What were some major mathematical breakthroughs in 2016? As the year is slowly coming to an end, I was wondering which great advances have there been in mathematics in the past 12 months. As researchers usually work in only a limited number of fields in mathematics, one often does not hear a lot of news about advances in other branches of mathematics. A person who works in complex analysis might not be aware of some astounding advances made in probability theory, for example. Since I am curious about other fields as well, even though I do not spend a lot of time reading about them, I wanted to hear about some major findings in distinct fields of mathematics. 
I know that the question posed by me does not allow a unique answer since it is asked in broad way. However, there are probably many interesting advances in all sorts of branches of mathematics that have been made this year, which I might have missed on and I would like to hear about them. Furthermore, I think it is sensible to get a nice overview about what has been achieved this year without digging through thousands of different journal articles. 
 A: The Non Existent Complex 6 Sphere by Michael Atiyah
"The possible existence of a complex structure on the 6-sphere has been a famous unsolved problem for over 60 years. In that time many "solutions" have been put forward, in both directions. Mistakes have always been found. In this paper I present a short proof of the non-existence, based on ideas developed, but not fully exploited, over 50 years ago. The only change in v2. is in section 3, where the notation has been clarified."
A: Logical induction is a probability theory for deductively limited systems. Formulating such a theory is a long open question, with previous attempts by for example Haim Gaifman.
Logical induction not only solves how deductively limited systems should assign credence to unproved arithmetical statements. It also allows deductively limited systems to reason about themselves, elegantly avoiding the associated Godelian/Lobian paradoxes.
"We present a computable algorithm that assigns probabilities to every logical statement in a given formal language, and refines those probabilities over time. For instance, if the language is Peano arithmetic, it assigns probabilities to all arithmetical statements, including claims about the twin prime conjecture, the outputs of long-running computations, and its own probabilities. We show that our algorithm, an instance of what we call a logical inductor, satisfies a number of intuitive desiderata, including: (1) it learns to predict patterns of truth and falsehood in logical statements, often long before having the resources to evaluate the statements, so long as the patterns can be written down in polynomial time; (2) it learns to use appropriate statistical summaries to predict sequences of statements whose truth values appear pseudorandom; and (3) it learns to have accurate beliefs about its own current beliefs, in a manner that avoids the standard paradoxes of self-reference. For example, if a given computer program only ever produces outputs in a certain range, a logical inductor learns this fact in a timely manner; and if late digits in the decimal expansion of $\phi$ are difficult to predict, then a logical inductor learns to assign $\approx 10\%$ probability to “the $n$th digit of $\phi$ is a 7” for large $n$. Logical inductors also learn to trust their future beliefs more than their current beliefs, and their beliefs are coherent in the limit (whenever $\phi\to\psi$, $P_\infty(\phi)\leq P_\infty(\psi)$, and so on); and logical inductors strictly dominate the universal semimeasure in the limit. 
These properties and many others all follow from a single logical induction criterion, which is motivated by a series of stock trading analogies. Roughly speaking, each logical sentence $\phi$ is associated with a stock that is worth \$1 per share if $\phi$ is true and nothing otherwise, and we interpret the belief-state of a logically uncertain reasoner as a set of market prices, where $P_n(\phi) = 50\% $ means that on day $n$, shares of $\phi$ may be bought or sold from the reasoner for 50¢. The logical induction criterion says (very roughly) that there should not be any polynomial-time computable trading strategy with finite risk tolerance that earns unbounded profits in that market over time. This criterion bears strong resemblance to the “no Dutch book” criteria that support both expected utility theory (von Neumann and Morgenstern 1944) and Bayesian probability theory (Ramsey 1931; de Finetti 1937)."
A: Personally, I was kind of fascinated by the solution to the Boolean Pythagorean triples problem which was finally solved in May. The problem asked whether or not the set of natural numbers $\mathbb{N}$ can "be divided into two parts, such that no part contains a triple $(a, b, c)$ with $a^2+b^2=c^2$". Heule, Kullmann and Marek managed to prove (with the help of a lot of computing power) that this is in fact not possible.
References:

Heule, Marijn J. H.; Kullmann, Oliver; Marek, Victor W. (2016-05-03). "Solving and Verifying the Boolean Pythagorean Triples problem via Cube-and-Conquer".
A: Don't know if this counts, as the proof was announced in late 2015. Tao's solution of the Erdős discrepancy problem was published in 2016. You can find it here; it was actually the first paper of the Discrete Analysis journal.
A: For me, I was stunned at the discovery that: Prime numbers have decided preferences about the final digits of the primes that immediately follow them. It was found that among the first billion primes, for instance, a prime ending in $9$ is almost $65$ percent more likely to be followed by a prime ending in $1$ than another prime ending in $9$. This groundbreaking research has been done by Soundararajan and Lemke Oliver. 

They believe that they have found that the biases they uncovered in consecutive primes come very close to what the prime k-tuples conjecture predicts. In other words, the most sophisticated conjecture mathematicians have about randomness in primes forces the primes to display strong biases.

You can read their paper here.
A: I don't really follow major breakthroughs, but my favorite paper this year was Raphael Zentner's Integer homology 3-spheres admit irreducible representations in $SL_2(\Bbb C)$. 
It has been known for quite some time, and is a corollary of the geometrization theorem, that much of the geometry and topology of 3-manifolds is hidden inside their fundamental group. In fact, as long as a (closed oriented) 3-manifold cannot be written as a connected sum of two other 3-manifolds, and is not a lens space $L(p,q)$, the fundamental group determines the entire 3-manifold entirely. (The first condition is not very serious - there is a canonical and computable decomposition of any 3-manifold into a connected sum of components that all cannot be reduced by connected sum any further.) A very special case of this is the Poincare conjecture, which says that a simply connected 3-manifold is homeomorphic to $S^3$. 
It became natural to ask how much you could recover from, instead of the fundamental group, its representation varieties $\text{Hom}(\pi_1(M), G)/\sim$, where $\sim$ identifies conjugate representations. This was particularly studied for $G = SU(2)$. Here is a still-open conjecture in this area, a sort of strengthening of the Poincare conjecture: if $M$ is not $S^3$, there is a nontrivial homomorphism $\pi_1(M) \to SU(2)$. (This is obvious when $H_1(M)$ is nonzero.) 
Zentner was able to resolve a weaker problem in the positive: every closed oriented 3-anifold $M$ other than $S^3$ has a nontrivial homomorphism $\pi_1(M) \to SL_2(\Bbb C)$. $SU(2)$ is a subgroup of $SL_2(\Bbb C)$, so this is not as strong. He does this in three steps.
1) Every hyperbolic manifold supports a nontrivial map $\pi_1(M) \to SL_2(\Bbb C)$; this is provided by the hyperbolic structure.
2) (This is the main part of the argument.) If $M$ is the "splice" of two nontrivial knots in $S^3$ (delete small neighborhoods of the two knots and glue the boundary tori together in an appropriate way), then there's a nontrivial homomorphism $\pi_1(M) \to SU(2)$.
3) Every 3-manifold with the homology of $S^3$ has a map of degree 1 to either a hyperbolic manifold, a Seifert manifold (which have long been known to have homomorphisms to $SL_2(\Bbb C)$, or the splice of two knots, and degree 1 maps are surjective on fundamental groups.
The approach to (2) is to write down the representation varieties of each knot complement and understand that the representation variety of the splice corresponds to a sort of intersection of these representation varieties. So he tries to prove that they absolutely must intersect. And now things get cool: there's a relationship between these representation varieties and solutions to a certain PDE on 4-manifolds called the "ASD equation". Zentner proves that if these things don't intersect, you can find a certain perturbation of this equation that has no solutions. But Kronheimer and Mrowka had previously proved that in the context that arises, that equation must have solutions, and so you derive your contradiction.
This lies inside the field of gauge theory, where one tries to understand manifolds by understanding certain PDEs on them. There's another gauge-theoretical invariant called instanton homology, which is the homology of a chain complex where the generators (sorta) correspond to representations $\pi_1(M) \to SU(2)$. (The differential counts solutions to a certain PDE, like before.) So there's another question, a strengthening of the one Zentner made partial progress towards: "If $M \neq S^3$, is $I_*(M)$ nonzero?" Who knows.
