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$^{\textbf{26}}\mathrm{Al}$ and 
$^{\textbf{60}}\mathrm{Fe}$ in the Cosmos and in the solar system
The cosmic evolution of the chemical elements from the Big Bang to the present time is driven by nuclear fusion reactions inside stars and stellar explosions. A cycle of matter recurrently re-processes metal-enriched stellar ejecta into the next generation of stars. The study of cosmic nucleosynthesis and this matter cycle requires the understanding of the physics of nuclear reactions, of the conditions at which the nuclear reactions are activated inside the stars and stellar explosions, of the stellar ejection mechanisms through winds and explosions, and of the transport of the ejecta towards the next cycle, from hot plasma to cold, star-forming gas. Due to the long timescales of stellar evolution, and because of the infrequent occurrence of stellar explosions, observational studies are challenging, as they have biases in time and space as well as different sensitivities related to the various astronomical methods. Here, we describe in detail the astrophysical and nuclear-physical processes involved in creating two radioactive isotopes useful in such studies, 
$^{26}\mathrm{Al}$ and 
$^{60}\mathrm{Fe}$. Due to their radioactive lifetime of the order of a million years, these isotopes are suitable to characterise simultaneously the processes of nuclear fusion reactions and of interstellar transport. We describe and discuss the nuclear reactions involved in the production and destruction of 
$^{26}\mathrm{Al}$ and 
$^{60}\mathrm{Fe}$, the key characteristics of the stellar sites of their nucleosynthesis and their interstellar journey after ejection from the nucleosynthesis sites. This allows us to connect the theoretical astrophysical aspects to the variety of astronomical messengers presented here, from stardust and cosmic-ray composition measurements, through observation of 
$\gamma$ rays produced by radioactivity, to material deposited in deep-sea ocean crusts and to the inferred composition of the first solids that have formed in the Solar System. We show that considering measurements of the isotopic ratio of 
$^{26}\mathrm{Al}$ to 
$^{60}\mathrm{Fe}$ eliminate some of the unknowns when interpreting astronomical results, and discuss the lessons learned from these two isotopes on cosmic chemical evolution. This review paper has emerged from an ISSI-BJ Team project in 2017–2019, bringing together nuclear physicists, astronomers, and astrophysicists in this inter-disciplinary discussion.
A duality theorem for the singularity category of a finite dimensional Gorenstein algebra is proved. It complements a duality on the category of perfect complexes, discovered by Happel. One of its consequences is an analogue of Serre duality, and the existence of Auslander–Reiten triangles for the 
$\mathfrak{p}$ -local and 
$\mathfrak{p}$ -torsion subcategories of the derived category, for each homogeneous prime ideal 
$\mathfrak{p}$ arising from the action of a commutative ring via Hochschild cohomology.
A duality theorem for the stable module category of representations of a finite group scheme is proved. One of its consequences is an analogue of Serre duality, and the existence of Auslander–Reiten triangles for the 
$\mathfrak{p}$-local and 
$\mathfrak{p}$-torsion subcategories of the stable category, for each homogeneous prime ideal 
$\mathfrak{p}$ in the cohomology ring of the group scheme.
