This is an extended version of my Nobel Lecture, delivered on December 8, 2021. I will
recall the genesis of the concept of multiple equilibria in natural sciences. I will then describe …

Glasses are amorphous solids whose constituent particles are caged by their neighbours
and thus cannot flow. This sluggishness is often ascribed to the free energy landscape …

We consider the problem of coloring the vertices of a large sparse random graph with a
given number of colors so that no adjacent vertices have the same color. Using the cavity …

Using the cavity equations of Mézard, Parisi, and Zecchina Science 297 (2002), 812;
Mézard and Zecchina, Phys Rev E 66 (2002), 056126 we derive the various threshold …

Note: This Perspective offers a progress report for the Simons collaboration “Cracking the
Glass Problem,” of which the authors are some of the principal investigators. While the …

We consider Ising mixed $ p $-spin glasses at high-temperature and without external field,
and study the problem of sampling from the Gibbs distribution $\mu $ in polynomial time. We …

We consider the theory of the glass phase and jamming of hard spheres in the large space
dimension limit. Building upon the exact expression for the free-energy functional obtained …

The problem of controllability of the dynamical state of a network is central in network theory
and has wide applications ranging from network medicine to financial markets. The driver …

In the first part of this paper, we derive the general replica equations that describe infinite-
dimensional hard spheres at any level of replica symmetry breaking (RSB) and in particular …

Using elementary rigorous methods we prove the existence of a clustered phase in the
random K-SAT problem, for K≥ 8. In this phase the solutions are grouped into clusters …