Notes

Here is the homepage for the book based on the 2007 Talbot workshop.

Talk Schedule

-- Monday --

1. Historic Overview. [Corbett]
Introduce general (co)homology theories and the examples of $K$-theory and complex cobordism. Define a genus as a multiplicative invariant of almost complex manifolds and give the example of the Todd Genus. State Quillen's theorem and establish the equivalence between genera with values in $R$, and formal group laws (FGL) over $R$, under certain conditions, given a genus $g:MU_* \to R$, one can find a homology theory $E_*(-)$ with $E_*=R$, represented by a spectrum $E$, and a map of ring spectra $MU \to E$ realizing $g$ at the level of homotopy. Define elliptic cohomology.

2. Elliptic curves and Modular forms. [Carl]
Introduce elliptic curves as cubic curves in $\mathbb{P}^2$, and as one dimensional abelian group schemes. Give a hint of why the two definitions are equivalent. Give some examples of elliptic curves. Define the canonical line bundle ω over the modui space of elliptic curves as an assignment {elliptic curve over some field} $\mapsto$ {one dimensional vector space over that field}. Define modular forms as sections of $\omega^n$ and give some examples of them. Describe the ring of integral modular forms.

3. Algebraic stacks. [Nick]
Explain the formalism of functor of points and introduce general stacks. Give the examples of the moduli space of elliptic curves $M_{Ell}$ and the moduli space of formal groups $M_{FG}$. Explain what a groupoid object in schemes is and how it represents a stack. Work out the groupoid representing $M_{Ell}$. Explain what a line bundle over a stack is and construct $\omega$ rigorously. Show that $\omega$ is pulled back from a map $M_{Ell}\to M_{FG}$.

4. The geometry of $M_{Ell}$. [Andre]
Introduce the notions of $p$-typical formal group law and explain why every formal group law over a field can be isomorphed to a $p$-typical one. State the classification of formal groups over algebraically closed fields. Introduce the notions of ordinary versus supersingular elliptic curves and give some examples at the primes 2 and 3. The stack of supersingular elliptic curves $M_{Ell}^{\mathit{ss}}$ is a zero dimensional substack of $M_{Ell}\otimes \Z/p$, and $M_{Ell}^{\mathit{ord}}$ is its open complement. The stack $M_{Ell}^{\mathit{ord}}$ still makes sense as an open substack of $M_{Ell} \otimes \Z_p$. Explain why $M_{Ell}^{\mathit{ss}}$ is not well defined as a substack of $M_{Ell}\otimes \Z_p$, but its formal neighborhood still makes sense.

-- Tuesday --

5. The Landweber exact functor theorem. [Henning]
Explain how one assigns a FG to an even periodic cohomology theory. Introduce $MU$ and its periodic version $MP$. State Quillen's theorem and explain why the groupoid $\mathrm{Spec}(MP_0MP)\to \mathrm{Spec}(MP_0)$ represents $M_{FG}$. State the Landweber exact functor theorem (LEFT). Define what it means for a map from $\mathrm{Spec}(R)$ to a stack to be flat. Explain why the statement of LEFT is equivalent to the following statement: If a map $MP_0\to R$ induces a flat map $\mathrm{Spec}(R) \to M_{FG}$, then the functor $X \to MP_{*}(X)\otimes_{P_0} R$ is a homology theory. Prove that the map $M_{Ell} \to M_{FG}$ is flat and use it to build a presheaf {flat maps from an affine scheme to $M_{Ell}$} $\to$ {even periodic homology theories}.

6. $E_\infty$ ring spectra. [Matthew]
Recall what a spectrum is and introduce the formalism of symmetric spectra. Explain what operads are, and give the examples of the $A_\infty$ and of the $E_\infty$ operads. Give two possible definitions of $E_\infty$ ring spectra: as spectra with a multiplication which is commutative up to infinitely many homotopies i.e. which is equipped with an action of an $E_\infty$ operad, and as a spectrum with a strictly commutative multiplication. Give a hint of why those two approaches are equivalent. Explain the philosophy of why {(co)homology theories} = {spectra} but {cohomology theories equipped with a commutative product} $\neq$ {$E_\infty$ ring spectra}.

7. Sheaves in homotopy theory. [Chris]
Explain what a sheaf on a stack is, and why it's not that different from a sheaf on a space. Define sheaf cohomology. If a presheaf takes values in a category in which homotopy limits make sense, then one can modify the descent condition to include higher homotopies on the $n$-tuple intersections: this is the homotopy meaningful replacement of the concept of sheaf. Setup the spectral sequence (SS) which, given a sheaf of spectra $F$ over $X$ computes the homotopy groups of the spectrum of global section π_{*}($F$($X)$). Its $E$_{2} term is the sheaf cohomology of the sheafification of the presheaf $U$ |\to π_{*}($F$($U)$). A special case of this SS will be used in talk 11 to compute π_{*}($F)$ .

8. The $F$ theorem. [Mike Hopkins]
This talk contains a survey of the second half of the workshop. The main theorem is about the existence of a sheaf of $E_\infty$ ring spectra $O^{top}$ over $M_{Ell}$ that recovers the presheaf of talk 5 at the level of the corresponding homology theories. The spectrum of global sections of that sheaf is $F$.

-- Wednesday --

9. The Hasse square. [Tilman]
Explain what it means to localize a spectrum with respect to a cohomology theory and give the examples of rationalization and $p$-completion. State Sullivan's arithmetic square. Define the Morava $K$-theories $K$($n)$ and the corresponding localization functors $L$_{$K$($n)$}. When applied to Landweber exact cohomology theories, $L$_{$K$($n)$} kills everything that is not of height $n$. When applied to the sheaf of spectra $O^{top}$ over $M_{Ell}$, explain why $L$_{$K$(2)} = (completion around the supersingular locus) and $L$_{$K$(1)} = (localization away from the supersingular locus). Prove that, given a $K$(1)v$K$(2)-local spectrum $X$, the Hasse square $L$_{$K$(2)} $X$ // $L$_{$K$(1)} $L$_{$K$(1)} $L$_{$K$(2)} $X$ is a homotopy pullback.

10. Model categories. [Andrew]
Introduce the formalism of cofibrantly generated model categories. Explain the notions of fibrant and cofibrant objects. Describe the model structure on the category of $E_\infty$ ring spectra. Given a model category $M$ and a space (or stack) $X$, describe the Jardine model category structure on the category of sheaves on $X$ with values in $M$. Show that the fibrant objects in that model category are sheaves in the sense of talk 7.

11. The big spectral sequence. [Mike Hopkins]
Mike will explain what the big spectral sequence $H^{s}(M_{Ell}\omega^{t}) \Rightarrow \pi_s(F)$ looks like, and give some applications.

12. Universal deformations. [Jacob]
Explain what it means to be a universal deformations of a FGL over a field $k$, and why it's equivalent to taking formal neighborhoods of points inside the stack $M_{FG}$. Give Lubin-Tate's explicit description of the universal deformation of a FGL. Define and describe the Morava $E$-theories $E_{n}$. Prove the Serre-Tate theorem according to which the formal neighborhood in $M_{Ell}$ of a supersingular curve is isomorphic to the formal neighborhood of the corresponding point in $M_{FG}$.

-- Thursday --

13. Goerss-Hopkins obstruction theory. [Vigleik]
The Goerss-Hopkins obstruction theory tries to answer the following question: Suppose $E_{*}(-)$ is a "good" homology theory and we have an $E_{*}$-algebra $A$ in the cateogry of $E_{*}E$ comodules ("good" means that spectrum $E$ has a homotopy commutative multiplication, that $E_{*}E$ is flat over $E_{*}$, and that the Adams condition is satisfied). Is there an $E_\infty$ ring spectrum $X$ with $E_{*}X=A$? More precisely, what is the moduli space of such $E_\infty$ ring spectra? To answer this question, one defines a new type of homotopy groups which have a good theory of Postnikov towers and of $k$-invariants. One then tries to build $X$ one Postnikov section at a time, and identify the obstructions to the existence of the right $k$-invariants. The answer turns out to lie in certain Andre-Quillen cohomology groups. This theory can also be used to compute the homotopy groups of the space of $E_\infty$ maps between two $E_\infty$ ring spectra. In talks 14 and 16, this theory will be specialized to the cases $E=E_{n}$, and $E=K(1)$ respectively.

14. The Hopkin-Miller theorem. [John F.]
Given a formal group law over a field, its universal deformation is always Landweber exact, i.e. the corresponding map $\mathrm{Spec}(R) \to M_{FG}$ is flat. Thus, we get a functor {formal group laws over fields} $\to$ {even periodic homology theories}, by first taking the universal deformation, and then applying LEFT. The Hopkins-Miller theorem says that one can lift that functor to the category of $E_\infty$ ring spectra. The existence of such a lift is proven by considering all those $E_\infty$ ring spectra which could possibly be in the image of the functor. They form a category, which is naturally equipped with a functor to the category of FG's over fields. Using the GH obstruction theory, one shows that that functor is a homotopy equivalence, and thus that it has an inverse. We then apply the Hopkins-Miller theorem to construct $L_{K(2)} O^{top}$ and $L_{K(2)} F$.

15. The string orientation. [Mike Hopkins]
In this talk, Mike will introduce Rezk's logarithm and construct the $O\langle 8\rangle$-orientation, which is a map of $E_\infty$ ring spectra $O\langle 8\rangle \to F$.

-- Friday --

16. $K(1)$-local obstruction theory. [Mike Hill]
When working inside the cateogry of $K(1)$-local spectra, the appropriate variant of GH obstruction theory is the one based on $K(1)$. This has the advantage that the algebra playing the role of the Dyer-Lashof algebra is very small: it's generated by a single operation $\theta : x \mapsto (\psi_{p}(x) - x^{p}) / p$. So our variant of GH obstruction theory now looks as follows: it starts with a $K(1)_{*}K(1)$-comodule algebra, and it tells us whether there exists a $K(1)$-local $E_\infty$ ring spectrum $X$, whose $K(1)$-homology $K(1)_{*}X$ has the above structure.

17. The construction of $F$. [Mark]
Even though we don't know yet what $O^{top}$ is, we know what $K(1)_{*}(O^{top})$ should look like, as a sheaf of $K(1)_{*}K(1)$-comodule algebras. Applying the GH obstruction theory in the category of (pre)sheaves of $K(1)$-local spectra, one shows that there's a unique possible value for $L_{K(1)} (O^{top})$. Similarly, one can compute $K(1)_{*} (L_K(2)) O^{top})$ as a sheaf of $K(1)_{*}K(1)$-comodule algebras. Applying again the GH obstruction theory, one checks that there's a unique possible map $L_{K(1)} O^{top}\to L_{K(2)} O^{top}$. One then defines the $p$-completion of $O^{top}$ to be the homotopy pullback of $L_{K(1)} O^{top} \to L_{K(1)} L_{K(2)} O^{top}\leftarrow L_{K(2)} O^{top}$. Finally, one uses Sullivan's arithmetic square to construct the global version of $O^{top}$.

18. Title to be announced. [Mike Hopkins]
In this talk, Mike could talk about the fiber of the logarithm, relations with number theory, things related to the work of Mazur and Wiles, or other stuff...