Is algebraic geometry applied or pure mathematics?

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From a comment by Tamas Gabal:

“This division into 'pure' and 'applied' mathematics is real, as it is understood and awkwardly enforced by the math departments in the US. How is algebraic geometry not 'applied' when so much of its development is motivated by theoretical physics?”

Of course, the division into the pure and applied mathematics is real. They are two rather different types of human activity in every respect (including the role of the “problems”). Contrary to what you think, it is hardly reflected in the structure of US universities. Both pure and applied mathematics belong to the same department (with few exceptions). This allows the university administrators to freely convert positions in the pure mathematics into positions in applied mathematics. They never do the opposite conversion.

Algebraic geometry is not applied. You will be not able to fool by such statement any dean or provost. I am surprised that it is, apparently, not obvious anymore. Here are some reasons.

1. First of all, the part of theoretical physics in which algebraic geometry is relevant is itself as pure as pure mathematics. It deals mostly with theories which cannot be tested experimentally: the required conditions existed only in the first 3 second after the Big Bang and, probably, only much earlier. The motivation for these theories is more or less purely esthetical, like in pure mathematics. Clearly, these theories are of no use in the real life.

2. Being motivated by outside questions does not turn any branch of mathematics into an applied branch. Almost all branches of mathematics started from some questions outside of it. To qualify as applied, a theory should be really applied to some outside problems. By the way, this is the main problem with what administrators call “applied mathematics”. While all “applied mathematicians” refer to applications as a motivation of their work, their results are nearly always useless. Moreover, usually they are predictably useless. In contrast, pure mathematicians cannot justify their research by applications, but their results eventually turn out to be very useful.

3. Algebraic geometry was developed as a part of pure mathematics with no outside motivation. What happens when it interacts with theoretical physics? The standard pattern over the last 30-40 years is the following. Physicists use they standard mode of reasoning to state, usually not precisely, some mathematical conjectures. The main tool of physicists not available to mathematicians is the Feynman integral. Then mathematicians prove these conjectures using already available tools from pure mathematics, and they do this surprisingly fast. Sometimes a proof is obtained before the conjecture is published. About 25 years ago I.M. Singer (of the Atiyah-Singer theorem fame) wrote an outline of what, he hoped, will result from the interaction of mathematics with the theoretical physics in the near future. In one phrase, one may say that he hoped for infinitely-dimensional geometry as nice and efficient as the finitely-dimensional geometry is. This would be a sort of replacement for the Feynman integral. Well, his hopes did not materialize. The conjectures suggested by physicists are still being proved by finitely-dimensional means; physics did not suggested any way even to make precise what kind of such infinitely-dimensional geometry is desired, and there is no interesting or useful genuinely infinitely-dimensional geometry. By “genuinely” I mean “not being essentially/morally equivalent to a unified sequence of finitely dimensional theories or theorems”.

To sum up, nothing dramatic resulted from the interaction of algebraic geometry and theoretical physics. I don not mean that nothing good resulted. In mathematics this interaction resulted in some quite interesting theorems and theories. It did not change the landscape completely, as Grothendieck’s ideas did, but it made it richer. As of physics, the question is still open. More and more people are taking the position that these untestable theories are completely irrelevant to the real world (and hence are not physics at all). There are no applications, and hence the whole activity cannot be considered as an applied one.


Next post: The role of the problems.

Hard, soft, and Bott periodicity - Reply to T. Gowers

Previous post: Reply to JSE.

This is a reply to a comment of T. Gowers.

Yesterday I left remarks about “hard” arguments and Bott periodicity without any comments. Here are few.

First, the meaning of the word "hard" varies from person to person. There is a fairly precise definition in analysis, due to Hardy. Still, I fail to use it for classification of, say, Lars Ahlfors work: is it hard or soft? I was told once that it is not "hard analysis", but wasn't told anything meaningful why. For me, Ahlfors is the greatest analyst of the last century (let us assume that the 20th century started around 1910, at least - in order to avoid hardly relevant comparisons).

Notice that the terminology is already fairly misleading: the opposite of “hard” is not “easy” (despite the hard analysis is assumed to be difficult and hard to work in). It is “soft”. What dichotomy is understood here? Clearly, finding the right definitions is not an easy thing; more often than not it is very difficult. The defined objects may turn out to either “hard” or “soft” depending on what we wanted. For definitions I see only one meaningful interpretation: objects are hard if they are rigid (like Platonic solids), and soft if they can be easily deformed (and the space of deformations is highly dimensional) without losing their key characteristics. It seems that “hard” theories are very often the ones dealing with “soft” objects.

But I suspect that the people using the hard-soft terminology will disagree with me. At the same time in the conceptual mathematics there is no such dichotomy at all and it is impossible to acquire any experience in using it.

The example of the Bott periodicity theorem is a good testing ground. The situation with the Bott periodicity is more or less opposite to what you wrote about it. First of all, it is not a black box to be used without opening it.

The first proof was based on the Morse theory for infinitely dimensional space of curves in some classical manifolds. (A nice idea of Morse reduces the problem at once to the finitely dimensional situation.) This is result is crucial for topological K-theory; it is build into its structure. But I never saw any detailed exposition when the original theorem of Bott was used as black box for developing the topological K-theory. The theorem seems to be too weak for this, it provides for one point space the result needed for a wide class of topological spaces. Probably, the machinery of algebraic topology allows deducing this particular result just from the result for one-point space, but this definitely cannot be done without reworking the Bott theorem in order to get a more explicit result first. Atiyah in his book uses another proof (due to him and Bott), which has the advantage of being “elementary” and giving the result for all reasonable spaces without any intermediaries, and the disadvantage of being the most obscure one. Later on, the K-theory (and, hence, the Bott periodicity) were used in order to prove the Atiyah-Singer index theorem. The index theorem has an advanced version, the index theorem for families (of elliptic operators).

In fact, really useful theorem is the index theorem for families, not the original index theorem. After the second proof of the index theorem was found, which imitated Grothendieck proof of the Grothendieck-Riemann-Roch theorem and allowed to prove the index theorem for families (the first one does not), Atiyah used it to give a new proof of the Bott periodicity. One may suspect a circular argument here, but there are none. A carefully excised fragment of this proof does not depend on the Bott periodicity, and if combined with an algebraic idea due to Atiyah, led to a new proof of Bott’s periodicity. This proof turned out to be the most important one. The subject of analytic K-theory is to a big extent an attempt to use the ideas of this proof in as general situation as possible, and to apply the results. The main point here that people working in analytic K-theory are not using Bott periodicity as a black box; they are thinking about what is really inside this box. By now we have at least 8 different (substantially different) proofs of Bott periodicity, and the progress in the questions related to the Bott periodicity usually requires rethinking the theorem and its proof, not using it as a black box. Perhaps, because of all this, people prefer to speak about the phenomenon of Bott periodicity and not about Bott’s theorem.



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