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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.]


1) "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.


2) Proof of the Razumov-Stroganov conjecture

http://arxiv.org/abs/arXiv:1003.3376

So the conjecturee 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.

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MathOverflow celebrated its first full year of existence. Gerhard "Ask Me About Something Obvious" Paseman, 2011.12.12 –  Gerhard Paseman Dec 12 '11 at 22:08
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I voted to reopen because I want to see answers. It should be made "community Wiki", though, since there is no unique answer. –  Mark Sapir Dec 13 '11 at 15:23
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No, the canonical way is to use meta. –  Mark Sapir Dec 13 '11 at 16:12
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I might vote to reopen if it were CW. –  Benjamin Steinberg Dec 13 '11 at 17:36
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Meta thread tea.mathoverflow.net/discussion/1245 Side note to OP. You can make the question CW (edit and tick box). General remark: Instead of discussing/suggesting CW it always seems more efficient to me to flag for moderators with this request. (In particular if there are already an answer(s) this is anyway important as else already answers will stay non CW) –  quid Dec 13 '11 at 19:15
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11 Answers

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

$$ |\Delta(E)| \geq c \frac{|E|}{\log |E|}. $$

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At Gil Kalai's blog Timothy Gowers also mentions Guth and Katz result: gilkalai.wordpress.com/2011/12/14 –  Alexander Chervov Dec 18 '11 at 11:07
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Ryan Williams proved a breakthrough result about circuit lower bounds, showing (for example) that NTIME[$2^n$] does not have non-uniform ACC circuits of polynomial size.

Francisco Santos disproved the Hirsch conjecture.

Tomas Rokicki, Herbert Kociemba, Morley Davidson, and John Dethridge proved that diameter of the Rubik's cube group is 20.

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@Timothy: You think that #3 is comparable with #1 or #2? There are quite a few etremely strong new results about making finite groups into expanders (hence small diameter). Am I right that #1 is a major new result in this area, similar in importance to Razborov's lower bounds results? –  Mark Sapir Dec 14 '11 at 6:55
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I beg for a little comment for non-experts on #1. "NTIME[2]", "ACC" - what is it ? What is the context (except general fact that is related to PvsNP). #3 is fun ! Thanks for posting it. –  Alexander Chervov Dec 14 '11 at 7:18
    
See Wiki articles on non-deterministic time complexity and curcuit complexity. –  Mark Sapir Dec 14 '11 at 9:57
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@Mark: I'd say #1 is in the same ballpark as Razborov's monotone circuit lower bounds, yes. Also, I think #3 is comparable in terms of "noteworthiness," which I take to be a distinct concept from mathematical depth. @Alexander: See the Complexity Zoo: qwiki.stanford.edu/index.php/Complexity_Zoo The context of the result is explained well by Ryan Williams himself in the introduction to the paper. –  Timothy Chow Dec 14 '11 at 15:03
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I would like to know other people answers, so perhaps other people will want my answer also. I can offer my opinion about the area of geometric/algorithmic/asymptotic group theory which I am doing. I should start with saying that my opinion most probably does not coincide with opinions of other people in my area which, I think, is normal. I may also forget something. Also by an achievement I mean a concrete result, not a theory. In my area, the "top" achievements of the last year are (IMHO, in no particular order)

  • Sela's continued work on Tarski-related problems. His series of 10 papers is now more than 1000 pages long and the latest (recent) results include a solution of an old Malcev's problem about elementary equivalence of free products of groups. That of course assuming it is correct: the solution is being checked.

  • Kharlampovich-Myasnikov's solution of another Malcev's problem, also related to Tarski problems, that proper subgroups of free non-Abelian groups cannot be defined by first order formulas.

  • Dani Wise's work on "cubulating" groups, i.e. embedding groups into Right Angled Artin Groups (again assuming it is correct). In particular his solution of an old problem by Baumslag: all 1-related groups with torsion are residually finite. Also his and Agol's results imply that hyperbolic Haken 3-manifold groups are virtually surface-by-cyclic which is a great result.

  • Igor Mineyev's (very short!) proof of the Strong Hanna Neumann conjecture (more than 40 years old).

  • Bestvina-Bromberg-Fujiwara's proof that mapping class groups have finite asymptotic dimension.

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Just out of curiosity, regarding your entry on Igor Mineyev's work: is this question now settled? –  Hailong Dao Dec 14 '11 at 3:09
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@Hailong : I can't speak for Mark, but Igor visited me earlier this year and we spent an afternoon picking his proof apart. I emerged completely convinced (and amazed). –  Andy Putman Dec 14 '11 at 5:10
    
@Andy: Thank you. I was actually asking about the Friedman's preprints, since they also claimed to prove HNC. –  Hailong Dao Dec 14 '11 at 5:28
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@Hailong: Igor's paper has been accepted in Annals of Math. –  Mark Sapir Dec 14 '11 at 6:28
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Mineyev's proof, as distilled by Warren Dicks, is easy enough to be digested in a few minutes. See, for instance, the following slides of a talk by Yago Antolin Pichel: personal.soton.ac.uk/am1t07/ggt/Antolin-talk.pdf . –  HJRW Dec 14 '11 at 10:54
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Duminil-Copin and Smirnov proved Nienhaus' conjecture that the connective constant of the self-avoiding walk on the honeycomb lattice is equal to $\sqrt{2 + \sqrt{2}}$.

http://arxiv.org/abs/1007.0575

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The Table of Contents for Current Developments in Mathematics, 2010 (link) lists:

  • The Arf-Kervaire problem in algebraic topology: Sketch of the proof by Michael A. Hill, Michael J. Hopkins, and Douglas C. Ravenel
  • On the Friedlander-Milnor conjecture for groups of small rank by Fabien Morel
  • Universal formulas for counting nodal curves on surfaces by Yu-Jong Tzeng
  • Some recent results on representations of p-adic special orthogonal groups by Jean-Loup Waldspurger
  • Wellposedness of the two- and three-dimensional full water wave problem by Sijue Wu

This website lists topic discussed for Current Developments in Mathematics in 2011.

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I need to know how they made that rotating Mobius strip! –  Steve D Dec 12 '11 at 22:29
    
Those damn "inquiring minds"... –  Igor Rivin Dec 12 '11 at 22:32
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you can ask the guy who created it, math.harvard.edu/~knill –  YangMills Dec 14 '11 at 20:23
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Although from 2009, the Kervaire invariant 1 problem is indeed worthy; the biggest breakthrough in algebraic topology in the last years. –  Lennart Meier Dec 16 '11 at 10:41
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The best-case theoretical complexity of matrix multiplication has been lowered from $n^{2.376}$ to $n^{2.373}$.

You may argue against its real-world importance, since it is not based on a ground-breaking new technique and is just 0.1% lower than the previous bound, but this is the first improvement on that exponent in 25 years, so I find it interesting.

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Thanks for posting this, because until now I had somehow missed the news that Andrew Stothers had already improved the Coppersmith-Winograd exponent back in 2010. I was only aware of the 2011 paper by Virginia Vassilevska Williams. –  Timothy Chow Apr 5 '12 at 15:10
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In differential geometry there have been two old conjectures both solved in spring 2012 within 1 month:

  1. A proof of the Willmore conjecture by Marques and Neves (arXiv:1202.6036) which states that the Willmore minimizer amongs immersed tori in $S^3$ is the Clifford torus up to Moebius transformation.
  2. A proof of the Lawson conjecture on less than 10 pages by Brendle (arXiv:1203.6597): The only embedded minimal torus in $S^3$ is the Clifford torus.

One should note that the proofs of the theorems rely on totally different methods.

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Tzeng's proof of the Goettsche's conjecture, which says the number of nodal curves in a linear system $|C|$ on a projective surface $S$ is given by a universal polynomial in the Chern numbers of $C$ and $S$ .

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Set theorists have started to seriously look at $C^*$-algebras and there have been several nice results in the last years. The most spectacular one is probably due to Farah, Phillips, and Weaver:

Whether all automorphism of the Calkin algebra (the quotient of the algebra of bounded operators on a separable Hilbert space by the ideal of compact operators) are inner automorphisms (i.e., conjugation by some unitary element) is independent over ZFC.

Philips and Weaver proved the existence of outer automorphisms assuming the continuum hypothesis (Duke Math J., 2009) and Farah showed the non-existence of outer automorphisms assuming Todorcevic's Open Coloring Axiom (Annals of Math, 2011). The interest of set theorists in this field was certainly increased by the Phillips-Weaver result.

There have been previous and clearly related results on automorphisms of the Boolean algebra $\mathcal P(\omega)/fin$, but the methods in the case of the Calkin algebra seem to be slightly different and also a bit more involved.

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This is very interesting, because now $C^\ast$-algebra papers (geometric analysis and so on) will have to include statements about which side of the fence they want to fall on. –  David Roberts Dec 21 '11 at 23:40
    
@David: actually, I don't see why. For instance, the Elliot classification program is IIRC concerned with separable C*-algebras. People who do index theory via C*-algebras are also probably not affected by this. (I agree that the results are spectacular and well worth a look, but I think your comment overstates things) –  Yemon Choi Dec 23 '11 at 15:06
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Loose analogy: despite Shelah's solution of Whitehead's problem in group cohomology, people can still write papers on group cohomology without including explicit statements about AC. –  Yemon Choi Dec 23 '11 at 15:08
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The proof of the Ehrenpreis Conjecture by Jeremy Kahn and Vladimir Markovic.

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AGT conjecture (String theory related to representation theory and algebraic geometry)

http://arxiv.org/abs/0906.3219

Liouville Correlation Functions from Four-dimensional Gauge Theories

Luis F. Alday, Davide Gaiotto, Yuji Tachikawa

"We conjecture an expression for the Liouville theory conformal blocks and correlation functions on a Riemann surface of genus g and n punctures as the Nekrasov partition function of a certain class of N=2 SCFTs recently defined by one of the authors."

As far as I understand it became big trend in hep-th, cited more than 200 times.

--

Nekrasov's paper introducing "Nekrasov partition function" is :

http://arxiv.org/abs/hep-th/0206161

Seiberg-Witten Prepotential From Instanton Counting

It conjectured mathematical formulation for what is known to physicst as "Seiberg-Witten
prepotential" for N=2 supersymmetric Yang-Mills theory. Conjecture has been proved later.

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as far as I understand Yuji Tachikawa is member of MO, so I hope he can give some comment explaining this to mathematicians... –  Alexander Chervov Dec 15 '11 at 17:19
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