Subexponential size $\mathbb{R}P^n$

A practical way to encode a manifold is to triangulate it. Among all possible triangulations it makes sense to look for the minimal one, which for the purpose of this talk means using the least number of vertices.
Consider now a family of manifolds such as $S^n$, $\mathbb{R}P^n$, $SO_n$, etc. A natural question is how the size of the minimal triangulation depends on $n$? Surprisingly, except for the trivial case of $S^n$, our best lower and upper bounds are very far apart.
For $\mathbb{R}P^n$ the current best lower and upper bounds are around $n^2$ and $\varphi^n$, respectively, where $\varphi$ is the golden ratio.
In this talk I will present the first triangulation of $\mathbb{R}P^n$ with a subexponential, $\sqrt{n}^\sqrt{n}$, number of vertices.
I will also speak about the lower bound and state a few related open problems.
This is a joint work with Karim Adiprasito and Roman Karasev.