Alexander Chervov asked over Mathoverflow about Noteworthy results in and around 2010 and some interesting results were offered in the answers. If you would like to mention additional results you can comment on them here. The only requirement is to explain what the result says and give links if possible.
My favorite result in 2010 is that Francisco Santos disproved the Hirsch-Conjecture (statet ’57). The Hirsch Conjectures says that a Polytope in dimension d with n facets can not have a graph diameter greater than n-d. Link to the arxiv-article: http://arxiv.org/abs/1006.2814?context=cs
Would it be possible to tag or categorize this post “PlanetMO” or “mathoverflow” so that it might appear via http://www.mathblogging.org/planetmo?
Thanks!
Two that particularly struck me (and many other people) were the Guth-Katz solution of the Erdos distance problem and Tom Sanders’s results on Freiman’s theorem and Roth’s theorem. It hardly feels necessary to state these results, since they’ve been discussed a great deal, including here on this blog, but since those are the rules, here goes. Guth and Katz proved that any
points in the plane must give rise to at least 
distinct distances, and Sanders proved (amongst other things) that the largest density of a subset of 
that does not contain an AP of length 3 is 
, to within a power of 
.
Thanks, Tim and Tim.