One of the main goals of the workshop will be to, through a collective effort, try to find answers to the following pertinent questions:

-What are the effects of self-consistency and how does the final solution depend on the initial guess and on the chosen self-consistent scheme?

This question has been recently elucidated in the case of a simplified model system, stressing the fact that the danger to run into unphysical solutions increases with the complexity of the equations to solve, as when including vertex corrections. This is a very important result for further developments beyond GW and work is in progress to understand what happens in realistic cases.

-What is the best strategy to find efficient vertex corrections?

GW, with the self-energy being of first order in W, is not expected to describe strong correlation. Higher orders in W could be added by iterating the equations, but this is technically and computationally difficult. Some alternative strategies will be discussed, such as combining GW with other approximations to the self-energy, such as the T-matrix, which works better in the low-density limit, where GW fails.

-What is the best way to deal with the strong-coupling limit and what can we learn from other methods?

The atomic limit is a true limitation for GW, and it is very difficult to formulate corrections that are able to capture this physics. However, much can be learnt from other methods, such as DMFT. We will discuss how to reformulate existing equations in order to set a common framework with other methods and to get unambiguous corrections from them.

-What are the limits of thinking in terms of the self-energy?

Up to now there does not exist a well-established strategy to obtain vertex corrections that systematically improve on GW. This motivates the choice of alternative formulations that do not approximate the self-energy but that directly approximate the GF. Some problems, such as those related to self-consistency, the inclusion of dynamical effects, the treatment of strong correlation, would then be eliminated. However, other problems may appear. Clarifying what are the pros and cons of this new strategy will be one of the central themes of the workshop.

-Which knowledge obtained from the one-body GF can be transfered to the calculation of higher-order GFs calculations?

There are common patterns in the equations for the various GFs. For example, the two-body GF fulfills a Dyson-like equation, the so-called Bethe-Salpeter equation, which has the same structure as the Dyson equation for the one-body GF. The identification of common patterns is very useful in order to apply the same strategy to similar problems. For example, the modus operandi for deriving the cumulant approximation to the 1-body GF, which describes multiple satellites in photoemission spectra, can be used to derive the corresponding approximation for the 2-body GF, which will describe multiple higher-order excitations in absorption spectra. We will address the opportunities to transfer the lessons learned for the one-body GF to higher-order quantities.