Q:

Noteworthy, but not so famous conjectures resolved recent years

Conjectures play important role in development of mathematics.
Mathoverflow gives an interaction platform for mathematicians from various fields, while in general it is not always easy to get in touch with what happens in the other fields.
Question What are the conjectures in your field proved or disproved (counterexample found) recent years, which are noteworthy, but not so famous outside your field ?
Answering the question you are welcome to give some comment for outsiders of your field which would help to appreciate the result.
Asking the question I keep in mind by "recent years" something like a dozen years before now, by a "conjecture" something which was  known as an open problem for something like at least dozen years before it was proved and I would say the result for which the Fields medal was awarded like a proof of fundamental lemma would not fit "not so famous", but on the other hand these might not be considered as strict criteria, and let us "assume a good will" of the answerer.

A:

Karim Adiprasito proved the g-conjecture for spheres in a preprint that was posted in December of last year: https://arxiv.org/abs/1812.10454.
This was probably considered the biggest open problem in the combinatorics of simplicial complexes. See Gil Kalai's blog post: https://gilkalai.wordpress.com/2018/12/25/amazing-karim-adiprasito-proved-the-g-conjecture-for-spheres/.

A:

Hinged dissections exist.
(See 3-piece dissection of square to equilateral triangle? for an animation of Dudeney's famous equilateral-triangle-to-square hinged dissection.)

Abbott, Timothy G., Zachary Abel, David Charlton, Erik D. Demaine, Martin L. Demaine, and Scott Duke Kominers. "Hinged dissections exist." Discrete & Computational Geometry 47, no. 1 (2012): 150-186.
  Springer link.

"Abstract. We prove that any finite collection of polygons of equal area has a common hinged dissection. That
is, for any such collection of polygons there exists a chain of polygons hinged at vertices that can be
folded in the plane continuously without self-intersection to form any polygon in the collection. This
result settles the open problem about the existence of hinged dissections between pairs of polygons that
goes back implicitly to 1864 and has been studied extensively in the past ten years."

          

The proof is not simple—as hinted by the above figure—but it is constructive.

A:

Konstantin Tikhomirov recently proved that the probability that a random $n\times n$ Bernoulli matrix $M_n$ with independent $\pm 1$ entries, and $$\mathbb{P}[M_{ij}=1]=p,\quad 1\leq i,j\leq n,$$  is singular is
$$
\mathbb{P}[M_n~\mathrm{is~singular}]=(1-p+o_n(1))^n
$$
for any fixed $p\in (0,1/2].$
This problem was considered by Komlos, Kahn-Komlos-Szemeredi, Bourgain, Tao-Vu etc., so I am unsure if it qualifies in terms of being not-so-famous. 
Nevertheless it was exciting reading about it in Gil Kalai's blog here .

