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In a workshop about the geometry of $\mathbb{F}_1$ I attended recently, it came up a question related to a mysterious but "not-so-secret-anymore" seminar about... an hypothetical Topological Langlands Correspondence!

I had never heard about this program; I have found this page via Google:

http://www.math.jhu.edu/~asalch/toplang/

I only know a bit about the Number-Theoretic Langlands Program, and I still have a hard time trying to understand what is happening in the Geometrical one, so I cannot even start to draw a global picture out of the information dispersed in that site.

So, the questions are: What do you know (or what can you infere from the web) about the Topological Langlands Correspondence? Which is the global picture? What are its analogies with the (original) Langlands Program? Is it doable, or just a "little game" for now? What has been proved until now? What implications would it have?

(Note: It is somewhat difficult to tag this one, feel free to retag it if you have a better understanding of the subject than I have!)

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3 Answers

up vote 21 down vote accepted

I would cautiously venture that there is not really something we could call a topological Langlands program to outsiders at this point. My objection is to the final word - we don't really know what we're doing. For example, I don't think we even have a conjecture at this point relating representations of something to something else, or have the right idea of L-functions that are supposed to play a role.

The "topological Langlands program" banner is more of an idea that the scattered pieces of number-theoretic content we see in stable homotopy theory should be part of a general framework. There is the appearance of groups whose orders are denominators of Bernoulli numbers in stable homotopy groups of spheres, Andrew Salch's calculations at higher chromatic levels that seem to be related to special values of L-functions, the relationship between K(1)-local orientation theory and measures p-adically interpolating Bernoulli numbers (see Mark Behrens' website for some material on this), the surprisingly canonical appearance of realizations of Lubin-Tate formal group laws due to work of Goerss-Hopkins-Miller in homotopy theory, et cetera, et cetera.

Perhaps at this point I'd say that we have a "topological Langlands program" program, whose goal is to figure out why on earth all this arithmetic data is entering homotopy theory and what form the overarching structure takes in a manner similar to the Langlands program itself.

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I'd love to know the answer to your question! Unfortunately, I don't have a clear idea myself about what Topological Langlands should be. The best I can say is that a number of elements that show up in the story of Langlands (especially local langlands, a la Harris and Taylor) show up in homotopy theory: things like Lubin-Tate formal groups, p-divisible groups, automorphic forms, Hecke operators.

One thought. The basic case of the Langlands correspondence is supposed to be class field theory. In topology, the analogue of class field theory (over the rationals, at least), appears (to me) to be the story about the "Adams conjecture", which was proved around 1969 by Quillen and Sullivan. See my answer at What kind of geometric operations "scale up" cohomology? for a (very brief) description of this.

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My response to this question (when it came up on n-category cafe) is here - further down that page is a more informative discussion by Jack Morava and Jim Borger about class field theory and homotopy theory (and some related work of Charles). But in any case it seems that there doesn't exist a "topological Langlands program" at this point, but rather for now as Charles says it's more that special Shimura varieties play a central role in both..

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