Before I turn to the main topic of this posting, a lecture by Jacob Lurie, I’d like to point to something else involving him, a comment and posting at Mathematics Without Apologies, a blog you should be following anyway. On the topic of the usefulness of “proof assistants”, I liked Lurie’s point that a major problem with this is:
Working in a formal system, more or less by definition, means that you can’t ignore steps which are routine and focus attention on the ones that contain the fundamental ideas.
But if you want to discuss this, it should be over there, the topic of this posting is something very different.
Last week I noticed that Lurie had given a talk at Harvard on “Categorifying Fourier Theory”, which is available here. I enjoyed watching it, ending up quite intrigued by the abstract picture he was painting, but rather discouraged by the lack of any example that would give insight into what it might be useful for. Neglecting to mention the example that explains why an abstract theorem is useful is unfortunately all-too-common practice among mathematicians. Perhaps in this case with Dick Gross, Jean-Pierre Serre and John Tate in the front row, he felt it unnecessary. Luckily though, he gave the same talk recently in Arizona, and there (in the question session) did give a fascinating motivating example.
His starting point was the Fourier transform, which one can generalize to any abelian group G, and think of as identifying complex functions on G with complex functions on the dual (or character) group G^. The standard Fourier transform is the case G=G^=R, Fourier series are the case G=U(1), G^=Z. He then went on to discuss two levels of abstraction, or categorification of this. The first identifies a representation of G on a vector space V with a function from G^ to vector spaces (the isotypic decomposition of the representation). The second identifies representations of G on categories with representations of G^ on categories.
It was this equivalence of representations on categories that was his main result, for which in Arizona he gave the example of G the group of invertible Laurent series. The idea is that this group can be identified with its dual group G^ (in some sense as algebraic groups), using the Weil symbol (for a definition and context, see here). Lurie’s claim that was new to me was that the equivalence in this case is essentially the GL(1) version of the general local geometric Langlands conjecture, which is supposed to be an equivalence of two representations on categories, for more general (non-abelian) groups G.
At least for me, understanding of some sophisticated mathematical phenomenon really starts when I understand the simplest example of the phenomenon. For the number field case of Langlands theory, my initial efforts to understand the subject didn’t lead anywhere until I realized that maybe it was best to first think about the local version, which was a statement about representation theory that I could make some sense of. I was hopeful that thinking about the simplest case of that, the abelian case, would give great insight, found though that the Abelian case is already quite non-trivial (local class field theory). For the geometric Langlands case, I found that the discussion of the local version in Edward Frenkel’s book was very helpful, but I always wondered about the abelian case. Now I’m hopeful that the abelian story is something that although I’ve never seen it, is well-understood, and that a helpful reader will point me to a reference.
Another reason for being interested in this particular topic is that it has some connection to the relationship between Langlands theory and QFT that first got me interested in all of this. Back in 1987 Witten wrote some fascinating papers giving an abstract formulation of free fermion theories on Riemann surfaces (see here and here) with tantalizing connections to what later became geometric Langlands. In this work the group of invertible Laurent series and the Weil symbol play a central role. There was also later work by Takhtajan on this, see here and here. I wonder why the most recent version of the last reference deletes the material on the multiplicative group case, which is the one Lurie mentions.
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