I think, some less known shops need an advertisement. Independent University of Moscow (here is a link http://ium.mccme.ru/english/ to an English web-site) is a wonderful place itself and also runs ...

In general, the math I do use -- I do not forget [here I think I repeat everybody who commented above], the only kind of math I do really forget is the math I do not use at all. This is not as sad, ...

For this topic in general, I really recommend a book of Anna Cannas da Silva Lectures on Symplectic Geometry. It's wonderfully written and very clear. You can read a proof of the theorem in the book ...

If you are Russian, everything was done for us by A.B. Sosinsky in his great book "How to write a mathematical article in English". Here is a link (http://www.ega-math.narod.ru/Quant/ABS.htm). The ...

I feel there is some implicit idea here but nobody formulated it, so let me try: a smooth closed surface has no filling geodesic if and only if its geodesic flow is integrable What do you think ...

The answer to my question (provided by BS) is the following: We have to change the action by looking at the group $G=U(n, \mathbb{C})$ and its action by conjugacy on pairs of Hermitian matrices. The ...

I'm repeating what was said above and adding some details for completeness. A little chunk of historiography: Zoll found all the surfaces with periodic geodesic flow in the case of the isometry group ...

I will probably say a banal thing but the interesting degree of regularity depends on the problem you consider. In the theory of foliations, the foliations themselves are usually considered smooth ...

I actually think that Hilbert's Third problem is one of the explainable for school guys. It's even more cool that it exists in such a famous list close to the problems that are so tempting and not yet ...

One of the ways of defining naturally a metric on the tangent bundle $TM$ is indeed (as Peter says above) a Sasaki metric defined in 1968. You can prove that $(TM, g_{\mathrm{Sas}})$ is flat if and ...

Thinking about my own question, I had another question that came to my mind. Take any real number and write its binary representation. With probability $1$ the proportion of zeroes and ones is the ...

The answer to your question is YES, and you can look up the proof in our article with A. Klimenko https://arxiv.org/pdf/1305.6746.pdf for example. The key idea is to mix monotonicity with Moebius ...

I feel that this is important that I answer you that one third of my thesis was devoted to this equation. :) I have two published articles on this equation (one in Russian though) and I am not alone. ...

If you read French, you should certainly look at this site : http://images.math.cnrs.fr/?lang=fr This was first aimed at teachers and large audience interested in maths but on practice this is read by ...

This is more a remark than an answer since it gives an answer to the question that Greg didn't ask but I find it a very interesting and less popular property of Calogero-Moser system! Wojciechowski ...

These videos are fun: http://www.etudes.ru/en/ Although they do not explain what the proof is but they show some wonderful mathmatical ideas: Arnold's rouble problem, polyhedra, etc... Lots of ...