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• Tags
  • #homotopy/stable-homotopy
• Refs:
  • Jacob Lurie. Chromatic homotopy theory. #resources/notes/lectures
    • An overview of the lecture titles can be found here and possible elucidations of the material can be found in the lecture notes
    • Lecture 1. #projects/to-read
    • Lecture 2.
    • Lecture 3.
    • Lecture 4.
    • Lecture 5.
    • Lecture 6.
    • Lecture 7.
    • Lecture 8.
    • Lecture 9.
    • Lecture 10.
    • Lecture 11.
    • Lecture 12.
    • Lecture 13.
    • Lecture 14.
    • Lecture 15.
    • Lecture 16.
    • Lecture 17.
    • Lecture 18.
    • Lecture 19.
    • Lecture 20.
    • Lecture 21.
    • Lecture 22.
    • Lecture 23.
    • Lecture 24.
    • Lecture 25.
    • Lecture 26.
    • Lecture 27.
    • Lecture 28.
    • Lecture 29.
    • Lecture 30.
    • Lecture 31.
    • Lecture 32.
    • Lecture 33.
    • Lecture 34.
    • Lecture 35.
  • Piotr Pstrągowski. Finite height chromatic homotopy theory. #resources/notes/lectures\
  • Lennart Meier. Elliptic homology and topological modular forms. #resources/notes/lectures
  • Michael Hopkins. Complex oriented cohomology theories and the language of stacks. #resources/notes/lectures
  • Sanath Devalapurkar. Chromatic homotopy theory. #resources/notes/lectures
  • Behrens Hopkins Hill #resources/papers
  • A lot of references in the bibliography here: http://math.uchicago.edu/~may/REU2019/REUPapers/Ni,Colin.pdf#page=22 #resources/summaries
• Links:
  • stable homotopy
  • phantom maps
  • chromatic homotopy theory
  • Fracture theorem
  • The chromatic spectral sequence
  • Nishida’s Theorem
  • Bousfield localization
  • Morava stabilizer group
    • Morava E theory
  • Formal group
    • Formal group
    • Lubin-Tate space
    • Lubin-Tate theory
  • Kervaire invariant 1
  • topological modular forms
  • Dieudonne module
  • Redshift
  • K-theory
  • p-divisible group
  • Balmer spectrum

chromatic homotopy theory

See tensor triangular geometry.

The chromatic view-point, which studies stable homotopy theory via its relationships to the moduli of formal groups, and related topics such as topological modular forms, use a sizable amount of (fairly abstract) algebraic geometry. And Lurie’s work on derived algebraic geometry was motivated in part by establishing foundations adequate to the task of defining equivariant forms of topological modular forms.

Kervaire and Milnor defined \(\Theta_n\) to be the group of homotopy spheres up to h-cobordism (where the group operation is given by connect sum).

By the h-cobordism theorem (\(n > 4\)) and Perelman’s proof of the Unsorted/Poincare conjectures (\(n = 3\)).

For \(n \neq 4\), \(\Theta_n = 0\) if and only if \(S^n\) has a unique smooth structure (i.e. there are no exotic spheres of dimension \(n\)).

We wish to consider the following question: For which \(n\) is \(\Theta_n = 0\)? The general belief is that there should be finitely many such \(n\), and these n should be concentrated in relatively low dimensions.

Relation to complex cobordism, complex oriented cohomology theories, and FGLs via the Landweber exact functor theorem.

See Morava E theory, Morava K theory, thick subcategory, tmf.

Relation to stacks:

The chromatic tower

Fix a prime \(p\) the chromatic tower of a Thom spectrum \(X\) is the tower of Bousfield localization: \begin{align*} X \rightarrow \cdots \rightarrow X _ { E ( n ) } \rightarrow X _ { E ( n - 1 ) } \rightarrow \cdots \rightarrow X _ { E ( 0 ) } \end{align*} where \(E(n)\) is the \(n\)th Johnson-Wilson spectrum \((E(0) = { \mathsf{H} }{\mathbf{Q}}\), by convention) with \begin{align*} E ( n ) _ { * } = \mathbb { Z } _ { ( p ) } \left[ v _ { 1 } , \dots , v _ { n - 1 } , v _ { n } ^ { \pm } \right] \end{align*} The fibers of the chromatic tower \begin{align*} M _ { n } X \rightarrow X _ { E ( n ) } \rightarrow X _ { E ( n - 1 ) } \end{align*} are called the monochromatic layers.

The spectral sequence associated to the chromatic tower is the chromatic spectral sequence \begin{align*} E _ { 1 } ^ { n , * } = \pi _ { * } M _ { n } X \Rightarrow \pi _ { * } X{ \scriptsize {}_{ \left[ { \scriptstyle { {p}^{-1}} } \right] } } \end{align*}

Let \(M_\ell\) denote the DM moduli stack of elliptic curves over \(\operatorname{Spec}({\mathbf{Z}})\).

For a commutative ring \(R\), the groupoid of \(R\)-points of \(M_\ell\) is the groupoid of elliptic curves over \(R\). This Unsorted/stacks MOC carries a line bundle \(\omega\) where for an elliptic curve \(C\), the fiber over \(C\) is given by \(\omega C = T^∗_e C,\) the tangent space of \(C\) at its basepoint \(e\).

The stack \(M_{\ell}\) admits a compactification \({ \mathcal{M}_{\mathrm{ell}} }\) whose \(R\) points are generalized elliptic curves. The space of integral modular forms of weight \(k\) is defined to be the space of sections \begin{align*} H ^ { 0 } \left( \mkern 1.5mu\overline{\mkern-1.5mu{ \mathcal{M}_{\mathrm{ell}} }\mkern-1.5mu}\mkern 1.5mu; \omega{ {}^{ \scriptscriptstyle\otimes_{k}^{n} } } \right) \end{align*} Motivated by the definition of integral modular forms and the descent spectral sequence in the case of \(U = M_\ell\) , the spectrum $\mathrm{TMF} $ (see TMF) is defined to be its global sections: \begin{align*} \mathrm{TMF} \coloneqq{\mathcal{O}}^ { {\mathsf{Top}}} \left( \mkern 1.5mu\overline{\mkern-1.5mu{ \mathcal{M}_{\mathrm{ell}} }\mkern-1.5mu}\mkern 1.5mu \right) \end{align*}

Chromatic convergence

Can be done on a thick subcategory:

Chromatic filtration

An application to K-theory:

The thick subcategory and nilpotent theorems

Relation to tensor triangular geometry and tmf

See tensor triangular geometry and tmf. Explaining the infamous chromatic stratification picture:

Applications