# Is the field of q-series 'dead'? [closed]

I had a discussion with my advisor about what am I interested as my future research direction and I said it is special functions and q-series. He laughed and said that the topic is essentially dead and the people who study it are dinosaurs. I'm really confused by this statement and don't know what to think. Is this area really dead and not worth pursuing a research in it?

• @Ycor The advior said this in public meeting of undergraduate students with the research staff. – Tyrell May 23 '19 at 11:11
• Anyway it was addressed to a few undergrads and I'm not sure it is very useful to overspread, apart from being offensive to possible people working on it. It's an advisor's responsibility not to send students in active areas, and they do it according to their own knowledge and possibly bias. I remember having been told to avoid some directions before PhD and this turned out, I think, to be good advice, and also heard that some directions are dead, which turned out to be false. It makes sense to ask other people as you do here about research in these directions, without copying all this. – YCor May 23 '19 at 11:31
• @YCor: There being no -1 for comments, I'm recording one here. Academic conversations are not commonly understood to be Deep Background, at least not where I am. Second opinions are almost always worth getting when considering research subjects to choose. MathOverflow is as good as any other place for getting these second opinions, and may be the most accessible one (the OP isn't necessarily at a major university and even then may be too shy to ask a question like this). – darij grinberg May 23 '19 at 13:14
• @YCor: I'm not sure I got your "as" right, but the tenor of your comment still seems to be that you are admonishing the OP for going public. And I just don't agree with that. – darij grinberg May 23 '19 at 13:21
• @darijgrinberg we indeed disagree on this. This was not considered as offensive towards the OP (and the OP doesn't claim this). Also I don't blame public accusation of the advisor, since it's anonymous. All the benefit I see from this is an anonymous depreciation of some field, and this can be offensive to people working in this field. – YCor May 23 '19 at 13:29

I cannot answer the question of whether the field of $$q$$-series is dead. I would make this a comment, but I lack sufficient reputation.

I am a Banach space theorist, and Banach space theory is not fully dead. However, it is sufficiently unfashionable as to make it terribly difficult for me to find a job (even with a very strong research record). In fact, I have been completely unable to find employment, and I am now forced to leave academia because of it. Granted, geographical factors, personal connections, and luck may all be different for you. However, I made the mistake of choosing my research area only because of its intellectual interest to me, without any practical consideration of the job market, and I would caution you against doing the same. If your advisor says the area is dead, I would listen.

I would also say that, if you like that area, you can study it as much as you want after you find a permanent position.

• I was in a similar situation, if you want to discuss it please drop me a line. – Tomasz Kania May 24 '19 at 5:52

Jehanne Dousse has recently obtained a sought-after CNRS position. From what I remember seeing, she was inundated with job interviews. This should disprove the death of q-series, at least as far as partition-like series are concerned. Certain subfields may be worse off.

If you want to discreetly ask specific people inside the field whether the field is alive enough to support a career, a good approach seems to me to state the question positively: Ask what the most interesting currently existing questions are, what the most exciting recent work is about, etc. With some luck you may get a good research project in return :)

• is it hard to get a CNRS position? – user140765 May 23 '19 at 14:19
• @kartop_man: From what I know, yes. These are permanent positions with no teaching required! – darij grinberg May 23 '19 at 14:24
• @kartop_man Darij is correct, these are extremely competitive and sought after. They don't carry any teaching duty, but usually you can choose to teach a light load for an increase in salary. Moreover after 3-5 years you can ask (and are usually granted) the possibility to move to any CNRS research unit, in France or abroad. To give an idea, this year there were 11 positions in total for all of pure and applied math (+ 3 designated for applied math only). The only downside is the salary, which is slightly lower than lecturers', which is itself not competitive internationally. – Najib Idrissi May 23 '19 at 15:34
• @Najib I'm a CNRS researcher and the time you have to wait if you want to move to a different institution is only one year. I have a young colleague who recently did that. Plus you can change topic as soon as you have the position. – Ratbert May 23 '19 at 20:01
• @Ratbert You can ask every year, but do you get the permission every year? I was at a meeting for newly recruited mathematicians last month, and the vice director of the Insmi told us that 3-5 years was the "reasonable" delay (but if you have a good reason they can make exceptions). – Najib Idrissi May 24 '19 at 9:25

The opinions that certain areas of mathematics are dead are frequently stated but in many cases incorrect. Some areas experience declines in activity and then revivals. Many examples can be given.

On the other hand, my recommendation to a PhD student would be: do what your adviser says. When you obtain a permanent position you will be free to pursue whatever topic you yourself find interesting. Until that time you have to take into account various things other than intrinsic interest of the topic.

Example. In the early 1980th my friend, then a graduate student, discussed with his adviser possible directions of research. The adviser himself contributed to classical complex functions theory and to holomorphic dynamics. He said that holomorphic dynamics is dead, and recommended to work in classical function theory (the area which was not the most popular at that time either). It is almost exactly at that time (1982) when the revolution in holomorphic dynamics happened, suddenly this become one of the most fashionable area in whole mathematics, and it preserves this status to this day. But in 1960-80 the adviser was almost alone, working in holomorphic dynamics.

The areas which experienced strong revival during my lifetime, after a long oblivion, are besides holomorphic dynamics, hyperbolic geometry, Kleinian groups, analytic theory of differential equations, especially Painleve equations, Schubert calculus, knot theory.

Remark. Just for fun: type such terms as "elliptic curve", "automorphic function", "modular form", "enumerative geometry", "Painleve", "Fatou" on Google ngram.

• "Many examples can be given" - maybe you could give at least one, so that the answer is more self-contained? – user140765 May 23 '19 at 18:14

The reports of the death of the field of $$q$$-series and special functions are greatly exaggerated. George Andrews wrote a book Q-Series on the subject published in 1986. He and Bruce Berndt last year completed the publication of a five volume edition about the results in Ramanujan's Lost Notebook. An important subject for Ramanujan was $$q$$-series and special functions. George Andrews is a former President of the AMS with many awards.

Ken Ono, the Vice President of the AMS, has done research on Rogers-Ramanujan identities, Mock theta functions, and recently he and his coworkers have proved the Umbral Moonshine Conjecture. In joint work with Jan Bruinier, he discovered a finite algebraic formula for computing partition numbers. The partition numbers are the coefficients of the reciprocal of the fundamental $$q$$-series $$(q;q)_\infty$$. There are many other examples that I could cite, but this should be enough.

I think that (properly understood) theory of $$q-$$series is one of the most fashionable areas of mathematics right now. One reason is the appearance of $$q-$$series in topology. Namely, Turaev-Viro TQFT associates to each $$3-$$manifold with boundary a $$q-$$special function, defined for $$q$$ being a root of unity. These functions are expected to be $$q-$$holonomic, so should satisfy interesting difference equations. The building block for these invariants is so-called quantum $$6j-$$symbol, which coincides with $${}_4F_3-$$ basic hypergeometric series with general parameters at $$z=1$$.