User misha belolipetsky - MathOverflow

Answer by Misha Belolipetsky for Theorems with unexpected conclusions

2011-04-23T22:10:19Z

The classical differential geometry results should definitely be mentioned here. Although it may seem not surprising for us, Gauss found his Theorema Egregium to be truly remarkable and unexpected. My favorite example is Gauss-Bonnet Theorem.

Answer by Misha Belolipetsky for Video lectures of mathematics courses available online for free

2011-04-08T12:11:36Z

LMS Durham Symposia have archive of videos online which can be found at http://www.maths.dur.ac.uk/events/Meetings/LMS/

For example, 2009 conference on model theory of fields has videos of the talks by Hrushovski, Kazhdan, Macintyre and Zilber, among the others.

Answer by Misha Belolipetsky for Perron number distribution

2011-03-28T15:50:19Z

You can ask a similar question about Pisot and Salem numbers. Last year together with my project student Charlie Scarr we were looking, in particular, at a possible connection between distribution of the roots inside the unit circle and the Mahler measure of a polynomial. We did not progress too far but Charlie made some interesting observations which can be found in his report http://www.maths.dur.ac.uk/~dma0mb/projects/C_Scarr.pdf

Closer to your question, I think it would be very interesting to check what kind of picture comes out if one restricts to the polynomials with small Mahler measure. A large list of such polynomials is available at Michael Mossinghoff's page on Lehmer's problem. It is in no way a random sample but the ordering by Mahler measure looks quite natural in this context.

Comment by Misha Belolipetsky

2011-10-07T03:34:59Z

It is still often called "Lehmer's question", but after a lot of computational as well as some theoretical results it has naturally turned into a conjecture. One can, of course, call it "Lehmer's question which we now treat as a conjecture". This would be more precise but is a bit too long.