The Gibbs measure of many disordered systems at low temperature may exhibit a very strong dependance on even tiny variations of temperature, usually called “temperature chaos”. I will discuss this question for Spin Glasses. I will report on a recent work with Eliran Subag (Courant) and Ofer Zeitouni (Weizmann and Courant), where we give a detailed geometric description of the Gibbs measure at low temperature, which in particular implies temperature chaos for a general class of spherical Spin Glasses at low temperature. This question has a very long past in the physics literature, and an interesting recent history in mathematics. Indeed, in 2015, Eliran Subag has given a very sharp description of the Gibbs measure for pure p-spin spherical Spin Glasses at low temperature, building on results on the complexity of these spin glasses by Auffinger-Cerny and myself. This description (close to the so-called Thouless-Anderson-Palmer picture) excludes the existence of temperature chaos for the pure p-spin!! The recent work gives an extension of this very detailed geometric description of the Gibbs measure to the case of general mixed models, and shows that in fact the pure p-spin is very singular.[-]

One fairly standard version of the Riemann Hypothesis (RH) is that a specific probability density on the real line has a moment generating function (Laplace transform) that as an analytic function on the complex plane has all its zeros pure imaginary. We'll review a series of results that span the period from the 1920's to 2018 concerning a perturbed version of the RH. In that perturbed version, due to Polya, the log of the probability density is modified by a kind of mass term (in quantum field theory language). This gives rise to an implicitly defined real constant known as the de Bruijn-Newman Constant, Lambda. The conjecture and now theorem (Newman 1976, Rodgers and Tau 2018) that Lambda is greater than or equal to zero is complementary to the RH which is equivalent to Lambda less than or equal to zero; The conjecture/theorem is a version of the dictum that the RH, if true, is only barely so. We'll also briefly discuss some connections with quantum field theory and the Lee-Yang circle theorem.[-]

Graphons and graphexes are limits of graphs which allow us to model and estimate properties of large-scale networks. In this pair of talks, we review the theory of dense graph limits, and give two alterative theories for limits of sparse graphs – one leading to unbounded graphons over probability spaces, and the other leading to bounded graphons (and graphexes) over sigma-finite measure spaces. Talk I, given by Jennifer, will review the general theory, highlight the unbounded graphons, and show how they can be used to consistently estimate properties of large sparse networks. This talk will also give an application of these sparse graphons to collaborative filtering on sparse bipartite networks. Talk II, given by Christian, will recast limits of dense graphs in terms of exchangeability and the Aldous Hoover Theorem, and generalize this to obtain sparse graphons and graphexes as limits of subgraph samples from sparse graph sequences. This will provide a dual view of sparse graph limits as processes and random measures, an approach which allows a generalization of many of the well-known results and techniques for dense graph sequences.[-]

Graphons and graphexes are limits of graphs which allow us to model and estimate properties of large-scale networks. In this pair of talks, we review the theory of dense graph limits, and give two alterative theories for limits of sparse graphs – one leading to unbounded graphons over probability spaces, and the other leading to bounded graphons (and graphexes) over sigma-finite measure spaces. Talk I, given by Jennifer, will review the general theory, highlight the unbounded graphons, and show how they can be used to consistently estimate properties of large sparse networks. This talk will also give an application of these sparse graphons to collaborative filtering on sparse bipartite networks. Talk II, given by Christian, will recast limits of dense graphs in terms of exchangeability and the Aldous Hoover Theorem, and generalize this to obtain sparse graphons and graphexes as limits of subgraph samples from sparse graph sequences. This will provide a dual view of sparse graph limits as processes and random measures, an approach which allows a generalization of many of the well-known results and techniques for dense graph sequences.[-]

Swiss-born mathematician Nicola Kistler was the first holder of the Jean-Morlet Chair for mathematical sciences at CIRM and, in that capacity, became the first visiting researcher in residence for six months at the Centre. His stay at CIRM lasted from early February till July 2013. He set up a program of mathematical events focusing on 'Probability', with the collaboration of Véronique Gayrard, local project leader working at Marseille's Laboratoire d'Analyse, Topologie, Probabilités (ex LATP - now I2M).
CIRM - Chaire Jean-Morlet 2013 - Aix-Marseille Université[-]