Fusing the principles of general relativity and quantum mechanics in a consistent theoretical framework still constitutes one of the main open challenges in theoretical physics today. Over the last decades, the gravitational asymptotic safety program has taken significant steps towards achieving this goal. A central virtue, driving the success of the approach, is its conservative nature: the program builds on well-established principles of quantum field theory and extends them in a rather minimalistic way. In this talk, we will review the key developments in the program, building up to its present status. I will also attempt to give an outlook on the challenges that need to still be addressed in the future.
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Correlated Electrons from Zero to Infinite Dimensions: Early Days of KOMET 7 in Mainz. I will talk about the problems in correlated electron systems that occupied us in the early days of the KOMET 7 research group headed by Peter van Dongen. These problems include quantum impurity problems as well as the dynamical mean field theory (DMFT), i.e., correlated electrons in the infinite-dimensional limit. At first glance, these two problems are very different because the impurity problem is in a sense zero-dimensional, whereas the DMFT is formally infinite-dimensional. However, the effective problems and solution methods of these two problems are losely related, and both approaches can be used to describe the behavior of real three-dimensional materials. In addition, a major activity of group members has been to develop and use matrix-product-state and tensor-network methods, especially the density matrix renormalization group. These methods are ideally suited to study quasi-one-dimensional and two-dimensional strongly correlated systems. They can be applied to a variety of systems ranging from transition-metal coumpounds such as the cuprates to organometallic materials such as Bechgaard salts as well as to quantum simulators formed from cold atomic gases on optical lattices.
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The fact that neutrinos are massive particles can be neatly explained by the existence of (at least two) additional neutrinos, called sterile neutrinos. Depending on their mass, sterile neutrinos can be looked for in various experiments. I will discuss how sterile neutrinos can be probed indirectly in neutrinoless double beta decay experiments and how this search compares to more direct probes and how it can test low-scale leptogenesis as a solution for the baryon number asymmetry of the universe.

Since neutrinos have no electric charges, they may be their own antiparticles, referred to as Majorana neutrinos, and thus violate lepton number conservation. Neutrinoless double beta decay would be a direct consequence, and the search for this decay mode is the most sensitive method to unravel the Majorana nature of neutrinos. By operating bare germanium diodes, enriched in Ge-76, in an active liquid argon shield, GERDA achieved an unprecedently low background index of 5.2∙10−4 counts/keV kg yr in the signal region and collected an exposure of 127 kg yr in a background-free regime. No signal was observed, and a limit on the half-life of 0νββ decay in Ge-76 is set at T1/2 > 1.8∙1026 yr (90 % C.L.) [1] and Majorana neutrino masses are constrained to mββ< 79–180meV (90\% C.L.). The LEGEND Collaboration builds on the success of GERDA and MJD, and develops a phased, Ge-76-based double-beta decay experimental program with a T1/2 - discovery potential beyond 1028 years. Its first stage, LEGEND-200, started data-taking in early 2023, and LEGEND-1000 is under preparation. The first results from LEGEND-200, based on 48.3 kg·yrs of data, were presented in June at the Neutrino 2024 conference in Milan and will be discussed.