## Section: New Results

### Universality in polytope phase transitions and message passing algorithms

In [5] , we consider a class of nonlinear mappings ${\U0001d5a5}_{A,N}$ in ${\mathbb{R}}^{N}$ indexed by symmetric random matrices $A\in {\mathbb{R}}^{N\times N}$ with independent entries. Within spin glass theory, special cases of these mappings correspond to iterating the TAP equations and were studied by Bolthausen [Comm. Math. Phys. 325 (2014) 333-366]. Within information theory, they are known as "approximate message passing" algorithms. We study the high-dimensional (large $N$) behavior of the iterates of $\U0001d5a5$ for polynomial functions $\U0001d5a5$, and prove that it is universal; that is, it depends only on the first two moments of the entries of $A$, under a sub-Gaussian tail condition. As an application, we prove the universality of a certain phase transition arising in polytope geometry and compressed sensing. This solves, for a broad class of random projections, a conjecture by David Donoho and Jared Tanner.