Integrability. What precisely is the role of the higher conservation laws? Which ones are important? How many do we need? When is a system "close" to being integrable"?

Thermalization and pre-thermalization. What are the mechanisms of thermalization? How does it happen in nearly integrable systems?

Disorder and many body localization.

Universal (slow) dynamics (Kibble-Zurek mechanism). To what extent the ideas of universality can be applied to dynamics?

Quench dynamics.

Applications: cold atoms, solid state systems, nonlinear optics, cosmology, ...

Numerical methods for quantum dynamics.

THERMALIZATION and PRE-THERMALIZATION

Definition of thermalization,

Connection between thermalization and kinetic equations (and the truncation of the BBGKY hierarchy)

Connection between thermalization and quantum chaos

Relaxation dynamic:

does it happen in different stages? (pre-thermalization)

what does pre-thermalization mean?

conditions for pre-thermalization? (close to integrability)

Emergence of effective bath in closed systems via dephasing/decoherence

(Pre)thermalization as a renormalization group transformation in time

UNIVERSALITY OF QUANTUM AND CLASSICAL DYNAMICS

Necessary conditions for observing universal dynamics. Kibble-Zurek scaling.

Geometric response in quantum and classical systems to slowly driven perturbations.

QUENCHES and TIME-DEPENDENT PROTOCOLS

Universal and macroscopic characterization of thermodynamic quantities during out-of-equilibrium processes:

entropy production

work/heat fluctuations

non-equilibrium phase transitions

driven systems and phase transitions

Classify types of not-equilibrium initial conditions:

initial state is the ground state of a different local Hamiltonian (sudden and continuous quench)

initial state has some highly excited degrees of freedom

Realize interesting Hamiltonians through time dependent protocols (long range interactions, topological states, extension of Floquet to almost periodic driving)

UNCONVENTIONAL TRANSITIONS

Survey of evidences for the many-body localization transition

Phase transitions in the time domain (time is the parameter that drives the transition)

INTEGRABLE and ALMOST INTEGRABLE SYSTEMS

How to define the "minimal" Generalized Gibbs Ensemble (GGE) which describes the steady state? (How many and which integrals of motion need to be included?)

What is the effect of breaking the translation invariance (either in the initial state or in the post-quench Hamiltonian) in free integrable theories

Universality away from equilibrium away from ergodicity: integrable, near integrable, mesoscopic systems

Definition of "almost" integrable system

EIGENSTATE THERMALIZATION HYPOTHESIS (ETH)

Does ETH work in the thermodynamic limit in systems with an infinitesimally small integrability breaking perturbation?

What does ETH imply for the two-time correlation function? Does it have predictive power?

Does ETH imply fluctuation-dissipation relations? Can we use ETH to prove fluctuation-dissipation away from equilibrium?

LAST BUT NOT LEAST

Does the fact that the system is closed (and retains phase coherence at all times) imply something? (interference in time domain, many-body echoes)

Numerical Tecniques for out-of-equilibrium systems

Program topicsNonequilibrium thermodynamics: non-equilibrium steady states of driven systems, dynamical phase transitions## Table of Contents

Integrability. What precisely is the role of the higher conservation laws? Which ones are important? How many do we need? When is a system "close" to being integrable"?Thermalization and pre-thermalization.What are the mechanisms of thermalization? How does it happen in nearly integrable systems?Disorder and many body localization.Universal (slow) dynamics (Kibble-Zurek mechanism).To what extent the ideas of universality can be applied to dynamics?Quench dynamics.Applications: cold atoms, solid state systems, nonlinear optics, cosmology, ...Numerical methods for quantum dynamics.## THERMALIZATION and PRE-THERMALIZATION

## UNIVERSALITY OF QUANTUM AND CLASSICAL DYNAMICS

## QUENCHES and TIME-DEPENDENT PROTOCOLS

## UNCONVENTIONAL TRANSITIONS

## INTEGRABLE and ALMOST INTEGRABLE SYSTEMS

## EIGENSTATE THERMALIZATION HYPOTHESIS (ETH)

## LAST BUT NOT LEAST