation referred to earlier.

Quillen’s work on Chromatic homotopy theory (continued) formal group laws and complex cobordism theory

Doug Ravenel

Complex cobordism It was seen then to contain one systematic pattern related to theory after Quillen Bott periodicity. The known homotopy groups of the stable Morava’s work Chromatic homotopy theory orthogonal group mapped to the unknown stable homotopy Elliptic cohomology and topological modular forms

groups of spheres by the Hopf-Whitehead J-homomorphism. Conclusion

1.28 It contained the rich arithmetic structure detected by the Novikov calculation referred to earlier.

Quillen’s work on Chromatic homotopy theory (continued) formal group laws and complex cobordism theory

Doug Ravenel

groups of spheres by the Hopf-Whitehead J-homomorphism. Conclusion

Its image was determined by Adams.

1.28 Quillen’s work on Chromatic homotopy theory (continued) formal group laws and complex cobordism theory

Doug Ravenel

groups of spheres by the Hopf-Whitehead J-homomorphism. Conclusion

Its image was determined by Adams. It contained the rich arithmetic structure detected by the Novikov calculation referred to earlier.

1.28 The aim of chromatic theory was ﬁnd a uniﬁed framework for such patterns. It was very successful. A milestone result in it is the Nilpotence Theorem of Ethan Devinatz, Mike Hopkins and Jeff Smith, proved in 1985.

Of this result Adams said At one time it seemed that homotopy theory was ut- terly without system; now it is almost proved that systematic effects predominate.

Quillen’s work on Chromatic homotopy theory (continued) formal group laws and complex cobordism theory

In the early 1970s some more Doug Ravenel

systematic patterns were found Quillen’s 6 page paper Enter the formal group law independently by Larry Smith Show it is universal The Brown-Peterson and Hirosi Toda. theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.29 It was very successful. A milestone result in it is the Nilpotence Theorem of Ethan Devinatz, Mike Hopkins and Jeff Smith, proved in 1985.

Of this result Adams said At one time it seemed that homotopy theory was ut- terly without system; now it is almost proved that systematic effects predominate.

Quillen’s work on Chromatic homotopy theory (continued) formal group laws and complex cobordism theory

In the early 1970s some more Doug Ravenel

The aim of chromatic theory was ﬁnd a uniﬁed framework for Complex cobordism theory after Quillen such patterns. Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.29 A milestone result in it is the Nilpotence Theorem of Ethan Devinatz, Mike Hopkins and Jeff Smith, proved in 1985.

Of this result Adams said At one time it seemed that homotopy theory was ut- terly without system; now it is almost proved that systematic effects predominate.

Quillen’s work on Chromatic homotopy theory (continued) formal group laws and complex cobordism theory

In the early 1970s some more Doug Ravenel

The aim of chromatic theory was ﬁnd a uniﬁed framework for Complex cobordism theory after Quillen such patterns. It was very successful. Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.29 Of this result Adams said At one time it seemed that homotopy theory was ut- terly without system; now it is almost proved that systematic effects predominate.

Quillen’s work on Chromatic homotopy theory (continued) formal group laws and complex cobordism theory

In the early 1970s some more Doug Ravenel

The aim of chromatic theory was ﬁnd a uniﬁed framework for Complex cobordism theory after Quillen such patterns. It was very successful. A milestone result in it Morava’s work is the Nilpotence Theorem of Ethan Devinatz, Mike Hopkins Chromatic homotopy theory Elliptic cohomology and and Jeff Smith, proved in 1985. topological modular forms Conclusion

1.29 At one time it seemed that homotopy theory was ut- terly without system; now it is almost proved that systematic effects predominate.

Quillen’s work on Chromatic homotopy theory (continued) formal group laws and complex cobordism theory

In the early 1970s some more Doug Ravenel

Of this result Adams said

1.29 now it is almost proved that systematic effects predominate.

Quillen’s work on Chromatic homotopy theory (continued) formal group laws and complex cobordism theory

In the early 1970s some more Doug Ravenel

Of this result Adams said At one time it seemed that homotopy theory was ut- terly without system;

1.29 Quillen’s work on Chromatic homotopy theory (continued) formal group laws and complex cobordism theory

In the early 1970s some more Doug Ravenel

Of this result Adams said At one time it seemed that homotopy theory was ut- terly without system; now it is almost proved that systematic effects predominate.

1.29 Every elliptic curve over a ring R has a formal group law attached to it. This means there is a homomorphism to R from MU∗. It is known that its mod p reduction (for any prime p) of this formal group law has height 1 or 2.

This led to the deﬁnition of the elliptic genus by Serge Ocha- nine in 1984 and the deﬁnition of elliptic cohomology by Peter Landweber, Bob Stong and me a few years later. Attempts to interpret the former analytically have been made by Ed Wit- ten, Stephan Stolz and Peter Teichner.

Quillen’s work on Elliptic cohomology and topological modular forms formal group laws and complex cobordism theory

For over a century elliptic curves have stood Doug Ravenel at the center of mathematics. Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.30 This means there is a homomorphism to R from MU∗. It is known that its mod p reduction (for any prime p) of this formal group law has height 1 or 2.

Quillen’s work on Elliptic cohomology and topological modular forms formal group laws and complex cobordism theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.30 It is known that its mod p reduction (for any prime p) of this formal group law has height 1 or 2.

Quillen’s work on Elliptic cohomology and topological modular forms formal group laws and complex cobordism theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.30 has height 1 or 2.

Quillen’s work on Elliptic cohomology and topological modular forms formal group laws and complex cobordism theory

Conclusion

1.30 This led to the deﬁnition of the elliptic genus by Serge Ocha- nine in 1984 and the deﬁnition of elliptic cohomology by Peter Landweber, Bob Stong and me a few years later. Attempts to interpret the former analytically have been made by Ed Wit- ten, Stephan Stolz and Peter Teichner.

Quillen’s work on Elliptic cohomology and topological modular forms formal group laws and complex cobordism theory

Conclusion

1.30 and the deﬁnition of elliptic cohomology by Peter Landweber, Bob Stong and me a few years later. Attempts to interpret the former analytically have been made by Ed Wit- ten, Stephan Stolz and Peter Teichner.

Quillen’s work on Elliptic cohomology and topological modular forms formal group laws and complex cobordism theory

For over a century elliptic curves have stood Doug Ravenel at the center of mathematics. Every elliptic Quillen’s 6 page paper curve over a ring R has a formal group law Enter the formal group law Show it is universal attached to it. This means there is a homo- The Brown-Peterson theorem morphism to R from MU∗. It is known that p-typicality its mod p reduction (for any prime p) of this Operations in BP-theory Complex cobordism formal group law has height 1 or 2. theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and This led to the deﬁnition of the elliptic genus by Serge Ocha- topological modular forms nine in 1984 Conclusion

1.30 Attempts to interpret the former analytically have been made by Ed Wit- ten, Stephan Stolz and Peter Teichner.

Quillen’s work on Elliptic cohomology and topological modular forms formal group laws and complex cobordism theory

1.30 Quillen’s work on Elliptic cohomology and topological modular forms formal group laws and complex cobordism theory

1.30 The main players here are Mike Hopkins, Haynes Miller, Paul Goerss and Jacob Lurie.

Algebraic geometers study objects like elliptic curves by looking at moduli spaces for them. Roughly speaking, the moduli space (or stack) ME`` for elliptic curves is a topological space with an elliptic curve attached to each point. The theory of elliptic curves is in a certain sense controlled by the geometry of this space.

Quillen’s work on Elliptic cohomology and topological modular forms formal group laws and complex (continued) cobordism theory

Doug Ravenel

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.31 Algebraic geometers study objects like elliptic curves by looking at moduli spaces for them. Roughly speaking, the moduli space (or stack) ME`` for elliptic curves is a topological space with an elliptic curve attached to each point. The theory of elliptic curves is in a certain sense controlled by the geometry of this space.

Quillen’s work on Elliptic cohomology and topological modular forms formal group laws and complex (continued) cobordism theory

Doug Ravenel

Quillen’s 6 page paper A deeper study of the connection between ellitpic curves and Enter the formal group law Show it is universal algebraic topology led to the theory of topological modular The Brown-Peterson theorem forms in the past decade. The main players here are Mike p-typicality Hopkins, Haynes Miller, Paul Goerss and Jacob Lurie. Operations in BP-theory Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.31 Roughly speaking, the moduli space (or stack) ME`` for elliptic curves is a topological space with an elliptic curve attached to each point. The theory of elliptic curves is in a certain sense controlled by the geometry of this space.

Quillen’s work on Elliptic cohomology and topological modular forms formal group laws and complex (continued) cobordism theory

Doug Ravenel

Conclusion

Algebraic geometers study objects like elliptic curves by looking at moduli spaces for them.

1.31 The theory of elliptic curves is in a certain sense controlled by the geometry of this space.

Quillen’s work on Elliptic cohomology and topological modular forms formal group laws and complex (continued) cobordism theory

Doug Ravenel

Conclusion

1.31 Quillen’s work on Elliptic cohomology and topological modular forms formal group laws and complex (continued) cobordism theory

Doug Ravenel

Conclusion

1.31 This collection is called a sheaf of rings OE`` over ME``.

Such a sheaf has a ring of global sections Γ(OE``), which encodes a lot of useful information. Elements of this ring are closely related to modular forms, which are complex analytic functions with certain arithmetic properties that have fascinated number theorists for over a century. They were a key ingredient in Wiles’ proof of Fermat’s Last Theorem.

Quillen’s work on Elliptic cohomology and topological modular forms formal group laws and complex (continued) cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem To each open subset in the moduli stack ME`` one can p-typicality associate a certain commutative ring of functions related to the Operations in BP-theory Complex cobordism corresponding collection of elliptic curves. theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.32 Such a sheaf has a ring of global sections Γ(OE``), which encodes a lot of useful information. Elements of this ring are closely related to modular forms, which are complex analytic functions with certain arithmetic properties that have fascinated number theorists for over a century. They were a key ingredient in Wiles’ proof of Fermat’s Last Theorem.

Quillen’s work on Elliptic cohomology and topological modular forms formal group laws and complex (continued) cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem To each open subset in the moduli stack ME`` one can p-typicality associate a certain commutative ring of functions related to the Operations in BP-theory Complex cobordism corresponding collection of elliptic curves. This collection is theory after Quillen called a sheaf of rings O over M . Morava’s work E`` E`` Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.32 Elements of this ring are closely related to modular forms, which are complex analytic functions with certain arithmetic properties that have fascinated number theorists for over a century. They were a key ingredient in Wiles’ proof of Fermat’s Last Theorem.

Quillen’s work on Elliptic cohomology and topological modular forms formal group laws and complex (continued) cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem To each open subset in the moduli stack ME`` one can p-typicality associate a certain commutative ring of functions related to the Operations in BP-theory Complex cobordism corresponding collection of elliptic curves. This collection is theory after Quillen called a sheaf of rings O over M . Morava’s work E`` E`` Chromatic homotopy theory Elliptic cohomology and topological modular forms

Such a sheaf has a ring of global sections Γ(OE``), which Conclusion encodes a lot of useful information.

1.32 They were a key ingredient in Wiles’ proof of Fermat’s Last Theorem.

Quillen’s work on Elliptic cohomology and topological modular forms formal group laws and complex (continued) cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem To each open subset in the moduli stack ME`` one can p-typicality associate a certain commutative ring of functions related to the Operations in BP-theory Complex cobordism corresponding collection of elliptic curves. This collection is theory after Quillen called a sheaf of rings O over M . Morava’s work E`` E`` Chromatic homotopy theory Elliptic cohomology and topological modular forms

Such a sheaf has a ring of global sections Γ(OE``), which Conclusion encodes a lot of useful information. Elements of this ring are closely related to modular forms, which are complex analytic functions with certain arithmetic properties that have fascinated number theorists for over a century.

1.32 Quillen’s work on Elliptic cohomology and topological modular forms formal group laws and complex (continued) cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem To each open subset in the moduli stack ME`` one can p-typicality associate a certain commutative ring of functions related to the Operations in BP-theory Complex cobordism corresponding collection of elliptic curves. This collection is theory after Quillen called a sheaf of rings O over M . Morava’s work E`` E`` Chromatic homotopy theory Elliptic cohomology and topological modular forms

1.32 We can think of E as an iceberg whose tip is R. The one associated with Γ(OE``) is known as tmf , the ring spectrum of topological modular forms. This ring spectrum is an iceberg whose tip is the classical theory of modular forms.

Quillen’s work on Elliptic cohomology and topological modular forms formal group laws and complex (continued) cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson Hopkins, Lurie et al. have found a way theorem p-typicality to enrich this theory by replacing ev- Operations in BP-theory

ery ring R in sight with a commutative Complex cobordism theory after Quillen ring spectrum E with suitable formal Morava’s work properties. Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.33 The one associated with Γ(OE``) is known as tmf , the ring spectrum of topological modular forms. This ring spectrum is an iceberg whose tip is the classical theory of modular forms.

Quillen’s work on Elliptic cohomology and topological modular forms formal group laws and complex (continued) cobordism theory

Doug Ravenel

1.33 This ring spectrum is an iceberg whose tip is the classical theory of modular forms.

Quillen’s work on Elliptic cohomology and topological modular forms formal group laws and complex (continued) cobordism theory

Doug Ravenel

ery ring R in sight with a commutative Complex cobordism theory after Quillen ring spectrum E with suitable formal Morava’s work properties. We can think of E as an Chromatic homotopy theory Elliptic cohomology and iceberg whose tip is R. The one asso- topological modular forms Conclusion ciated with Γ(OE``) is known as tmf , the ring spectrum of topological modular forms.

1.33 Quillen’s work on Elliptic cohomology and topological modular forms formal group laws and complex (continued) cobordism theory

Doug Ravenel

1.33 Quillen’s work on Elliptic cohomology and topological modular forms formal group laws and complex (continued) cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem p-typicality Operations in BP-theory

Complex cobordism theory after Quillen Morava’s work Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion The 2-primary homotopy of tmf illustrated by Andre Henriques.

1.34 It led to a chain of discov- eries that is unabated to this day.

Thank you!

Quillen’s work on Conclusion formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem Quillen’s work on formal p-typicality Operations in BP-theory

group laws and complex Complex cobordism cobordism opened a new theory after Quillen Morava’s work era in algebraic topology. Chromatic homotopy theory Elliptic cohomology and topological modular forms

Conclusion

1.35 Thank you!

Quillen’s work on Conclusion formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem Quillen’s work on formal p-typicality Operations in BP-theory

1.35 Quillen’s work on Conclusion formal group laws and complex cobordism theory

Doug Ravenel

Quillen’s 6 page paper Enter the formal group law Show it is universal The Brown-Peterson theorem Quillen’s work on formal p-typicality Operations in BP-theory

Thank you!

1.35