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.