The interplay between the geometry of formal groups and homotopy theory has been a guiding influence since the work of Morava in '70s, and it has been a thread in homotopy theory ever since. The current language of algebraic geometry allows us to make very concise and natural statements about this relationship.

In these lectures, I will discuss how the geometry of the moduli stack M of smooth one-dimensional formal groups dictates the chromatic structure of stable homotopy theory. At a prime, there is an essentially unique filtration of M by the open substacks U(n) of formal groups of height no more than n and the resulting decomposition of coherent sheaves on M gives exactly the chromatic filtration. Topics may include formal groups, the relationship between quasi-coherent sheaves and complex cobordism, the role of coordinates, the height filtration, closed points and Lubin-Tate deformation theory, and algebraic chromatic convergence. Since M is in some sense very large and cumbersome, I hope also to give some discussion of small (ie Deligne-Mumford) stacks over M; these include the moduli stack of elliptic curves and certain Shimura varities, thus bringing us to the current research of Hopkins, Miller, Lurie, Behrens, Lawson, Naumann, Ravenel, Hovey, Strickland, and many others.

Let me remark that I am merely the expositor here, building on the work of many people, especially Jack Morava and Mike Hopkins.

Suggested reading:

• Behrens, Mark, "A modular description of the {$K(2)$}-local sphere at
  the prime 3", Topology 45 No. 2 (2006), 343--402.
• Goerss, Paul, "(Pre-)sheaves of ring spectra over the moduli stack of
  formal group laws" , Axiomatic, enriched and motivic homotopy
  theory,NATO Sci. Ser. II Math. Phys. Chem., 131, 101-131, Kluwer Acad.
  Publ., Dordecht 2004.
• Hopkins, M. J., "Algebraic topology and modular forms", Proceedings of
  the International Congress of Mathematicians, Vol. I (Beijing, 2002),
  291--317, Higher Ed. Press, Beijing, 2002.
• Hopkins, M. J. and Gross, B. H., The rigid analytic period mapping,
  Lubin-Tate space, and stable homotopy theory, Bull. Amer. Math. Soc.
  (N.S.), 30 No. 1 (1994), 76-86.
• Hovey, Mark and Strickland, Neil, "Comodules and Landweber exact
  homology theories", Adv. Math., 192 No. 2 (2005), 427-456.
• Naumann, Niko, "Comodule categories and the geometry of the stack of
  formal groups", available at http://front.math.ucdavis.edu/math.AT/0503308.
• Smithling, Brian, "On the moduli stack of commutative, 1-parameter
  formal Lie groups" available at http://www.math.uchicago.edu/~bds/fg.pdf