Inspired by the successes of algebraic quantum statistical mechanics in dealing with some fundamental nonequilibrium questions, we investigate the relation between ``approach to equilibrium'' (sometimes called the Zeroth Law) and the Second Law. Short of being able to provide a new example of non-trivial and physically pertinent system approaching equilibrium, we bring some partial answers to a question raised by David Ruelle in 1967.

This is a joint work with Vojkan Jaksic and Clément Tauber.

Message passage is a method to approximately compute marginal distributions of probability distributions having an underlying graphical structure. One of its first appearances was in decoding algorithms for error correcting codes. More recently, versions of it where used as an alternative to LASSO type methods in compressed sensing. Depending on the underlying graphical structure, it is possible to reduce the algorithmic complexity by an approximation step which results in what is called "Approximate Message Passage". The efficiency of the corresponding algorithms in compressed sensing was first observed numerically by Donoho and Montanari. A theoretical understanding was first obtained in spin glass theory where it is closely related to the Thouless-Anderson-Palmer equations. More recently, it lead also to a new proof of the Gardner formula for the perceptron, extending results previously obtained by Talagrand. This is joint work with Shuta Nakajima, Nike Sun, and Changji Xu. The talk is mainly an overview.

We consider the problem of particle transport in a class of quasi-one-dimensional channels with polygonal boundaries, and look at the statistics of displacement, first passage time and first returns. Polygonal channels of this type are an example of pseudo-chaotic billiards due to the presence of conic singularities. Transport is characterised by a strong anomalous diffusion. We explore the consequences this has on the residence time and return to the origin of finitely long channels.

In one-dimensional real and complex quadratic dynamics, the Collet-Eckmann condition guarantees that expansion beats contraction, on average, and the dynamics shares many properties with that of the expanding circle map $z \mapsto z^2$. For transcendental maps, the situation is less clear. I will give a survey of some recent and less recent results, focusing on the exponential family $z \mapsto \lambda \exp(z)$.

After a short mathematical introduction to repeated quantum measurement processes, we will focus on the probability distribution of the sequence of measurement outcomes. We will limit ourselves to the case where all measurements take values in a finite set. As we shall see through a series of examples, the resulting distributions range from familiar (i.i.d., Markov, ...) measures to highly singular, non-Gibbsian ones. We will explore some of the singularities from the point of view of the large deviations of entropy production.

We study the long-time behavior of localized solutions to linear or semilinear parabolic equations in the whole space R^n, assuming that the diffusion matrix depends on the space variable x and has a finite limit along any ray as |x| goes to infinity. Under smallness conditions in the nonlinear case, we prove convergence to a self-similar solution whose profile is entirely determined by the asymptotic diffusion matrix. The rate of convergence itself depends on the continuity and coercivity properties of the limiting matrix. Our proof relies on energy estimates for the diffusion equation in self-similar variables, involving not only the difference between the solution and the self-similar profile, but also an antiderivative which satisfies a linear elliptic problem. This is a joint work with R. Joly and G. Raugel.

DNA sequencing is a powerful tool to study the past, it can provide evidence for ancient migrations, lifestyle changes, diseases and struggles. Furthermore, humans never travel alone, they bring their parasites (pathogenic or not) on board. We have studied the evolution of Mycobacterium Leprae, the bacteria responsible for leprosy, obtaining evidence of an epidemic dating back 100'000 years, with a few surprising observations along the way.

We shall concentrate on mathematics and the use of properly human intelligence, with the use of natural languages, the written scientific legacy of earlier times (and perhaps some noncreative use of computers). We exclude possibly non-human intelligent beings on other planets and creatively `intelligent' computers.

In this talk, based on joint works with Nicholas Cook, Huy Tuan Pham and Sohom Bhattacharya, I will discuss recent developments in the emerging theory of nonlinear large deviations, focusing on sharp upper tails for counts of several fixed subgraphs in a large sparse random graph (such as Erdős–Rényi or uniformly d-regular). These results allow in turn to determine the typical structure of samples from an associated class of Gibbs measures, known as Exponential Random Graph Models, which are widely used in the analysis of social networks.

I will discuss the long time behavior of a random splitting model of the Euler equations. I will explain how the random switching ensures a that there exists a unique invariant measure with respect to the Hausdorff measure on the surface defined by constant energy and enstrophy. I will then explain our proof that they system almost surely has positive Lyapunov exponent. This is joint work with Omar Melikechi and Andrea Agazzi

A personal overview of the dynamics and the structure of the phase space of convex billiards in the plane and more general surfaces

We introduce a family of divergences between probability measures which interpolate between the 1-Wasserstein metrics and a f-divergences (for example relative entropy) and which inherit desirable properties from both of them. We demonstrate the usefulness of those objects in the context of generative modeling, i.e. the construction of a sampling model for systems known only through finite data. We discuss in particular generative adversarial networks as well Lipschitz-regularized gradient flows. An important feature of our approach is the ability of handling singular target distributions.

We consider a 2D directed polymer in random environment in the critical regime (where temperature is proportional to the logarithm of the length of the polymer. We focus on the weak disorder phase and evaluate exponential moments of the partition function. Joint work with Clement Cosco.

Protein is a matter of dual nature. As a physical object, a protein molecule is a folded chain of amino acids with diverse biochemistry. But it is also a point along an evolutionary trajectory determined by the function performed by the protein within a hierarchy of interwoven interaction networks of the cell, the organism, and the population. Therefore, a physical theory of proteins needs to unify both aspects, the biophysical and the evolutionary. Specifically, it should provide a model of how the DNA gene is mapped into the functional phenotype of the protein. A physical approach to the protein problem will be described, focusing on a mechanical framework that treats proteins as evolvable condensed matter: Mutations introduce localized perturbations in the gene, which are translated to localized perturbations in the protein matter. A natural tool to examine how mutations shape the phenotype is the Green function, which maps the evolutionary linkage among mutations in the gene (termed epistasis) to cooperative physical interactions among the amino acids in the protein. The mechanistic view can be applied to examine basic questions of protein evolution and design.

We report on recent progress on the problem of building a stochastic process that admits the hypothetical Yang-Mills measure as its invariant measure. One interesting feature of our construction is that it preserves gauge-covariance in the limit even though it is broken by our UV regularisation. This is based on joint work with Ajay Chandra, Ilya Chevyrev, and Hao Shen.

The aim of this talk is to review the current knowledge on the stationary solutions of the Navier-Stokes equations in the whole space R³ and mainly in the whole plane R². Contrary to the Cauchy problem for which the R³ domain is more complicated than R², for the stationary problem, this is the opposite: R² is the most difficult case. In the first part, I will discuss the construction of weak solutions by topological methods, and in the second part, I will show how scale invariance can be used to construct perturbative solutions. Open problems will be discussed and I will also present some numerical results and conjectures.

Advances in the technology of brain imaging data acquisition, the broadly accessible computational power nowadays available, and advances in mathematical algorithms to analyze data allow us to analyze the brain activity of patients under different states of consciousness: general anesthesia, coma, different states of sleep, awake, deep meditation and under the effects of psychedelic drugs. This new scenario in brain science allows starting to approach the problem of understanding aspects of the consciousness from the perspective of Science, Technology, Engineering, and Mathematics (STEAM). In this talk, I will broadly introduce this very modern approach to the study of the brain, the relevance, and the goals of this research field. I will also show some results obtained recently using ideas from non-equilibrium statistical mechanics.

We consider the asymptotic properties of matrix products that appear in the study of the Hofstadter model (of an electron moving on a lattice) and the associated almost Mathieu operators. In a restricted setup that is characterized by a symmetry, we show that critical behavior occurs and is universal in an open neighborhood of the almost Mathieu family. This behavior is governed by a periodic orbit of a renormalization transformation. Other periodic orbits govern supercritical behavior.

Breather solutions are spatially localized, temporally periodic solutions of partial differential equations and lattice systems. They are extremely rare for partial differential equations with constant coefficients but are known to exist in some PDEs with inhomogeneous coefficients. In this talk I’ll describe recent work with Prof. Shuguan Ji of Northeast Normal University (China) establishing the existence of such solutions in two new classes of nonlinear wave equations. The proof consists of an analysis of the gaps in the spectrum of the linear part of the equation, coupled with a variational analysis to control the effects of the nonlinear terms.

Parameter-dependent quantum Hamiltonians give rise to energy bands and an adiabatic connection on the vector bundle of states of each band, and when the parameter space is a compact surface the band topology is determined by the Chern class, that determines physical properties like the quantum Hall conductance, and the existence and chirality of edge states. When the quantum system is disordered or displays complex dynamics, one may model it statistically by a random matrix. Here we study the statistics of the adiabatic curvature and Chern numbers of random matrix fields defined on a two-sphere. We define a parametric correlation length, and argue that the curvature correlations obey universal scaling laws, decaying rapidly for large separations; the universality hypothesis is supported with Monte-Carlo simulations. On the other hand, we show that the correlations diverge inversely with the separation when it is small. We derive a universal expression for the Chern number variance which grows inversely with the square of the parametric correlation length when it is small, and show that the Chern number distribution approaches a Gaussian in this regime. Monte-Carlo calculations indicate that when the correlation length is not small, the Chern number distribution is still universal but does not follow a power law, and is not Gaussian. Finally we study numerically the joint distribution of the gap Chern numbers, showing that, contrary to conjecture, the gap Chern numbers are weakly correlated, and the correlation decays as a power of the spectral distance between the gaps. Joint work with Or Swartzberg and Michael Wilkinson

The talk presents a theory combining gravitation and electromagnetism. The theory is formulated in a fluid dynamical framework and the dynamical equations are therefore first order in time. the motion being generated by a vector field. The vector field is the sum of two vector fields, a Hamiltonian vector field that is associated with the energy function being derived using Hamilton's principle of least action and a gradient vector field associated with a dissipation function being the gradient thereof. A model of a physical system is thus defined by specification of the energy function including an expression for the gravitational energy and the dissipation function. The theory differs from the Einstein-Maxwell theory in that the equations of motion satisfy the integral laws of conservation of energy and momentum and the second law of thermodynamics. Only the derivation of the Hamiltonian vector field is presented in this talk.

The stochastic dynamics of general networks of chemical reactions can be modelled, at the microscopic level, as Markov jump processes where each jump represents the occurrence of a reaction. In recent years, a growing body of literature has investigated the connection between the dynamical properties of such process and the structure of the underlying network of reactions. An open problem in this sense is to prove (or disprove) that the jump process associated to weakly reversible networks (those whose connected components are all strongly connected) is positive recurrent. We answer this conjecture in the affirmative in two dimensions.
This is joint work with Jonathan Mattingly, David Anderson and Daniele Cappelletti.