About the title

About the title

I changed the title of the blog on March 20, 2013 (it used to have the title “Notes of an owl”). This was my immediate reaction to the news the T. Gowers was presenting to the public the works of P. Deligne on the occasion of the award of the Abel prize to Deligne in 2013 (by his own admission, T. Gowers is not qualified to do this).

The issue at hand is not just the lack of qualification; the real issue is that the award to P. Deligne is, unfortunately, the best compensation to the mathematical community for the 2012 award of Abel prize to Szemerédi. I predicted Deligne before the announcement on these grounds alone. I would prefer if the prize to P. Deligne would be awarded out of pure appreciation of his work.



I believe that mathematicians urgently need to stop the growth of Gowers's influence, and, first of all, his initiatives in mathematical publishing. I wrote extensively about the first one; now there is another: to take over the arXiv overlay electronic journals. The same arguments apply.



Now it looks like this title is very good, contrary to my initial opinion. And there is no way back.
Showing posts with label applications of mathematics. Show all posts
Showing posts with label applications of mathematics. Show all posts

Friday, August 23, 2013

Is algebraic geometry applied or pure mathematics?

Previous post: About some ways to work in mathematics.

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.

Tuesday, June 11, 2013

New comments to the post "What is mathematics?"

Previous post: What is combinatorics and what this blog is about according to Igor Pak.


There is a new thread of comments to the post "What is mathematics?" started by Sandro Magi. The post is dated April 3; this thread started on May 31. The thread is concerned with only one claim in that post: proofs are not needed at all for applications of mathematics.

Unfortunately, the very first phrase of Sandro Magi set the tone for the rest of the discussion: "This is blatantly false". I do not like to discuss things in such a manner: with a total lack of cooperation. The combinatorialists at Gowers's blog are much more friendly even after a direct attack on their field. But, I believe that the reason is not any kind of malice of either party. This dialog is a good illustration of the near impossiblity of people thinking linearly and verbally to understand people thinking visually. In this case the dialog of a mathematician (every mathematician thinks at least partially visually) and a software engineer turned out to be impossible. I encountered the same sort of difficulties while discussing essentially any other subject, from the movies to the current affairs. I see also this lack of understanding of visual and "the big picture" issues in the design and functionality of almost all the software.

Still, it seems to me that there are some important ideas in that discussion. Of course, it would be better to give a coherent exposition. But an attempt to write it would take a lot of time, and who knows when it would be ready.

If somebody wants to comment on any issue there, I suggest to post comments here; this will result in a more clear structure of comments. As an additional benefit for the next 30 days the comments here are not moderated; they are moderated at that post. This rule is subject to change without notice. :-) I would like to ask Sandro Magi to continue our discussion in comments to "What is mathematics?" and not here (of course, he is under not obligation to continue); then the whole discussion will be at the same place.


Next post: 2014 Fields medalists?.