**This is a virtual, online-only workshop.**

The GW approximation in many-body perturbation theory is the state of the art method for computing theoretical photoemission spectra in moderately sized systems. GW fits nicely between density functional theory and wave function methods when comparing accuracy or feasible system size. For weakly- to moderately-correlated systems, GW can now be applied to systems with hundreds of atoms and gives good agreement with experimental photoemission spectra.

The GW approximation was proposed by Hedin in 1965 [1]. It is based on the many-body Green's function formalism and approximates the electronic self-energy as the product of G, the single-particle Green's function, with W, the screened Coulomb interaction. GW is an approximation. It is the neglect of a certain class of Feynman diagrams from the exact theory, the so-called vertex corrections, that make the GW problem tractable. The many-body formalism behind GW gives the theory formal backing to compute excited states, though ground state energies are also accessible with GW. Today, GW is a standard approach and has been added to the major electronic structure codes in the last decade. However, challenges abound and a widespread adoption of the GW method requires further development. It is therefore a prudent time to convene at a dedicated GW workshop to address the most pressing challenges.

The most important challenge is how to push the theory to larger systems and develop numerically efficient algorithms, especially as the exascale computing frontier approaches. Developments cover the range from the numerical type, focused entirely on accelerating an ordinary GW calculation, to more physically motivated algorithm development. These physically motivated developments can include additional physics beyond ordinary GW or are acceleration methods inspired by physical, rather than purely numerical, reasoning. Most developments require aspects of both.

Numerical developments have proceeded in several directions [2]. One of them is the reduction of the computational prefactor while maintaining canonical scaling. Among these techniques are methods which avoid summations over empty states with the Sternheimer and related techniques [3]. Others address a reduction of scaling with respect to system size [4, 5] using imaginary time approaches. Recent efforts to push GW to large systems also include stochastic GW approaches [6]. A more physically motivated development for large systems is to embed a GW calculation in a surrounding medium treated at a lower level of theory [8]. Other avenues of development include real time solutions to the GW equations, continued development of parallelization algorithms, and the accurate prediction and understanding of core levels.

Many of these developments can be combined, in principle, and the developers would benefit from making contact with each other. There are also common threads to all development, like the preparation for exascale computing. The first exascale computational resource in the EU has been approved, and a substantial amount of its resources could be allocated to electronic structure theory. Roundtable discussions can address long standing challenges for GW calculations including the convergence of low-dimensional systems in three-dimensional codes, calculations for solids in localized basis sets, and memory distribution algorithms.