The True Heirs To Ramanujan

If you are in need of some light relief, you could do worse than peruse the opinions of Doron Zeilberger, who, if viewed strictly through the lens of these ramblings, appears to have a relationship with theoretical mathematics something along the lines of the Unabomber’s relationship with technology. (Let me add that my feelings in real life about Zeilberger is that he has proved some amazing theorems, an opinion which I am absolutely sure is not reciprocated.)

Zeilberger’s opinion 151 is somewhat of a doozy, calling out Ken Ono as a member of the “fancy math gang” who “stole” Ramanujan. First, the idea that one could imagine what Ramanujan thought of the modern theory of mock theta functions (or any other part of mathematics influenced by his legacy) is pure BS. Second, once you create something in mathematics, it transcends its creator; what Ramanujan actually would have thought of his legacy is mostly irrelevant. Maybe Robert Langlands thinks we are all fools for not taking up the double-bitted axe and the cross-cut saw and devoting our lives to the trace formula, and maybe he’s right, but in reality the Langlands program will go in a very different direction to what Langlands anticipated and most likely be better for it. Third, it’s hard to maintain any legitimacy making criticisms of the way modern mathematics is done while insisting that you don’t actually know any “fancy” mathematics, whether that is true or not. But finally, and this is what is most amusing about this opinion, is that I am usually inclined to criticise Ken on precisely the same point, except in the exact opposite direction. That is, my frustration with the theory of mock modular forms is not that it’s too fancy, but rather that it’s nowhere near fancy enough! The subject is crying out for a treatment which incorporates representation theory, where “shadows” are related to the (reducible) principal series which has a discrete series quotient, and where the amazing special structures which have been discovered are given a more algebraic framework rather than the subject as it currently stands: a wild west of crazy q-series, Lerch sums, and indefinite theta series. Whether such an approach would actually be useful is hard to guarantee in advance, but the relevance for the classical theory of modular forms cannot be overestimated.

The process of mathematics is as follows. You discover or observe some phenomenon, and you try to explain it. While trying to explain it, you may come up with a more general theory which explains not only the original example, but also an entire family of examples. And usually at this point, the general theory is more interesting than the original example, because it has more explanatory power to explain why things are true. That doesn’t mean the original example is no longer interesting, but it has to be viewed in the more general context. There is room in mathematics for crazy unique examples that don’t fit a pattern and very general theory. But there are no “heirs” to Ramanujan, because mathematics doesn’t work that way. The fact that Ramanujan’s name will always be linked to Deligne (a mathematician of a quite different sort, to say the least) is testament to that.

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8 Responses to The True Heirs To Ramanujan

  1. TG says:

    Not quite mock modular forms, but this paper might be the kind of thing you (or the subject) are crying out for:

    https://arxiv.org/abs/1609.06999

  2. Jeff Harvey says:

    Matt Emerton made the same complaint to me a couple of years ago regarding mock modular forms and representation theory. I wonder though if things are really so bad. Certainly there are at least as many ways to produce modular forms as mock modular forms so the “wild-west” characterization seems no less true of modular forms. I would also expect that if there was a nice representation theoretic description then there would be analogs of mock modular forms for groups other than SL(2,R), but I’m not aware of any such constructions.

  3. Sander Zwegers says:

    On Zeilberger’s opinion 151: I don’t want to comment on other people’s work, so I’ll restrict myself to my own. When I was doing my PhD very few people were interested in mock theta functions and the topic was definitely not considered fancy (and probably not even interesting). I pursued the topic anyway, because I really appreciated the (classical) theory of modular forms and because my supervisor (Don Zagier) had the feeling there could be something interesting going on with these strange functions of Ramanujan. My actual thesis, which “allegedly” “made sense” out of Ramanujan’s functions, is very concrete and formula based. It mainly builds upon work of Jacobi, Lerch, Mordell, Watson and maybe Hecke (so things that were known in Ramanujan’s time or shortly thereafter). Not exactly what you would call modern or “fancy” mathematics!! Then to be called a “fancy mathematician” by Zeilberger is quite amusing (but absurd). On the question “Would Ramanujan have liked the math I do?”: I don’t know and I don’t care!

    On this post: Personally I definitely prefer concrete stuff, but if other people can put things in a very general context (and thus make them more “fancy”), then that’s great! I don’t think that currently it’s really that much of a “wild west”, though. For example indefinite theta series are a natural generalization of classical theta series for positive definite quadratic forms. Both the classical and indefinite theta series provide a rich source of (mock) modular forms.

    • Dear Sander, certainly when I use that turn of phrase I am being a little provocative, and I definitely think some of the work on mock modular forms — yours included — has been very exciting. It’s just that certain approaches to the problem have been (mostly) ignored to this point, and hopefully that will change. It’s certainly not the case that *everyone* in the field should become more abstract. I’m mostly just making the (anodyne) comment that most areas of mathematics benefit from a multitude of approaches, and wrapping up that message with a little persiflage.

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