You Could've Invented Tmf

The ﬁnite stable homotopy category Chromatic homotopy theory Topological modular forms Fun with tmf

You could’ve invented tmf .

Aaron Mazel-Gee

University of California, Berkeley [email protected]

April 21, 2013

Aaron Mazel-Gee You could’ve invented tmf . The ﬁnite stable homotopy category Chromatic homotopy theory Topological modular forms Fun with tmf Overview

Aaron Mazel-Gee You could’ve invented tmf . 2 Chromatic homotopy theory

3 Topological modular forms

4 Fun with tmf

The ﬁnite stable homotopy category Chromatic homotopy theory Topological modular forms Fun with tmf Overview

1 The ﬁnite stable homotopy category

Aaron Mazel-Gee You could’ve invented tmf . 3 Topological modular forms

4 Fun with tmf

The ﬁnite stable homotopy category Chromatic homotopy theory Topological modular forms Fun with tmf Overview

1 The ﬁnite stable homotopy category

2 Chromatic homotopy theory

Aaron Mazel-Gee You could’ve invented tmf . 4 Fun with tmf

The ﬁnite stable homotopy category Chromatic homotopy theory Topological modular forms Fun with tmf Overview

1 The ﬁnite stable homotopy category

2 Chromatic homotopy theory

3 Topological modular forms

Aaron Mazel-Gee You could’ve invented tmf . The ﬁnite stable homotopy category Chromatic homotopy theory Topological modular forms Fun with tmf Overview

1 The ﬁnite stable homotopy category

2 Chromatic homotopy theory

3 Topological modular forms

4 Fun with tmf

Aaron Mazel-Gee You could’ve invented tmf . The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf

1. The ﬁnite stable homotopy category

Aaron Mazel-Gee You could’ve invented tmf . stable ﬁnite CW complexes

This is called the ﬁnite stable homotopy category, denoted SHCﬁn.

Let’s do some topology!

So, what is topology? Topology is the study of topological spaces.

But this is hard.

Let’s make things a bit easier for ourselves, and study the

homotopy category of topological spaces.

The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf The category SHCfin

Aaron Mazel-Gee You could’ve invented tmf . stable ﬁnite CW complexes

This is called the ﬁnite stable homotopy category, denoted SHCﬁn.

So, what is topology? Topology is the study of topological spaces.

But this is hard.

Let’s make things a bit easier for ourselves, and study the

homotopy category of topological spaces.

The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf The category SHCfin

Let’s do some topology!

Aaron Mazel-Gee You could’ve invented tmf . stable ﬁnite CW complexes

This is called the ﬁnite stable homotopy category, denoted SHCﬁn.

Topology is the study of topological spaces.

But this is hard.

Let’s make things a bit easier for ourselves, and study the

homotopy category of topological spaces.

The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf The category SHCfin

Let’s do some topology!

So, what is topology?

Aaron Mazel-Gee You could’ve invented tmf . stable ﬁnite CW complexes

This is called the ﬁnite stable homotopy category, denoted SHCﬁn.

But this is hard.

Let’s make things a bit easier for ourselves, and study the

homotopy category of topological spaces.

The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf The category SHCfin

Let’s do some topology!

So, what is topology? Topology is the study of topological spaces.

Aaron Mazel-Gee You could’ve invented tmf . stable ﬁnite CW complexes

This is called the ﬁnite stable homotopy category, denoted SHCﬁn.

Let’s make things a bit easier for ourselves, and study the

homotopy category of topological spaces.

The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf The category SHCfin

Let’s do some topology!

So, what is topology? Topology is the study of topological spaces.

But this is hard.

Aaron Mazel-Gee You could’ve invented tmf . stable ﬁnite CW complexes

This is called the ﬁnite stable homotopy category, denoted SHCﬁn.

The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf The category SHCfin

Let’s do some topology!

So, what is topology? Topology is the study of topological spaces.

But this is hard.

Let’s make things a bit easier for ourselves, and study the

homotopy category of topological spaces.

Aaron Mazel-Gee You could’ve invented tmf . stable ﬁnite CW complexes

This is called the ﬁnite stable homotopy category, denoted SHCﬁn.

The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf The category SHCfin

Let’s do some topology!

So, what is topology? Topology is the study of topological spaces.

But this is hard.

Let’s make things a bit easier for ourselves, and study the

homotopy category of topological spaces.

Aaron Mazel-Gee You could’ve invented tmf . stable ﬁnite

This is called the ﬁnite stable homotopy category, denoted SHCﬁn.

The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf The category SHCfin

Let’s do some topology!

So, what is topology? Topology is the study of topological spaces.

But this is hard.

Let’s make things a bit easier for ourselves, and study the

homotopy category of topological spaces. CW complexes

Aaron Mazel-Gee You could’ve invented tmf . stable ﬁnite

This is called the ﬁnite stable homotopy category, denoted SHCﬁn.

The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf The category SHCfin

Let’s do some topology!

So, what is topology? Topology is the study of topological spaces.

But this is hard.

Let’s make things a bit easier for ourselves, and study the

homotopy category of topological spaces. CW complexes

Aaron Mazel-Gee You could’ve invented tmf . stable

This is called the ﬁnite stable homotopy category, denoted SHCﬁn.

The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf The category SHCfin

Let’s do some topology!

So, what is topology? Topology is the study of topological spaces.

But this is hard.

Let’s make things a bit easier for ourselves, and study the

homotopy category of topological spaces. ﬁnite CW complexes

Aaron Mazel-Gee You could’ve invented tmf . stable

This is called the ﬁnite stable homotopy category, denoted SHCﬁn.

The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf The category SHCfin

Let’s do some topology!

So, what is topology? Topology is the study of topological spaces.

But this is hard.

Let’s make things a bit easier for ourselves, and study the

homotopy category of topological spaces. ﬁnite CW complexes

Aaron Mazel-Gee You could’ve invented tmf . This is called the ﬁnite stable homotopy category, denoted SHCﬁn.

The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf The category SHCfin

Let’s do some topology!

So, what is topology? Topology is the study of topological spaces.

But this is hard.

Let’s make things a bit easier for ourselves, and study the

stable homotopy category of topological spaces. ﬁnite CW complexes

Aaron Mazel-Gee You could’ve invented tmf . This is called the ﬁnite stable homotopy category, denoted SHCﬁn.

The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf The category SHCfin

Let’s do some topology!

So, what is topology? Topology is the study of topological spaces.

But this is hard.

Let’s make things a bit easier for ourselves, and study the

stable homotopy category of topological spaces. ﬁnite CW complexes

Aaron Mazel-Gee You could’ve invented tmf . The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf The category SHCfin

Let’s do some topology!

So, what is topology? Topology is the study of topological spaces.

But this is hard.

Let’s make things a bit easier for ourselves, and study the

stable homotopy category of topological spaces. ﬁnite CW complexes

This is called the ﬁnite stable homotopy category, denoted SHCﬁn.

Aaron Mazel-Gee You could’ve invented tmf . Recall that the suspension of a space X is given by

(0, x1) (0, x2) ΣX = [0, 1] X ∼ × (1, x1) (1, x2). ∼ Given a map X Y , we can suspend it to a map ΣX ΣY . → → This gives us a system

[X , Y ] [ΣX , ΣY ] [Σ2X , Σ2Y ] [Σ3X , Σ3Y ] . → → → → · · · The Freudenthal suspension theorem tells us that this system always stabilizes. So for two finite CW complexes X and Y , we define n n Hom fin (X , Y ) = lim [Σ X , Σ Y ]. SHC n→∞

The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf What does it mean to stabilize?

Aaron Mazel-Gee You could’ve invented tmf . Given a map X Y , we can suspend it to a map ΣX ΣY . → → This gives us a system

The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf What does it mean to stabilize?

Recall that the suspension of a space X is given by

(0, x1) (0, x2) ΣX = [0, 1] X ∼ × (1, x1) (1, x2). ∼

Aaron Mazel-Gee You could’ve invented tmf . This gives us a system

The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf What does it mean to stabilize?

Recall that the suspension of a space X is given by

(0, x1) (0, x2) ΣX = [0, 1] X ∼ × (1, x1) (1, x2). ∼ Given a map X Y , we can suspend it to a map ΣX ΣY . → →

Aaron Mazel-Gee You could’ve invented tmf . The Freudenthal suspension theorem tells us that this system always stabilizes. So for two finite CW complexes X and Y , we define n n Hom fin (X , Y ) = lim [Σ X , Σ Y ]. SHC n→∞

The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf What does it mean to stabilize?

Recall that the suspension of a space X is given by

(0, x1) (0, x2) ΣX = [0, 1] X ∼ × (1, x1) (1, x2). ∼ Given a map X Y , we can suspend it to a map ΣX ΣY . → → This gives us a system

[X , Y ] [ΣX , ΣY ] [Σ2X , Σ2Y ] [Σ3X , Σ3Y ] . → → → → · · ·

Aaron Mazel-Gee You could’ve invented tmf . So for two finite CW complexes X and Y , we define n n Hom fin (X , Y ) = lim [Σ X , Σ Y ]. SHC n→∞

The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf What does it mean to stabilize?

Recall that the suspension of a space X is given by

(0, x1) (0, x2) ΣX = [0, 1] X ∼ × (1, x1) (1, x2). ∼ Given a map X Y , we can suspend it to a map ΣX ΣY . → → This gives us a system

[X , Y ] [ΣX , ΣY ] [Σ2X , Σ2Y ] [Σ3X , Σ3Y ] . → → → → · · · The Freudenthal suspension theorem tells us that this system always stabilizes.

Recall that the suspension of a space X is given by

(0, x1) (0, x2) ΣX = [0, 1] X ∼ × (1, x1) (1, x2). ∼ Given a map X Y , we can suspend it to a map ΣX ΣY . → → This gives us a system

Aaron Mazel-Gee You could’ve invented tmf . (Recall that [ΣX , Z] is always a group (for the same reason that n π1 is a group), and that [Σ X , Z] is always an abelian group for n 2 (for the same reason that π≥2 is an abelian group).) ≥

The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf

The set n n Hom ﬁn (X , Y ) = lim [Σ X , Σ Y ] SHC n→∞ is in fact an abelian group.

Aaron Mazel-Gee You could’ve invented tmf . The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf

The set n n Hom ﬁn (X , Y ) = lim [Σ X , Σ Y ] SHC n→∞ is in fact an abelian group.

(Recall that [ΣX , Z] is always a group (for the same reason that n π1 is a group), and that [Σ X , Z] is always an abelian group for n 2 (for the same reason that π≥2 is an abelian group).) ≥

Aaron Mazel-Gee You could’ve invented tmf . Then, for example,

−i −j n−i n−j Hom ﬁn (Σ X , Σ Y ) = lim [Σ X , Σ Y ]. SHC n→∞

So, the objects of SHCﬁn are the ﬁnite CW complexes and their formal desuspensions.

The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf

And as long as we only need to be able to deﬁne maps between arbitrarily high suspensions of our ﬁnite CW complexes, we may as well allow desuspensions too.

Aaron Mazel-Gee You could’ve invented tmf . So, the objects of SHCﬁn are the ﬁnite CW complexes and their formal desuspensions.

The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf

And as long as we only need to be able to deﬁne maps between arbitrarily high suspensions of our ﬁnite CW complexes, we may as well allow desuspensions too.

Then, for example,

−i −j n−i n−j Hom ﬁn (Σ X , Σ Y ) = lim [Σ X , Σ Y ]. SHC n→∞

Aaron Mazel-Gee You could’ve invented tmf . The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf

And as long as we only need to be able to deﬁne maps between arbitrarily high suspensions of our ﬁnite CW complexes, we may as well allow desuspensions too.

Then, for example,

−i −j n−i n−j Hom ﬁn (Σ X , Σ Y ) = lim [Σ X , Σ Y ]. SHC n→∞

So, the objects of SHCﬁn are the ﬁnite CW complexes and their formal desuspensions.

Aaron Mazel-Gee You could’ve invented tmf . We would like to study the global structure of the finite stable homotopy category. What structure does it carry? A subcategory is called thick if it is closed under mapping cones, retracts, and weak equivalences. So, the “kernel” of any co/homology theory is thick. We can use the smash product to define ideal subcategories and prime ideal subcategories, exactly as in ring theory. Given a category with this structure, we can define a space Spec( ) in muchC the same way an algebraic geometer defines the primeC spectrum of a ring:

Spec( ) = P : P a thick prime ideal subcategory . C { ⊂ C } This packages a lot of information really cleanly.

The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf The geometry of SHCfin

Aaron Mazel-Gee You could’ve invented tmf . What structure does it carry? A subcategory is called thick if it is closed under mapping cones, retracts, and weak equivalences. So, the “kernel” of any co/homology theory is thick. We can use the smash product to define ideal subcategories and prime ideal subcategories, exactly as in ring theory. Given a category with this structure, we can define a space Spec( ) in muchC the same way an algebraic geometer defines the primeC spectrum of a ring:

Spec( ) = P : P a thick prime ideal subcategory . C { ⊂ C } This packages a lot of information really cleanly.

The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf The geometry of SHCfin

We would like to study the global structure of the ﬁnite stable homotopy category.

Aaron Mazel-Gee You could’ve invented tmf . A subcategory is called thick if it is closed under mapping cones, retracts, and weak equivalences. So, the “kernel” of any co/homology theory is thick. We can use the smash product to define ideal subcategories and prime ideal subcategories, exactly as in ring theory. Given a category with this structure, we can define a space Spec( ) in muchC the same way an algebraic geometer defines the primeC spectrum of a ring:

Spec( ) = P : P a thick prime ideal subcategory . C { ⊂ C } This packages a lot of information really cleanly.

The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf The geometry of SHCfin

We would like to study the global structure of the ﬁnite stable homotopy category. What structure does it carry?

Aaron Mazel-Gee You could’ve invented tmf . We can use the smash product to define ideal subcategories and prime ideal subcategories, exactly as in ring theory. Given a category with this structure, we can define a space Spec( ) in muchC the same way an algebraic geometer defines the primeC spectrum of a ring:

Spec( ) = P : P a thick prime ideal subcategory . C { ⊂ C } This packages a lot of information really cleanly.

The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf The geometry of SHCfin

Aaron Mazel-Gee You could’ve invented tmf . Given a category with this structure, we can deﬁne a space Spec( ) in muchC the same way an algebraic geometer deﬁnes the primeC spectrum of a ring:

Spec( ) = P : P a thick prime ideal subcategory . C { ⊂ C } This packages a lot of information really cleanly.

The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf The geometry of SHCfin

We would like to study the global structure of the ﬁnite stable homotopy category. What structure does it carry? A subcategory is called thick if it is closed under mapping cones, retracts, and weak equivalences. So, the “kernel” of any co/homology theory is thick. We can use the smash product to deﬁne ideal subcategories and prime ideal subcategories, exactly as in ring theory.

Aaron Mazel-Gee You could’ve invented tmf . Spec( ) = P : P a thick prime ideal subcategory . C { ⊂ C } This packages a lot of information really cleanly.

The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf The geometry of SHCfin

Aaron Mazel-Gee You could’ve invented tmf . This packages a lot of information really cleanly.

The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf The geometry of SHCfin

Spec( ) = P : P a thick prime ideal subcategory . C { ⊂ C }

Spec( ) = P : P a thick prime ideal subcategory . C { ⊂ C } This packages a lot of information really cleanly.

Aaron Mazel-Gee You could’ve invented tmf . There is a beautiful answer, given by the nilpotence theorem of Devinatz–Hopkins–Smith.

P∞,(2) P∞,(3) P∞,(5) P∞,(7) ··· ...... ···

P3,(2) P3,(3) P3,(5) P3,(7) ··· N0 ∪ {∞} P2,(2) P2,(3) P2,(5) P2,(7) ···

P1,(2) P1,(3) P1,(5) P1,(7) ···

P0 Spec(Z)

This is exciting! But to explain what the subcategories Pn,(p) are, we’ll have to talk about...

The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf So, what is Spec(SHCfin)?

Aaron Mazel-Gee You could’ve invented tmf . P∞,(2) P∞,(3) P∞,(5) P∞,(7) ··· ...... ···

P3,(2) P3,(3) P3,(5) P3,(7) ··· N0 ∪ {∞} P2,(2) P2,(3) P2,(5) P2,(7) ···

P1,(2) P1,(3) P1,(5) P1,(7) ···

P0 Spec(Z)

This is exciting! But to explain what the subcategories Pn,(p) are, we’ll have to talk about...

The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf So, what is Spec(SHCfin)?

There is a beautiful answer, given by the nilpotence theorem of Devinatz–Hopkins–Smith.

Aaron Mazel-Gee You could’ve invented tmf . This is exciting! But to explain what the subcategories Pn,(p) are, we’ll have to talk about...

The finite stable homotopy category Chromatic homotopy theory The category SHCfin Topological modular forms The geometry of SHCfin Fun with tmf So, what is Spec(SHCfin)?

There is a beautiful answer, given by the nilpotence theorem of Devinatz–Hopkins–Smith.

P∞,(2) P∞,(3) P∞,(5) P∞,(7) ··· ...... ···

P3,(2) P3,(3) P3,(5) P3,(7) ··· N0 ∪ {∞} P2,(2) P2,(3) P2,(5) P2,(7) ···

P1,(2) P1,(3) P1,(5) P1,(7) ···

P0 Spec(Z)

There is a beautiful answer, given by the nilpotence theorem of Devinatz–Hopkins–Smith.

P∞,(2) P∞,(3) P∞,(5) P∞,(7) ··· ...... ···

P3,(2) P3,(3) P3,(5) P3,(7) ··· N0 ∪ {∞} P2,(2) P2,(3) P2,(5) P2,(7) ···

P1,(2) P1,(3) P1,(5) P1,(7) ···

P0 Spec(Z)

This is exciting! But to explain what the subcategories Pn,(p) are, we’ll have to talk about... Aaron Mazel-Gee You could’ve invented tmf . The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

2. Chromatic homotopy theory

Aaron Mazel-Gee You could’ve invented tmf . The story of chromatic homotopy theory begins with formal group ∞ laws, and the story of formal group laws begins with CP .

∞ Recall that CP is a classifying space for complex line bundles; ∞ that is, it carries a universal line bundle Luniv CP , and there is a natural isomorphism ↓

∞ line bundles over X ∼= [X , CP ]. { ∗ } f Luniv f ↔

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf Formal group laws in topology

Aaron Mazel-Gee You could’ve invented tmf . ∞ Recall that CP is a classifying space for complex line bundles; ∞ that is, it carries a universal line bundle Luniv CP , and there is a natural isomorphism ↓

∞ line bundles over X ∼= [X , CP ]. { ∗ } f Luniv f ↔

The story of chromatic homotopy theory begins with formal group ∞ laws, and the story of formal group laws begins with CP .

Aaron Mazel-Gee You could’ve invented tmf . ∞ that is, it carries a universal line bundle Luniv CP , and there is a natural isomorphism ↓

∞ line bundles over X ∼= [X , CP ]. { ∗ } f Luniv f ↔

The story of chromatic homotopy theory begins with formal group ∞ laws, and the story of formal group laws begins with CP .

∞ Recall that CP is a classifying space for complex line bundles;

Aaron Mazel-Gee You could’ve invented tmf . and there is a natural isomorphism

∞ line bundles over X ∼= [X , CP ]. { ∗ } f Luniv f ↔

The story of chromatic homotopy theory begins with formal group ∞ laws, and the story of formal group laws begins with CP .

∞ Recall that CP is a classifying space for complex line bundles; ∞ that is, it carries a universal line bundle Luniv CP , ↓

Aaron Mazel-Gee You could’ve invented tmf . The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf Formal group laws in topology

The story of chromatic homotopy theory begins with formal group ∞ laws, and the story of formal group laws begins with CP .

∞ Recall that CP is a classifying space for complex line bundles; ∞ that is, it carries a universal line bundle Luniv CP , and there is a natural isomorphism ↓

∞ line bundles over X ∼= [X , CP ]. { ∗ } f Luniv f ↔

Aaron Mazel-Gee You could’ve invented tmf . (A pair of line bundles L1, L2 X is classiﬁed by a map ↓ f ∞ ∞ X CP CP −→ × ∗ ∗ such that L1 ∼= f Luniv,1 and L2 ∼= f Luniv,2. By assumption,

∞ ∞ µ ∞ CP CP CP × −→ ∗ classiﬁes Luniv,1 Luniv,2 = µ Luniv , so the composite ⊗ ∼ f ∞ ∞ µ ∞ X CP CP CP −→ × −→ ∗ classiﬁes L1 L2 = (µf ) Luniv .) ⊗ ∼

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

By Yoneda’s lemma, the natural operation of tensor product of ∞ ∞ µ ∞ two line bundles is classiﬁed by a map CP CP CP . × −→

Aaron Mazel-Gee You could’ve invented tmf . By assumption,

∞ ∞ µ ∞ CP CP CP × −→ ∗ classiﬁes Luniv,1 Luniv,2 = µ Luniv , so the composite ⊗ ∼ f ∞ ∞ µ ∞ X CP CP CP −→ × −→ ∗ classiﬁes L1 L2 = (µf ) Luniv .) ⊗ ∼

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

By Yoneda’s lemma, the natural operation of tensor product of ∞ ∞ µ ∞ two line bundles is classiﬁed by a map CP CP CP . × −→

(A pair of line bundles L1, L2 X is classiﬁed by a map ↓ f ∞ ∞ X CP CP −→ × ∗ ∗ such that L1 ∼= f Luniv,1 and L2 ∼= f Luniv,2.

Aaron Mazel-Gee You could’ve invented tmf . so the composite

f ∞ ∞ µ ∞ X CP CP CP −→ × −→ ∗ classiﬁes L1 L2 = (µf ) Luniv .) ⊗ ∼

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

By Yoneda’s lemma, the natural operation of tensor product of ∞ ∞ µ ∞ two line bundles is classiﬁed by a map CP CP CP . × −→

(A pair of line bundles L1, L2 X is classiﬁed by a map ↓ f ∞ ∞ X CP CP −→ × ∗ ∗ such that L1 ∼= f Luniv,1 and L2 ∼= f Luniv,2. By assumption,

∞ ∞ µ ∞ CP CP CP × −→ ∗ classiﬁes Luniv,1 Luniv,2 = µ Luniv , ⊗ ∼

By Yoneda’s lemma, the natural operation of tensor product of ∞ ∞ µ ∞ two line bundles is classiﬁed by a map CP CP CP . × −→

(A pair of line bundles L1, L2 X is classiﬁed by a map ↓ f ∞ ∞ X CP CP −→ × ∗ ∗ such that L1 ∼= f Luniv,1 and L2 ∼= f Luniv,2. By assumption,

∞ ∞ µ ∞ CP CP CP × −→ ∗ classiﬁes Luniv,1 Luniv,2 = µ Luniv , so the composite ⊗ ∼ f ∞ ∞ µ ∞ X CP CP CP −→ × −→ ∗ classiﬁes L1 L2 = (µf ) Luniv .) ⊗ ∼

Aaron Mazel-Gee You could’ve invented tmf . ∗ Let’s see what happens when we apply H ( ; Z). − ∗ ∞ Recall that H (CP ) = Z[[t]]; by the K¨unnethformula, ∗ ∞ ∞ ∼ H (CP CP ) = Z[[x, y]]. So, the ring homomorphism × ∼ ∗ ∞ ∗ ∞ ∞ H (CP ) H (CP CP ) → × is entirely described by

t F (x, y) Z[[x, y]]. 7→ ∈ What is F (x, y)?

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

∞ ∞ ∞ What does this map CP CP CP buy us? × →

Aaron Mazel-Gee You could’ve invented tmf . ∗ ∞ Recall that H (CP ) = Z[[t]]; by the K¨unnethformula, ∗ ∞ ∞ ∼ H (CP CP ) = Z[[x, y]]. So, the ring homomorphism × ∼ ∗ ∞ ∗ ∞ ∞ H (CP ) H (CP CP ) → × is entirely described by

t F (x, y) Z[[x, y]]. 7→ ∈ What is F (x, y)?

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

∞ ∞ ∞ What does this map CP CP CP buy us? × → ∗ Let’s see what happens when we apply H ( ; Z). −

Aaron Mazel-Gee You could’ve invented tmf . So, the ring homomorphism

∗ ∞ ∗ ∞ ∞ H (CP ) H (CP CP ) → × is entirely described by

t F (x, y) Z[[x, y]]. 7→ ∈ What is F (x, y)?

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

∞ ∞ ∞ What does this map CP CP CP buy us? × → ∗ Let’s see what happens when we apply H ( ; Z). − ∗ ∞ Recall that H (CP ) = Z[[t]]; by the K¨unnethformula, ∗ ∞ ∞ ∼ H (CP CP ) = Z[[x, y]]. × ∼

Aaron Mazel-Gee You could’ve invented tmf . What is F (x, y)?

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

∞ ∞ ∞ What does this map CP CP CP buy us? × → ∗ Let’s see what happens when we apply H ( ; Z). − ∗ ∞ Recall that H (CP ) = Z[[t]]; by the K¨unnethformula, ∗ ∞ ∞ ∼ H (CP CP ) = Z[[x, y]]. So, the ring homomorphism × ∼ ∗ ∞ ∗ ∞ ∞ H (CP ) H (CP CP ) → × is entirely described by

t F (x, y) Z[[x, y]]. 7→ ∈

t F (x, y) Z[[x, y]]. 7→ ∈ What is F (x, y)?

Aaron Mazel-Gee You could’ve invented tmf . We also need to remember that for any two line bundles L1 and L2,

c1(L1 L2) = c1(L1) + c1(L2). ⊗ Since F (x, y) just encodes how the ﬁrst Chern class behaves under tensor product, it follows that F (x, y) = x + y.

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

To determine the map

∗ ∞ ∗ ∞ ∞ H (CP ) H (CP CP ) → × t F (x, y), 7→ we need to remember that the element t also goes by another name: the ﬁrst Chern class.

Aaron Mazel-Gee You could’ve invented tmf . Since F (x, y) just encodes how the ﬁrst Chern class behaves under tensor product, it follows that F (x, y) = x + y.

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

To determine the map

∗ ∞ ∗ ∞ ∞ H (CP ) H (CP CP ) → × t F (x, y), 7→ we need to remember that the element t also goes by another name: the ﬁrst Chern class. We also need to remember that for any two line bundles L1 and L2,

c1(L1 L2) = c1(L1) + c1(L2). ⊗

To determine the map

c1(L1 L2) = c1(L1) + c1(L2). ⊗ Since F (x, y) just encodes how the ﬁrst Chern class behaves under tensor product, it follows that F (x, y) = x + y.

Aaron Mazel-Gee You could’ve invented tmf . A ring-valued cohomology theory E ∗ is called complex-oriented if we have an isomorphism

∗ ∞ ∗ E (CP ) ∼= (E (pt))[[t]] = E∗[[t]]; we call the element t a generalized ﬁrst Chern class.

∗ ∞ ∞ We’ll always have E (CP CP ) = E∗[[x, y]], and so once again ∞ ∞ ×∞ ∼ the map CP CP CP induces a map E∗[[t]] E∗[[x, y]], × → → and once again this is determined by t FE (x, y) E∗[[x, y]]. 7→ ∈

What can we say about FE (x, y)?

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

Let’s generalize this a bit.

Aaron Mazel-Gee You could’ve invented tmf . ∗ ∞ ∞ We’ll always have E (CP CP ) = E∗[[x, y]], and so once again ∞ ∞ ×∞ ∼ the map CP CP CP induces a map E∗[[t]] E∗[[x, y]], × → → and once again this is determined by t FE (x, y) E∗[[x, y]]. 7→ ∈

What can we say about FE (x, y)?

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

Let’s generalize this a bit.

A ring-valued cohomology theory E ∗ is called complex-oriented if we have an isomorphism

∗ ∞ ∗ E (CP ) ∼= (E (pt))[[t]] = E∗[[t]]; we call the element t a generalized ﬁrst Chern class.

Aaron Mazel-Gee You could’ve invented tmf . What can we say about FE (x, y)?

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

Let’s generalize this a bit.

A ring-valued cohomology theory E ∗ is called complex-oriented if we have an isomorphism

∗ ∞ ∗ E (CP ) ∼= (E (pt))[[t]] = E∗[[t]]; we call the element t a generalized ﬁrst Chern class.

Let’s generalize this a bit.

A ring-valued cohomology theory E ∗ is called complex-oriented if we have an isomorphism

∗ ∞ ∗ E (CP ) ∼= (E (pt))[[t]] = E∗[[t]]; we call the element t a generalized ﬁrst Chern class.

What can we say about FE (x, y)?

Aaron Mazel-Gee You could’ve invented tmf . L C = L FE (x, 0) = x (unitality) ⊗ ∼ ⇒ L1 L2 = L2 L1 FE (x, y) = FE (y, x) (commutativity) ⊗ ∼ ⊗ ⇒ (L1 L2) L3 = ⊗ ⊗ ∼ L1 (L2 L3) FE (FE (x, y), z) ⊗ ⊗ ⇒ = FE (x, FE (y, z)) (associativity)

These are the three deﬁning properties for FE to be a (1-dimensional commutative) formal group law over the ring E∗.

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

The power series FE (x, y) E∗[[x, y]] enjoys certain properties coming from analogous properties∈ of the tensor product of line bundles.

Aaron Mazel-Gee You could’ve invented tmf . L1 L2 = L2 L1 FE (x, y) = FE (y, x) (commutativity) ⊗ ∼ ⊗ ⇒ (L1 L2) L3 = ⊗ ⊗ ∼ L1 (L2 L3) FE (FE (x, y), z) ⊗ ⊗ ⇒ = FE (x, FE (y, z)) (associativity)

These are the three deﬁning properties for FE to be a (1-dimensional commutative) formal group law over the ring E∗.

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

The power series FE (x, y) E∗[[x, y]] enjoys certain properties coming from analogous properties∈ of the tensor product of line bundles.

L C = L FE (x, 0) = x (unitality) ⊗ ∼ ⇒

Aaron Mazel-Gee You could’ve invented tmf . (L1 L2) L3 = ⊗ ⊗ ∼ L1 (L2 L3) FE (FE (x, y), z) ⊗ ⊗ ⇒ = FE (x, FE (y, z)) (associativity)

These are the three deﬁning properties for FE to be a (1-dimensional commutative) formal group law over the ring E∗.

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

The power series FE (x, y) E∗[[x, y]] enjoys certain properties coming from analogous properties∈ of the tensor product of line bundles.

L C = L FE (x, 0) = x (unitality) ⊗ ∼ ⇒ L1 L2 = L2 L1 FE (x, y) = FE (y, x) (commutativity) ⊗ ∼ ⊗ ⇒

Aaron Mazel-Gee You could’ve invented tmf . These are the three deﬁning properties for FE to be a (1-dimensional commutative) formal group law over the ring E∗.

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

The power series FE (x, y) E∗[[x, y]] enjoys certain properties coming from analogous properties∈ of the tensor product of line bundles.

L C = L FE (x, 0) = x (unitality) ⊗ ∼ ⇒ L1 L2 = L2 L1 FE (x, y) = FE (y, x) (commutativity) ⊗ ∼ ⊗ ⇒ (L1 L2) L3 = ⊗ ⊗ ∼ L1 (L2 L3) FE (FE (x, y), z) ⊗ ⊗ ⇒ = FE (x, FE (y, z)) (associativity)

The power series FE (x, y) E∗[[x, y]] enjoys certain properties coming from analogous properties∈ of the tensor product of line bundles.

These are the three deﬁning properties for FE to be a (1-dimensional commutative) formal group law over the ring E∗.

Aaron Mazel-Gee You could’ve invented tmf . The formal group law FHZ(x, y) = x + y associated to singular cohomology is called the additive formal group law, denoted Ga, which is the germ of the additive group, denoted Ga. b

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

One can obtain formal group laws as germs of algebraic groups, much as one can obtain Lie algebras as germs of Lie groups.

The formal group law FHZ(x, y) = x + y associated to singular cohomology is called the additive formal group law, denoted Ga, which is the germ of the additive group, denoted Ga. b

Aaron Mazel-Gee You could’ve invented tmf . Recall that KU0(X ) is the group-completion of the monoid (Vect (X ), ), with multiplication given by . C ⊕ ⊗ ∗ ∞ K-theory is complex-oriented: KU (CP ) ∼= KU∗[[t]], where t = [Luniv ] [C] = [Luniv ] 1. So with x = [L1] 1 and − ∗ − − y = [L2] 1, since µ Luniv = L1 L2 we compute that − ⊗ FKU (x, y) = [L1 L2] 1 ⊗ − = [L1] [L2] 1 · − = ([L1] [L2] [L1] [L2] + 1) + [L1] 1 + [L2] 1 · − − − − = ([L1] 1) ([L2] 1) + ([L1] 1) + ([L2] 1) − · − − − = xy + x + y.

This is called the multiplicative formal group law, denoted Gm, which is the germ of the multiplicative group, denoted Gm. We’ll come back to this. b

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf Another example: complex K-theory.

Aaron Mazel-Gee You could’ve invented tmf . ∗ ∞ K-theory is complex-oriented: KU (CP ) ∼= KU∗[[t]], where t = [Luniv ] [C] = [Luniv ] 1. So with x = [L1] 1 and − ∗ − − y = [L2] 1, since µ Luniv = L1 L2 we compute that − ⊗ FKU (x, y) = [L1 L2] 1 ⊗ − = [L1] [L2] 1 · − = ([L1] [L2] [L1] [L2] + 1) + [L1] 1 + [L2] 1 · − − − − = ([L1] 1) ([L2] 1) + ([L1] 1) + ([L2] 1) − · − − − = xy + x + y.

This is called the multiplicative formal group law, denoted Gm, which is the germ of the multiplicative group, denoted Gm. We’ll come back to this. b

Recall that KU0(X ) is the group-completion of the monoid (Vect (X ), ), with multiplication given by . C ⊕ ⊗

Aaron Mazel-Gee You could’ve invented tmf . So with x = [L1] 1 and ∗ − y = [L2] 1, since µ Luniv = L1 L2 we compute that − ⊗ FKU (x, y) = [L1 L2] 1 ⊗ − = [L1] [L2] 1 · − = ([L1] [L2] [L1] [L2] + 1) + [L1] 1 + [L2] 1 · − − − − = ([L1] 1) ([L2] 1) + ([L1] 1) + ([L2] 1) − · − − − = xy + x + y.

This is called the multiplicative formal group law, denoted Gm, which is the germ of the multiplicative group, denoted Gm. We’ll come back to this. b

Aaron Mazel-Gee You could’ve invented tmf . = [L1] [L2] 1 · − = ([L1] [L2] [L1] [L2] + 1) + [L1] 1 + [L2] 1 · − − − − = ([L1] 1) ([L2] 1) + ([L1] 1) + ([L2] 1) − · − − − = xy + x + y.

This is called the multiplicative formal group law, denoted Gm, which is the germ of the multiplicative group, denoted Gm. We’ll come back to this. b

Aaron Mazel-Gee You could’ve invented tmf . This is called the multiplicative formal group law, denoted Gm, which is the germ of the multiplicative group, denoted Gm. We’ll come back to this. b

Recall that KU0(X ) is the group-completion of the monoid (Vect (X ), ), with multiplication given by . C ⊕ ⊗ ∗ ∞ K-theory is complex-oriented: KU (CP ) ∼= KU∗[[t]], where t = [Luniv ] [C] = [Luniv ] 1. So with x = [L1] 1 and − ∗ − − y = [L2] 1, since µ Luniv = L1 L2 we compute that − ⊗ FKU (x, y) = [L1 L2] 1 ⊗ − = [L1] [L2] 1 · − = ([L1] [L2] [L1] [L2] + 1) + [L1] 1 + [L2] 1 · − − − − = ([L1] 1) ([L2] 1) + ([L1] 1) + ([L2] 1) − · − − − = xy + x + y.

This is called the multiplicative formal group law, denoted Gm, which is the germ of the multiplicative group, denoted Gm. We’ll come back to this. b Aaron Mazel-Gee You could’ve invented tmf . In fact, there is a partial inverse (i.e. it’s not deﬁned on all formal group laws) given by the Landweber exact functor theorem.

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

Returning to the general theory, we have a functor

E 7→ FE (x, y) ∈ E∗[[x, y]]

complex-oriented formal group laws . cohomology theories { }

Returning to the general theory, we have a functor

E 7→ FE (x, y) ∈ E∗[[x, y]]

complex-oriented formal group laws . cohomology theories { }

In fact, there is a partial inverse (i.e. it’s not deﬁned on all formal group laws) given by the Landweber exact functor theorem.

Aaron Mazel-Gee You could’ve invented tmf . To define the Morava K-theories, we first need to define the height of a formal group law.

Given a formal group law F (x, y) R[[x, y]], we deﬁne its n-series ∈ [n]F (x) R[[x]] inductively by ∈

[1]F (x) = x [n] (x) = F (x, [n 1] (x)). F − F This classiﬁes “n-fold addition”. From the axioms, we know that F (x, y) = x + y + , and so [n]F (x) = nx + . ··· ···

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf The Morava K-theories

Aaron Mazel-Gee You could’ve invented tmf . Given a formal group law F (x, y) R[[x, y]], we deﬁne its n-series ∈ [n]F (x) R[[x]] inductively by ∈

[1]F (x) = x [n] (x) = F (x, [n 1] (x)). F − F This classiﬁes “n-fold addition”. From the axioms, we know that F (x, y) = x + y + , and so [n]F (x) = nx + . ··· ···

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf The Morava K-theories

To define the Morava K-theories, we first need to define the height of a formal group law.

Aaron Mazel-Gee You could’ve invented tmf . This classiﬁes “n-fold addition”. From the axioms, we know that F (x, y) = x + y + , and so [n]F (x) = nx + . ··· ···

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf The Morava K-theories

To define the Morava K-theories, we first need to define the height of a formal group law.

Given a formal group law F (x, y) R[[x, y]], we deﬁne its n-series ∈ [n]F (x) R[[x]] inductively by ∈

[1]F (x) = x [n] (x) = F (x, [n 1] (x)). F − F

Aaron Mazel-Gee You could’ve invented tmf . If R = k is a ﬁeld of characteristic p, then the ﬁrst term of [p]F (x) ph vanishes. In fact, we’ll always have [p]F (x) = ux + for u k× and h 1, and this integer h is called the height··· of F . ∈ ≥

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf The Morava K-theories

To define the Morava K-theories, we first need to define the height of a formal group law.

Given a formal group law F (x, y) R[[x, y]], we deﬁne its n-series ∈ [n]F (x) R[[x]] inductively by ∈

[1]F (x) = x [n] (x) = F (x, [n 1] (x)). F − F This classiﬁes “n-fold addition”. From the axioms, we know that F (x, y) = x + y + , and so [n]F (x) = nx + . ··· ···

Aaron Mazel-Gee You could’ve invented tmf . ph In fact, we’ll always have [p]F (x) = ux + for u k× and h 1, and this integer h is called the height··· of F . ∈ ≥

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf The Morava K-theories

To define the Morava K-theories, we first need to define the height of a formal group law.

Given a formal group law F (x, y) R[[x, y]], we deﬁne its n-series ∈ [n]F (x) R[[x]] inductively by ∈

[1]F (x) = x [n] (x) = F (x, [n 1] (x)). F − F This classiﬁes “n-fold addition”. From the axioms, we know that F (x, y) = x + y + , and so [n]F (x) = nx + . ··· ···

If R = k is a ﬁeld of characteristic p, then the ﬁrst term of [p]F (x) vanishes.

To define the Morava K-theories, we first need to define the height of a formal group law.

Given a formal group law F (x, y) R[[x, y]], we deﬁne its n-series ∈ [n]F (x) R[[x]] inductively by ∈

[1]F (x) = x [n] (x) = F (x, [n 1] (x)). F − F This classiﬁes “n-fold addition”. From the axioms, we know that F (x, y) = x + y + , and so [n]F (x) = nx + . ··· ···

If R = k is a ﬁeld of characteristic p, then the ﬁrst term of [p]F (x) ph vanishes. In fact, we’ll always have [p]F (x) = ux + for u k× and h 1, and this integer h is called the height··· of F . ∈ ≥ Aaron Mazel-Gee You could’ve invented tmf . This is realized in topology by a complex-oriented cohomology theory called the nth Morava K-theory, denoted K(n, p).

Here are a few edge cases.

H1,p = (Gm)Fp , and so K(1, p) = KU/p (mod p complex K-theory). b H∞,p = (Ga)Fp , and so K( , p) = HFp (mod p singular cohomology). ∞

Even thoughb there’s no H0,p, it turns out that we can reasonably deﬁne K(0, p) = HQ (rational singular cohomology) for any prime p.

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

th Over Fp itself, for each height n [1, ] we have the n Honda ∈ ∞ pn formal group law, denoted Hn,p, with p-series [p]Hn,p (x) = x .

Aaron Mazel-Gee You could’ve invented tmf . Here are a few edge cases.

H1,p = (Gm)Fp , and so K(1, p) = KU/p (mod p complex K-theory). b H∞,p = (Ga)Fp , and so K( , p) = HFp (mod p singular cohomology). ∞

Even thoughb there’s no H0,p, it turns out that we can reasonably deﬁne K(0, p) = HQ (rational singular cohomology) for any prime p.

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

th Over Fp itself, for each height n [1, ] we have the n Honda ∈ ∞ pn formal group law, denoted Hn,p, with p-series [p]Hn,p (x) = x . This is realized in topology by a complex-oriented cohomology theory called the nth Morava K-theory, denoted K(n, p).

Aaron Mazel-Gee You could’ve invented tmf . H1,p = (Gm)Fp , and so K(1, p) = KU/p (mod p complex K-theory). b H∞,p = (Ga)Fp , and so K( , p) = HFp (mod p singular cohomology). ∞

Even thoughb there’s no H0,p, it turns out that we can reasonably deﬁne K(0, p) = HQ (rational singular cohomology) for any prime p.

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

Here are a few edge cases.

Aaron Mazel-Gee You could’ve invented tmf . H∞,p = (Ga)Fp , and so K( , p) = HFp (mod p singular cohomology). ∞

Even thoughb there’s no H0,p, it turns out that we can reasonably deﬁne K(0, p) = HQ (rational singular cohomology) for any prime p.

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

Here are a few edge cases.

H1,p = (Gm)Fp , and so K(1, p) = KU/p (mod p complex K-theory). b

Aaron Mazel-Gee You could’ve invented tmf . Even though there’s no H0,p, it turns out that we can reasonably deﬁne K(0, p) = HQ (rational singular cohomology) for any prime p.

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

Here are a few edge cases.

H1,p = (Gm)Fp , and so K(1, p) = KU/p (mod p complex K-theory). b H∞,p = (Ga)Fp , and so K( , p) = HFp (mod p singular cohomology). ∞ b

Here are a few edge cases.

H1,p = (Gm)Fp , and so K(1, p) = KU/p (mod p complex K-theory). b H∞,p = (Ga)Fp , and so K( , p) = HFp (mod p singular cohomology). ∞

Even thoughb there’s no H0,p, it turns out that we can reasonably deﬁne K(0, p) = HQ (rational singular cohomology) for any prime p.

Aaron Mazel-Gee You could’ve invented tmf . That is, they are essentially all the homology theories whose kernel is not just a thick subcategory, but is also a prime ideal subcategory.

† The set {K(n, p)}n,p plays the same role for ring-valued cohomology theories as the set {Q} ∪ {Fp}p plays for ordinary rings: it is the set of prime ﬁelds.

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

The Morava K-theories represent essentially all† of the homology theories that have K¨unneth isomorphisms (instead of just a K¨unnethshort exact sequence, or worse).

Aaron Mazel-Gee You could’ve invented tmf . † The set {K(n, p)}n,p plays the same role for ring-valued cohomology theories as the set {Q} ∪ {Fp}p plays for ordinary rings: it is the set of prime ﬁelds.

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

The Morava K-theories represent essentially all† of the homology theories that have K¨unneth isomorphisms (instead of just a K¨unnethshort exact sequence, or worse).

That is, they are essentially all the homology theories whose kernel is not just a thick subcategory, but is also a prime ideal subcategory.

The Morava K-theories represent essentially all† of the homology theories that have K¨unneth isomorphisms (instead of just a K¨unnethshort exact sequence, or worse).

That is, they are essentially all the homology theories whose kernel is not just a thick subcategory, but is also a prime ideal subcategory.

† The set {K(n, p)}n,p plays the same role for ring-valued cohomology theories as the set {Q} ∪ {Fp}p plays for ordinary rings: it is the set of prime ﬁelds.

Aaron Mazel-Gee You could’ve invented tmf . In fact, the point Pn,(p) is nothing more or less than the kernel of K(n, p)∗!

P∞,(2) P∞,(3) P∞,(5) P∞,(7) ··· ...... ···

P3,(2) P3,(3) P3,(5) P3,(7) ··· N0 ∪ {∞} P2,(2) P2,(3) P2,(7) P2,(5) ···

P1,(2) P1,(3) P1,(5) P1,(7) ···

P0 Spec(Z)

However, as fantastic as the Morava K-theories are for giving us information at the various points of Spec(SHCﬁn), they do not tell us how to stitch that information back together.

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

Recall that the points of the space Spec(SHCﬁn) are the thick prime ideal subcategories of SHCﬁn.

Aaron Mazel-Gee You could’ve invented tmf . P∞,(2) P∞,(3) P∞,(5) P∞,(7) ··· ...... ···

P3,(2) P3,(3) P3,(5) P3,(7) ··· N0 ∪ {∞} P2,(2) P2,(3) P2,(7) P2,(5) ···

P1,(2) P1,(3) P1,(5) P1,(7) ···

P0 Spec(Z)

However, as fantastic as the Morava K-theories are for giving us information at the various points of Spec(SHCﬁn), they do not tell us how to stitch that information back together.

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

Recall that the points of the space Spec(SHCﬁn) are the thick ﬁn prime ideal subcategories of SHC . In fact, the point Pn,(p) is nothing more or less than the kernel of K(n, p)∗!

Aaron Mazel-Gee You could’ve invented tmf . However, as fantastic as the Morava K-theories are for giving us information at the various points of Spec(SHCﬁn), they do not tell us how to stitch that information back together.

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

Recall that the points of the space Spec(SHCﬁn) are the thick ﬁn prime ideal subcategories of SHC . In fact, the point Pn,(p) is nothing more or less than the kernel of K(n, p)∗!

P∞,(2) P∞,(3) P∞,(5) P∞,(7) ··· ...... ···

P3,(2) P3,(3) P3,(5) P3,(7) ··· N0 ∪ {∞} P2,(2) P2,(3) P2,(7) P2,(5) ···

P1,(2) P1,(3) P1,(5) P1,(7) ···

P0 Spec(Z)

Recall that the points of the space Spec(SHCﬁn) are the thick ﬁn prime ideal subcategories of SHC . In fact, the point Pn,(p) is nothing more or less than the kernel of K(n, p)∗!

P∞,(2) P∞,(3) P∞,(5) P∞,(7) ··· ...... ···

P3,(2) P3,(3) P3,(5) P3,(7) ··· N0 ∪ {∞} P2,(2) P2,(3) P2,(7) P2,(5) ···

P1,(2) P1,(3) P1,(5) P1,(7) ···

P0 Spec(Z)

However, as fantastic as the Morava K-theories are for giving us information at the various points of Spec(SHCﬁn), they do not tell us how to stitch that information back together.

Aaron Mazel-Gee You could’ve invented tmf . To define the Morava E-theories, we first need to define a deformation of a formal group law.

Let k be a perfect ﬁeld of characteristic p, let F be a formal group law over k, and let (A, m) be a complete local ring with projection π A A/m to its residue ﬁeld. −→ A deformation of F from k to A is a formal group law F over A i and a map k A/m such that π∗F = i ∗F over A/m. In algebraic geometry, the−→ picture looks like this:

[picture]

These collect into DefF /k (A).

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf The Morava E-theories

Aaron Mazel-Gee You could’ve invented tmf . Let k be a perfect ﬁeld of characteristic p, let F be a formal group law over k, and let (A, m) be a complete local ring with projection π A A/m to its residue ﬁeld. −→ A deformation of F from k to A is a formal group law F over A i and a map k A/m such that π∗F = i ∗F over A/m. In algebraic geometry, the−→ picture looks like this:

[picture]

These collect into DefF /k (A).

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf The Morava E-theories

To define the Morava E-theories, we first need to define a deformation of a formal group law.

Aaron Mazel-Gee You could’ve invented tmf . A deformation of F from k to A is a formal group law F over A i and a map k A/m such that π∗F = i ∗F over A/m. In algebraic geometry, the−→ picture looks like this:

[picture]

These collect into DefF /k (A).

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf The Morava E-theories

To define the Morava E-theories, we first need to define a deformation of a formal group law.

Let k be a perfect ﬁeld of characteristic p, let F be a formal group law over k, and let (A, m) be a complete local ring with projection π A A/m to its residue ﬁeld. −→

Aaron Mazel-Gee You could’ve invented tmf . In algebraic geometry, the picture looks like this:

[picture]

These collect into DefF /k (A).

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf The Morava E-theories

To define the Morava E-theories, we first need to define a deformation of a formal group law.

Aaron Mazel-Gee You could’ve invented tmf . These collect into DefF /k (A).

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf The Morava E-theories

To define the Morava E-theories, we first need to define a deformation of a formal group law.

[picture]

To define the Morava E-theories, we first need to define a deformation of a formal group law.

[picture]

These collect into DefF /k (A).

Aaron Mazel-Gee You could’ve invented tmf . [extend picture]

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

In fact, there is a universal deformation, which is a formal group law F living over the Lubin–Tate ring LTF /k such that

cts e DefF /k (A) Hom (LTF /k , A) ' f ∗F f ←[ (taking only local ring homomorphisms).e

In fact, there is a universal deformation, which is a formal group law F living over the Lubin–Tate ring LTF /k such that

cts e DefF /k (A) Hom (LTF /k , A) ' f ∗F f ←[ (taking only local ring homomorphisms).e [extend picture]

Aaron Mazel-Gee You could’ve invented tmf . For all n < , its universal deformation Hn,p is realized in topology by∞ a complex-oriented cohomology theory called the nth Morava E-theory, denoted En,p. This livese over the ring

LTn,p = LTHn,p/Fp = Zp[[u1,..., un−1]]. Here are a few edge cases. ∧ LT1,p = Zp and H1,p = (Gm)Zp , and so E1,p = KUp (p-completed complex K-theory).

Again, even thoughe there’sb no H0,p and hence no H0,p, we can reasonably deﬁne E0,p = HQ (rational singular cohomology) for any prime p. e

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

Recall that we have the Honda formal group law Hn,p over Fp.

Aaron Mazel-Gee You could’ve invented tmf . This lives over the ring

LTn,p = LTHn,p/Fp = Zp[[u1,..., un−1]]. Here are a few edge cases. ∧ LT1,p = Zp and H1,p = (Gm)Zp , and so E1,p = KUp (p-completed complex K-theory).

Again, even thoughe there’sb no H0,p and hence no H0,p, we can reasonably deﬁne E0,p = HQ (rational singular cohomology) for any prime p. e

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

Recall that we have the Honda formal group law Hn,p over Fp.

For all n < , its universal deformation Hn,p is realized in topology by∞ a complex-oriented cohomology theory called the nth Morava E-theory, denoted En,p. e

Aaron Mazel-Gee You could’ve invented tmf . Here are a few edge cases. ∧ LT1,p = Zp and H1,p = (Gm)Zp , and so E1,p = KUp (p-completed complex K-theory).

Again, even thoughe there’sb no H0,p and hence no H0,p, we can reasonably deﬁne E0,p = HQ (rational singular cohomology) for any prime p. e

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

Recall that we have the Honda formal group law Hn,p over Fp.

For all n < , its universal deformation Hn,p is realized in topology by∞ a complex-oriented cohomology theory called the nth Morava E-theory, denoted En,p. This livese over the ring

LTn,p = LTHn,p/Fp = Zp[[u1,..., un−1]].

Aaron Mazel-Gee You could’ve invented tmf . ∧ LT1,p = Zp and H1,p = (Gm)Zp , and so E1,p = KUp (p-completed complex K-theory).

Again, even thoughe there’sb no H0,p and hence no H0,p, we can reasonably deﬁne E0,p = HQ (rational singular cohomology) for any prime p. e

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

Recall that we have the Honda formal group law Hn,p over Fp.

For all n < , its universal deformation Hn,p is realized in topology by∞ a complex-oriented cohomology theory called the nth Morava E-theory, denoted En,p. This livese over the ring

LTn,p = LTHn,p/Fp = Zp[[u1,..., un−1]]. Here are a few edge cases.

Aaron Mazel-Gee You could’ve invented tmf . Again, even though there’s no H0,p and hence no H0,p, we can reasonably deﬁne E0,p = HQ (rational singular cohomology) for any prime p. e

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

Recall that we have the Honda formal group law Hn,p over Fp.

For all n < , its universal deformation Hn,p is realized in topology by∞ a complex-oriented cohomology theory called the nth Morava E-theory, denoted En,p. This livese over the ring

LTn,p = LTHn,p/Fp = Zp[[u1,..., un−1]]. Here are a few edge cases. ∧ LT1,p = Zp and H1,p = (Gm)Zp , and so E1,p = KUp (p-completed complex K-theory). e b

Recall that we have the Honda formal group law Hn,p over Fp.

For all n < , its universal deformation Hn,p is realized in topology by∞ a complex-oriented cohomology theory called the nth Morava E-theory, denoted En,p. This livese over the ring

LTn,p = LTHn,p/Fp = Zp[[u1,..., un−1]]. Here are a few edge cases. ∧ LT1,p = Zp and H1,p = (Gm)Zp , and so E1,p = KUp (p-completed complex K-theory).

Again, even thoughe there’sb no H0,p and hence no H0,p, we can reasonably deﬁne E0,p = HQ (rational singular cohomology) for any prime p. e

Aaron Mazel-Gee You could’ve invented tmf . This is reﬂected in topology!

For any space X ,(En,p)∗X = 0 if and only if K(i, p)∗X = 0 for i [0, n]. So, the kernel of En,p is P0, P ,..., P . ∈ { 1,(p) n,(p)}

P∞,(2) P∞,(3) P∞,(5) P∞,(7) ··· ...... ···

P3,(2) P3,(3) P3,(5) P3,(7) ··· N0 ∪ {∞} P2,(2) P2,(3) P2,(7) P2,(5) ···

P1,(2) P1,(3) P1,(7) P1,(5) ···

P0 Spec(Z)

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

There’s a sense in which a deformation of Hn can have any height up to n.

Aaron Mazel-Gee You could’ve invented tmf . For any space X ,(En,p)∗X = 0 if and only if K(i, p)∗X = 0 for i [0, n]. So, the kernel of En,p is P0, P ,..., P . ∈ { 1,(p) n,(p)}

P∞,(2) P∞,(3) P∞,(5) P∞,(7) ··· ...... ···

P3,(2) P3,(3) P3,(5) P3,(7) ··· N0 ∪ {∞} P2,(2) P2,(3) P2,(7) P2,(5) ···

P1,(2) P1,(3) P1,(7) P1,(5) ···

P0 Spec(Z)

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

There’s a sense in which a deformation of Hn can have any height up to n.

This is reﬂected in topology!

Aaron Mazel-Gee You could’ve invented tmf . So, the kernel of En,p is P0, P ,..., P . { 1,(p) n,(p)}

P∞,(2) P∞,(3) P∞,(5) P∞,(7) ··· ...... ···

P3,(2) P3,(3) P3,(5) P3,(7) ··· N0 ∪ {∞} P2,(2) P2,(3) P2,(7) P2,(5) ···

P1,(2) P1,(3) P1,(7) P1,(5) ···

P0 Spec(Z)

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

There’s a sense in which a deformation of Hn can have any height up to n.

This is reﬂected in topology!

For any space X ,(En,p)∗X = 0 if and only if K(i, p)∗X = 0 for i [0, n]. ∈

Aaron Mazel-Gee You could’ve invented tmf . P∞,(2) P∞,(3) P∞,(5) P∞,(7) ··· ...... ···

P3,(2) P3,(3) P3,(5) P3,(7) ··· N0 ∪ {∞} P2,(2) P2,(3) P2,(7) P2,(5) ···

P1,(2) P1,(3) P1,(7) P1,(5) ···

P0 Spec(Z)

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

There’s a sense in which a deformation of Hn can have any height up to n.

This is reﬂected in topology!

For any space X ,(En,p)∗X = 0 if and only if K(i, p)∗X = 0 for i [0, n]. So, the kernel of En,p is P0, P ,..., P . ∈ { 1,(p) n,(p)}

There’s a sense in which a deformation of Hn can have any height up to n.

This is reﬂected in topology!

For any space X ,(En,p)∗X = 0 if and only if K(i, p)∗X = 0 for i [0, n]. So, the kernel of En,p is P0, P ,..., P . ∈ { 1,(p) n,(p)}

P∞,(2) P∞,(3) P∞,(5) P∞,(7) ··· ...... ···

P3,(2) P3,(3) P3,(5) P3,(7) ··· N0 ∪ {∞} P2,(2) P2,(3) P2,(7) P2,(5) ···

P1,(2) P1,(3) P1,(7) P1,(5) ···

P0 Spec(Z)

Aaron Mazel-Gee You could’ve invented tmf . But what about arithmetic globalization?

The ﬁnite stable homotopy category Formal group laws in topology Chromatic homotopy theory The Morava K-theories Topological modular forms The Morava E-theories Fun with tmf

Thus, the Morava E-theories aﬀord us a notion of chromatic globalization, i.e. of stitching together information in the chromatic direction.

But what about arithmetic globalization?

3. Topological modular forms

Aaron Mazel-Gee You could’ve invented tmf . We can see the En,p as telling us how to globalize in the chromatic direction; what’s much more subtle is the question of how to globalize in the arithmetic direction.

A global height-n theory should allow us to recover the En,p at all primes p.

P∞,(2) P∞,(3) P∞,(5) P∞,(7) ··· ...... ···

P3,(2) P3,(3) P3,(5) P3,(7) ··· N0 ∪ {∞} P2,(2) P2,(3) P2,(5) P2,(7) ··· P1,(2) P1,(3) P1,(5) P1,(7) ···

P0 Spec(Z)

The ﬁnite stable homotopy category Towards tmf Chromatic homotopy theory So what is this sheaf, anyways? Topological modular forms ...and how is it constructed? Fun with tmf Towards tmf

Aaron Mazel-Gee You could’ve invented tmf . what’s much more subtle is the question of how to globalize in the arithmetic direction.

A global height-n theory should allow us to recover the En,p at all primes p.

P∞,(2) P∞,(3) P∞,(5) P∞,(7) ··· ...... ···

P3,(2) P3,(3) P3,(5) P3,(7) ··· N0 ∪ {∞} P2,(2) P2,(3) P2,(5) P2,(7) ··· P1,(2) P1,(3) P1,(5) P1,(7) ···

P0 Spec(Z)

The ﬁnite stable homotopy category Towards tmf Chromatic homotopy theory So what is this sheaf, anyways? Topological modular forms ...and how is it constructed? Fun with tmf Towards tmf

We can see the En,p as telling us how to globalize in the chromatic direction;

Aaron Mazel-Gee You could’ve invented tmf . A global height-n theory should allow us to recover the En,p at all primes p.

P∞,(2) P∞,(3) P∞,(5) P∞,(7) ··· ...... ···

P3,(2) P3,(3) P3,(5) P3,(7) ··· N0 ∪ {∞} P2,(2) P2,(3) P2,(5) P2,(7) ··· P1,(2) P1,(3) P1,(5) P1,(7) ···

P0 Spec(Z)

The ﬁnite stable homotopy category Towards tmf Chromatic homotopy theory So what is this sheaf, anyways? Topological modular forms ...and how is it constructed? Fun with tmf Towards tmf

We can see the En,p as telling us how to globalize in the chromatic direction; what’s much more subtle is the question of how to globalize in the arithmetic direction.

Aaron Mazel-Gee You could’ve invented tmf . P∞,(2) P∞,(3) P∞,(5) P∞,(7) ··· ...... ···

P3,(2) P3,(3) P3,(5) P3,(7) ··· N0 ∪ {∞} P2,(2) P2,(3) P2,(5) P2,(7) ··· P1,(2) P1,(3) P1,(5) P1,(7) ···

P0 Spec(Z)

The ﬁnite stable homotopy category Towards tmf Chromatic homotopy theory So what is this sheaf, anyways? Topological modular forms ...and how is it constructed? Fun with tmf Towards tmf

We can see the En,p as telling us how to globalize in the chromatic direction; what’s much more subtle is the question of how to globalize in the arithmetic direction.

A global height-n theory should allow us to recover the En,p at all primes p.

We can see the En,p as telling us how to globalize in the chromatic direction; what’s much more subtle is the question of how to globalize in the arithmetic direction.

A global height-n theory should allow us to recover the En,p at all primes p.

P∞,(2) P∞,(3) P∞,(5) P∞,(7) ··· ...... ···

P3,(2) P3,(3) P3,(5) P3,(7) ··· N0 ∪ {∞} P2,(2) P2,(3) P2,(5) P2,(7) ··· P1,(2) P1,(3) P1,(5) P1,(7) ···

P0 Spec(Z)

Aaron Mazel-Gee You could’ve invented tmf . For all p, E0,p is just HQ, rational cohomology. So, obviously we can take HQ as a global height-0 theory!

∧ Next, E1,p is KUp , p-completed complex K-theory. What object ∧ might allow us to recover the KUp at all primes p? Well, let’s just not p-complete the darn thing! We can take KU as a global height-1 theory.

The ﬁnite stable homotopy category Towards tmf Chromatic homotopy theory So what is this sheaf, anyways? Topological modular forms ...and how is it constructed? Fun with tmf

So, what are some examples of global height-n theories?

Aaron Mazel-Gee You could’ve invented tmf . So, obviously we can take HQ as a global height-0 theory!

The ﬁnite stable homotopy category Towards tmf Chromatic homotopy theory So what is this sheaf, anyways? Topological modular forms ...and how is it constructed? Fun with tmf

So, what are some examples of global height-n theories?

For all p, E0,p is just HQ, rational cohomology.

Aaron Mazel-Gee You could’ve invented tmf . ∧ Next, E1,p is KUp , p-completed complex K-theory. What object ∧ might allow us to recover the KUp at all primes p? Well, let’s just not p-complete the darn thing! We can take KU as a global height-1 theory.

The ﬁnite stable homotopy category Towards tmf Chromatic homotopy theory So what is this sheaf, anyways? Topological modular forms ...and how is it constructed? Fun with tmf

So, what are some examples of global height-n theories?

For all p, E0,p is just HQ, rational cohomology. So, obviously we can take HQ as a global height-0 theory!

Aaron Mazel-Gee You could’ve invented tmf . What object ∧ might allow us to recover the KUp at all primes p? Well, let’s just not p-complete the darn thing! We can take KU as a global height-1 theory.

The ﬁnite stable homotopy category Towards tmf Chromatic homotopy theory So what is this sheaf, anyways? Topological modular forms ...and how is it constructed? Fun with tmf

So, what are some examples of global height-n theories?

For all p, E0,p is just HQ, rational cohomology. So, obviously we can take HQ as a global height-0 theory!

∧ Next, E1,p is KUp , p-completed complex K-theory.

Aaron Mazel-Gee You could’ve invented tmf . Well, let’s just not p-complete the darn thing! We can take KU as a global height-1 theory.

The ﬁnite stable homotopy category Towards tmf Chromatic homotopy theory So what is this sheaf, anyways? Topological modular forms ...and how is it constructed? Fun with tmf

So, what are some examples of global height-n theories?

For all p, E0,p is just HQ, rational cohomology. So, obviously we can take HQ as a global height-0 theory!

∧ Next, E1,p is KUp , p-completed complex K-theory. What object ∧ might allow us to recover the KUp at all primes p?

Aaron Mazel-Gee You could’ve invented tmf . We can take KU as a global height-1 theory.

The ﬁnite stable homotopy category Towards tmf Chromatic homotopy theory So what is this sheaf, anyways? Topological modular forms ...and how is it constructed? Fun with tmf

So, what are some examples of global height-n theories?

For all p, E0,p is just HQ, rational cohomology. So, obviously we can take HQ as a global height-0 theory!

∧ Next, E1,p is KUp , p-completed complex K-theory. What object ∧ might allow us to recover the KUp at all primes p? Well, let’s just not p-complete the darn thing!

So, what are some examples of global height-n theories?

For all p, E0,p is just HQ, rational cohomology. So, obviously we can take HQ as a global height-0 theory!

Aaron Mazel-Gee You could’ve invented tmf . We deﬁne a quasicoherent sheaf of cohomology theories over X = Spec(Z), which we denote : we take the global sections to be KU (X ) = KU, KU and then by quasicoherence, evaluation on X ∧ X yields p ⊂ ∧ ∧ (Xp ) = KU Zp = KUp . KU ⊗ (Quasicoherence just means that web have this tensoring-up formula.)

This corresponds to the sheaf of formal groups over X determined ∧ by Gm, which evaluates as Gm(Xp ) = (Gm)Zp = H1,p.

b b b e

The ﬁnite stable homotopy category Towards tmf Chromatic homotopy theory So what is this sheaf, anyways? Topological modular forms ...and how is it constructed? Fun with tmf

In order to go further, let’s reimagine this a bit.

Aaron Mazel-Gee You could’ve invented tmf . we take the global sections to be (X ) = KU, KU and then by quasicoherence, evaluation on X ∧ X yields p ⊂ ∧ ∧ (Xp ) = KU Zp = KUp . KU ⊗ (Quasicoherence just means that web have this tensoring-up formula.)

This corresponds to the sheaf of formal groups over X determined ∧ by Gm, which evaluates as Gm(Xp ) = (Gm)Zp = H1,p.

b b b e

The ﬁnite stable homotopy category Towards tmf Chromatic homotopy theory So what is this sheaf, anyways? Topological modular forms ...and how is it constructed? Fun with tmf

In order to go further, let’s reimagine this a bit.

We deﬁne a quasicoherent sheaf of cohomology theories over X = Spec(Z), which we denote : KU

Aaron Mazel-Gee You could’ve invented tmf . and then by quasicoherence, evaluation on X ∧ X yields p ⊂ ∧ ∧ (Xp ) = KU Zp = KUp . KU ⊗ (Quasicoherence just means that web have this tensoring-up formula.)

This corresponds to the sheaf of formal groups over X determined ∧ by Gm, which evaluates as Gm(Xp ) = (Gm)Zp = H1,p.

b b b e

The ﬁnite stable homotopy category Towards tmf Chromatic homotopy theory So what is this sheaf, anyways? Topological modular forms ...and how is it constructed? Fun with tmf

In order to go further, let’s reimagine this a bit.

We deﬁne a quasicoherent sheaf of cohomology theories over X = Spec(Z), which we denote : we take the global sections to be KU (X ) = KU, KU

Aaron Mazel-Gee You could’ve invented tmf . This corresponds to the sheaf of formal groups over X determined ∧ by Gm, which evaluates as Gm(Xp ) = (Gm)Zp = H1,p.

b b b e

The ﬁnite stable homotopy category Towards tmf Chromatic homotopy theory So what is this sheaf, anyways? Topological modular forms ...and how is it constructed? Fun with tmf

In order to go further, let’s reimagine this a bit.

We deﬁne a quasicoherent sheaf of cohomology theories over X = Spec(Z), which we denote : we take the global sections to be KU (X ) = KU, KU and then by quasicoherence, evaluation on X ∧ X yields p ⊂ ∧ ∧ (Xp ) = KU Zp = KUp . KU ⊗ (Quasicoherence just means that web have this tensoring-up formula.)

In order to go further, let’s reimagine this a bit.

This corresponds to the sheaf of formal groups over X determined ∧ by Gm, which evaluates as Gm(Xp ) = (Gm)Zp = H1,p.

b Aaron Mazel-Geeb You could’veb invented tmfe. The previous case was easy, since all the H1,p came from the same formal group Gm. The H2,p are much more complicated! e However, we haveb a clue:e Recall that we can obtain a formal group as the germ of (1-dimensional commutative) algebraic group.

Besides Ga and Gm, what other 1-dimensional commutative algebraic groups are there, anyways?

The ﬁnite stable homotopy category Towards tmf Chromatic homotopy theory So what is this sheaf, anyways? Topological modular forms ...and how is it constructed? Fun with tmf

So, to get a global height-2 theory, we should look for some object with a sheaf of formal groups which contains as sections the H2,p for all primes p. e

Aaron Mazel-Gee You could’ve invented tmf . The H2,p are much more complicated!

However, we have a clue:e Recall that we can obtain a formal group as the germ of (1-dimensional commutative) algebraic group.

Besides Ga and Gm, what other 1-dimensional commutative algebraic groups are there, anyways?

The ﬁnite stable homotopy category Towards tmf Chromatic homotopy theory So what is this sheaf, anyways? Topological modular forms ...and how is it constructed? Fun with tmf

Aaron Mazel-Gee You could’ve invented tmf . However, we have a clue: Recall that we can obtain a formal group as the germ of (1-dimensional commutative) algebraic group.

Besides Ga and Gm, what other 1-dimensional commutative algebraic groups are there, anyways?

The ﬁnite stable homotopy category Towards tmf Chromatic homotopy theory So what is this sheaf, anyways? Topological modular forms ...and how is it constructed? Fun with tmf

Aaron Mazel-Gee You could’ve invented tmf . Recall that we can obtain a formal group as the germ of (1-dimensional commutative) algebraic group.

Besides Ga and Gm, what other 1-dimensional commutative algebraic groups are there, anyways?

The ﬁnite stable homotopy category Towards tmf Chromatic homotopy theory So what is this sheaf, anyways? Topological modular forms ...and how is it constructed? Fun with tmf

So, to get a global height-2 theory, we should look for some object with a sheaf of formal groups which contains as sections the H2,p for all primes p. e The previous case was easy, since all the H1,p came from the same formal group Gm. The H2,p are much more complicated! e However, we haveb a clue:e

Aaron Mazel-Gee You could’ve invented tmf . Besides Ga and Gm, what other 1-dimensional commutative algebraic groups are there, anyways?

The ﬁnite stable homotopy category Towards tmf Chromatic homotopy theory So what is this sheaf, anyways? Topological modular forms ...and how is it constructed? Fun with tmf

Besides Ga and Gm, what other 1-dimensional commutative algebraic groups are there, anyways?

Aaron Mazel-Gee You could’ve invented tmf . Very luckily, some elliptic curves over Fp have formal group of height 2! Such elliptic curves are called supersingular. (The rest, which have height 1, are called ordinary.)

For each prime p, there is a moduli of deformations of supersingular elliptic curves, denoted ss , and its canonical Mell,p sheaf of formal groups does indeed contain H2,p.

So, we might hope to bring these all togethere and deﬁne a sheaf of cohomology theories over ss . p Mell,p `

The ﬁnite stable homotopy category Towards tmf Chromatic homotopy theory So what is this sheaf, anyways? Topological modular forms ...and how is it constructed? Fun with tmf

As it turns out, there is only one other sort of 1-dimensional commutative algebraic group besides Ga and Gm: the elliptic curves.

Aaron Mazel-Gee You could’ve invented tmf . Such elliptic curves are called supersingular. (The rest, which have height 1, are called ordinary.)

For each prime p, there is a moduli of deformations of supersingular elliptic curves, denoted ss , and its canonical Mell,p sheaf of formal groups does indeed contain H2,p.

So, we might hope to bring these all togethere and deﬁne a sheaf of cohomology theories over ss . p Mell,p `

The ﬁnite stable homotopy category Towards tmf Chromatic homotopy theory So what is this sheaf, anyways? Topological modular forms ...and how is it constructed? Fun with tmf

As it turns out, there is only one other sort of 1-dimensional commutative algebraic group besides Ga and Gm: the elliptic curves.

Very luckily, some elliptic curves over Fp have formal group of height 2!

Aaron Mazel-Gee You could’ve invented tmf . (The rest, which have height 1, are called ordinary.)

For each prime p, there is a moduli of deformations of supersingular elliptic curves, denoted ss , and its canonical Mell,p sheaf of formal groups does indeed contain H2,p.

So, we might hope to bring these all togethere and deﬁne a sheaf of cohomology theories over ss . p Mell,p `

The ﬁnite stable homotopy category Towards tmf Chromatic homotopy theory So what is this sheaf, anyways? Topological modular forms ...and how is it constructed? Fun with tmf

As it turns out, there is only one other sort of 1-dimensional commutative algebraic group besides Ga and Gm: the elliptic curves.

Very luckily, some elliptic curves over Fp have formal group of height 2! Such elliptic curves are called supersingular.

Aaron Mazel-Gee You could’ve invented tmf . For each prime p, there is a moduli of deformations of supersingular elliptic curves, denoted ss , and its canonical Mell,p sheaf of formal groups does indeed contain H2,p.

So, we might hope to bring these all togethere and deﬁne a sheaf of cohomology theories over ss . p Mell,p `

The ﬁnite stable homotopy category Towards tmf Chromatic homotopy theory So what is this sheaf, anyways? Topological modular forms ...and how is it constructed? Fun with tmf

As it turns out, there is only one other sort of 1-dimensional commutative algebraic group besides Ga and Gm: the elliptic curves.

Very luckily, some elliptic curves over Fp have formal group of height 2! Such elliptic curves are called supersingular. (The rest, which have height 1, are called ordinary.)

Aaron Mazel-Gee You could’ve invented tmf . So, we might hope to bring these all together and deﬁne a sheaf of cohomology theories over ss . p Mell,p `