Chromatic Filtration of Burnside Ring I just attended a seminar on the chromatic filtration of the Burnside ring.  I understood it relatively well, but at no point did anyone give an explicit definition of what a chromatic filtration actually is, and I have had some trouble finding one.  Assuming I have the necessary categorical background, i.e. knowledge of filtrations, etc. can anyone give me an explanation of what makes a filtration "chromatic"?
 A: There is no, as far as i know, general chromatic filtration for object of type X. There is a chromatic filtration of the stable homotopy groups of spheres. You could try to make sense of some directed system that is "chromatic" associated to a space/spectrum by looking at the maps between its localizations with respect to Morava K-theory (or Morava E-theory or Johnson-Wilson, or what have you) and how they fit together, but i think this is essentially the chromatic convergence thm, (there are finite type and p-local hypotheses).
An important ingredient to note is Carlsson's Thm/the Segal conjecture, which states, i believe, that stable cohomotopy is the completed Burnside ring. Maybe you can feed some of the chromatic ideas through all the hard work that went into the Segal Conjecture and get something that way.
But maybe that is not it at all. A note about Aarons answer, the chromatic filtration does not come from the ANSS, although it does break up the way you say though. Ravenel spends some time in Nilpotence and Periodicity talking about this stuff. I think it has to do with self maps inducing multiplication by $v_n$ in $K(n)_*(X)$, and you filter the self maps of X that way. The $v_n$ are tied to the theory of formal group laws intimately. The way I think of the chromatic filtration, and this could be off, is by what level of (complex oriented) cohomology theory i need to detect that a self map is not trivial, where by level i mean height of the formal group law corresponding to the particular complex oriented theory. The height of the formal group laws govern the filtration and how it behaves. (This is how I think of it, but in light of the below, it is sort of junk, I will leave it because it is less technical.)
I believe the paper Jack linked to is where they develop the chromatic spectral sequence which converges to the $E_2$ term of ANSS which is crazy hard to compute (i dont think anyone knows all the 2 line, and if they do then no one knows all of the 3 line, and by all i mean out to $t-s=50$) So there is a filtration on the $E_2$ term of ANSS that gives rise to this spectral sequence, and I think this is the chromatic filtration.
(Also, it is a technical term, and it does not just mean periodic.)
EDIT:
I just found the following sentence while reading the review of "$v_n$ periodic elements in ring spectra and applications to bordism theory" by Hovey (the review is by stong):
"Let $R$ be a ring spectrum. If $v \in \pi_k R$, $v$ is called a $v_n$ element if $K(n)_*(v) \in K(n)_*(R)$ is a unit and $K(i)_*(v) \in K(i)_*(R)$ is nilpotent for all $i \neq n$."
I will be back in a bit to check it against a reading of Ravenel's orange book.
A: Expanding on Jack's answer, there is a whole exciting field known as "chromatic homotopy theory".  I don't know much about it, I'll say a few slogans since I noticed you also asked the question "geometry or topology".  Hopefully none of them are wrong.  (If so, somebody please correct me!)
Chromatic homotopy theory somehow slices the Adams-Novikov spectral sequence based on levels of periodicity (hence the name "chromatic" -- it's supposed to conjure up the image of light refracting through a prism or something). [cf. Sean's answer for a much better informed account.]


*

*Level 0 is ordinary cohomology with $\mathbb{Q}$ coefficients.  There, the interaction of geometry and homotopy theory is given by integration.

*Level 1 is K-theory.  There, the interaction of geometry and homotopy theory is given by the Atiyah-Singer index theorem.

*Level 2 is elliptic cohomology (and something called topological modular forms, which somehow loosely is the "universal elliptic cohomology theory").  It is unknown how to bridge the gap with geometry, but this is something people are working on.

*All the other finite levels are unknown.  A few people are trying to figure out what level 3 should be.

*Level $\infty$ is complex cobordism.

*All the levels (so far?) are complex-oriented.  The complex cobordism spectrum $MU$ carries the universal formal group law, and complex orientations of a spectrum $E$ are obtained as multiplicative maps of spectra $MU\rightarrow E$.  (In fact, $(MU_*,MU_*MU)$ corepresents the moduli stack of formal groups!)


If you want to actually learn some things about this, try:


*

*Hopkins' notes, "Complex Oriented Cohomology Theories and the Language of Stacks"

*class on chromatic homotopy theory taught by Lurie

*Lurie: "A Survey of Elliptic Cohomology"
A: There are joint papers by Benedict Gross (a number theorist) and Mike Hopkins (a topologist) on formal groups and the chromatic tower, that may be more accessible as a quick introduction to the concepts. They try to explain some of this material to non-topologists, and their exposition from almost 20 years ago does not rely on the machinery of stacks, "brave new algebra" or infinity-categories.  The summary paper in the AMS Bulletin has a dictionary relating stable homotopy and quasicoherent sheaves, in particular they explain that the chromatic filtration is analogous to the Cousin complex.  See:
http://www.ams.org/journals/bull/1994-30-01/S0273-0979-1994-00438-0/
A: It appears to be non-technical, and roughly equivalent to "periodic".

An explanation is given by Ravenel.

The "original paper" appears to be:

Miller, Haynes R.; Ravenel, Douglas C.; Wilson, W. Stephen. "Periodic phenomena in the Adams-Novikov spectral sequence."
  Ann. Math. (2) 106 (1977), no. 3, 469–516.
  MR458423
JSTOR

but the term is not used there.  These sorts of things were for homotopy of spheres, a major topic in algebraic topology.
Presumably the filtration discussed for the Burnside ring had a similar ambience.
