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 …

Coalescent theory is the study of random processes where particles may join each other to
form clusters as time evolves. These notes provide an introduction to some aspects of the …

Three series of lectures were given at the 32nd Probability Summer School in Saint-Flour
(July 7–24, 2002), by the Professors Pitman, Tsirelson and Werner …

The celebrated Parisi solution of the Sherrington-Kirkpatrick model for spin glasses is one of
the most important achievements in the field of disordered systems. Over the last three …

We consider the high-dimensional inference problem where the signal is a low-rank
symmetric matrix which is corrupted by an additive Gaussian noise. Given a probabilistic …

Spin glasses are disordered magnetic systems that have led to the development of
mathematical tools with an array of real-world applications, from airline scheduling to neural …

This self-contained book is a graduate-level introduction for mathematicians and for
physicists interested in the mathematical foundations of the field, and can be used as a …

In this paper we prove that the support of a random measure on the unit ball of a separable
Hilbert space that satisfies the Ghirlanda-Guerra identities must be ultrametric with …

The Potts spin glass is a generalization of the Sherrington–Kirkpatrick (SK) model that
allows for spins to take more than two values. Based on a novel synchronization …

In neural network's literature, Hebbian learning traditionally refers to the procedure by which
the Hopfield model and its generalizations store archetypes (ie, definite patterns that are …