The goal of this question is to compile a list of noteworthy mathematical achievements from about 2010 (so somewhat but not too far in the past).
In particular, this is meant to include (but not limited to) results considered of significant importance in the respective mathematical subfield, but that might not yet be widely known throughout the community.
Compiling such a list is inevitably a bit subjective, yet this can also be seen as a merit, at least as long as one keeps this implicit subjectivenes in mind.
Thus the specific question to be answered is;
Which mathematical achievements from about 2010 do you find particularly noteworthy?
This is perhaps too broad a question. So, a way to proceed could be that people answering focus on their respective field(s) of expertise and document that they did so in the answer (for an example see Mark Sapir's answer).
As my own candidates let me mention two things:
[Note: original version of the question by Alexander Chervov, so these are Alexander Chervov's candidates.]
- "Polar coding" (Actually it is earlier than 2010, but I asked for "around 2010")
introduction of "Polar coding" http://arxiv.org/abs/0807.3917 by E.Arikan.
New approach to construct error-correcting codes with very good
properties ("capacity achieving").
Comparing the other two recent and popular approaches
turbo-codes (http://en.wikipedia.org/wiki/Turbo_code)
and
LDPC codes (http://en.wikipedia.org/wiki/LDPC)
Polar coding promises much simpler decoding procedures,
although currently (as far as I know) they have not yet achieved same
good characteristics
as LDPC and turbo, it might be a matter of time.
It became very hot topic of research in information theory these days
- just in arxiv 436 items found on the key-word "polar codes".
I was surpised how fast such things can go from theory to practice –
turbo codes were invented in 1993 and adopted in e.g. mobile
communication standards within 10 years. So currently yours
smartphones use it.
Proof of the Razumov-Stroganov conjecture
http://arxiv.org/abs/arXiv:1003.3376
So the conjecture lies in between mathematical physics (integrable systems) and combinatorics. There was much interest in it recent years.
Let me quote D. Zeilberger (http://dimacs.rutgers.edu/Events/2010/abstracts/zeil.html):
In 1995, Doron Zeilberger famously proved the alternating sign matrix conjecture (given in 1996, a shorter proof by Greg Kuperberg). In 2001, Razumov and Stroganov made an even more amazing conjecture, that has recently been proved by Luigi Cantini and Andrea Sportiello. I will sketch the amazing conjecture and the even more amazing proof, that is based on brilliant ideas of Ben Wieland.
Best Answer
The biggest result in my field in 2010 was the solution to the Erdos-distance problem in the plane by Guth and Katz. This result was quite a breakthrough, and it was a surprise to many. Specifically, they proved the following conjecture of Erdos.
For $E \subset \mathbb{R}^n$ put $\Delta(E) = \lbrace |x - y| : x,y \in E \rbrace$, where $| \cdot |$ denotes Euclidean distance. Then for finite sets $E \subset \mathbb{R}^2$, there exists a universal constant such that
$$ |\Delta(E)| \geq c \frac{|E|}{\log |E|}. $$