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A lot of "big" names in analysis (and other fields) seem to be doing some form of combinatorics (without any order, some examples are Tim Gowers, Terence Tao and Jean Bourgain).

So, looking a bit around makes me conclude that combinatorics is a huge field. There must be one "kind" which is the most fruitful in analysis. What kind is this? What is a good introduction to this?

Edit: I forgot, analysis is also a big field. I mean more in the direction of harmonic analysis and PDE.


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@Timothy: Thanks, but the wikipedia page does not suggest that is useful in analysis, only that it is done by analytic methods. – Jonas Teuwen Dec 12 '10 at 1:27
I see no reason that the partial order of combinatorics by its applicability to analysis needs to have a maximum element. – Qiaochu Yuan Dec 12 '10 at 1:32
This question reminds of Sperner's Lemma and Brouwer's Fixed point theorem. – Aryabhata Dec 12 '10 at 1:34
up vote 2 down vote accepted

With regard to combinatorics & harmonic analysis, you might find this interesting. This work by Terence Tao (whom you mentioned) sheds light on all three of combinatorics, analysis, and PDEs. Hope this helps!

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Your question seems to ask what combinatorics is fruitful in analysis. I think the trend (at least with those you have mentioned) is the application of analysis to combinatorics

You might be most interested in arithmetic combinatorics. I'd recommend the page from Tao's course here. Alan Frieze has a page with many good links here.

I have also heard good things (and keep intending to read more closely) Richard Stanley's text Enumerative Combinatorics, which is different "flavor" of combinatorics than the links above.

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Thanks. I am also aware of combinatorics problems in harmonic analysis, like the problem where they try to prove that John-Nirenberg (in BMO) can be "dimension-free". Looking at Terence Tao's website and following the link "Geometric combinatorics" I find the following: the topics seem to be quite analytical in nature. Thanks for the other suggestions, I will look at them. – Jonas Teuwen Dec 12 '10 at 1:37

Phillipe Flajolet's Analytic Combinatorics is a good resource for this. (first book on the page, pdf)

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That looks like it is about combinatorics using analytic methods, and not analysis using combinatorical methods? – Jonas Teuwen Dec 14 '10 at 21:57

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