Periodic Reporting for period 3 - GenGeoHol (Non AdS holography and generalized geometric structures) Reporting period: 2020-09-01 to 2022-02-28 Summary of the context and overall objectives of the project The twentieth century saw two mayor revolutions in the area of theoretical physics. The first was the understanding that the force of gravity can be described as a consequence of a geometric description of space and time called General Relativity. The second was the realization that, at the smallest scales, the rules that guide the behavior of subatomic particles are that of quantum mechanics (as opposed to classical mechanics). While these new paradigms have been extremely successful and have been corroborated in experiments at the smallest and largest possible scales they have raised important questions about the nature of reality at the most fundamental level. In particular, these two discoveries, together, have important and puzzling consequences for the basic building blocks of space and time. A window into the nature of this problem is provided by interesting objects called black holes. These are compact astrophysical objects from which light cannot escape. Therefore, we cannot see what happens inside them. But rather surprisingly, General Relativity and Quantum Mechanics put together predict that the degrees of freedom of these objects lie only on their surface and not inside. This became the most important discovery in physics at the end of the twentieth century and goes by the name of the Holographic Principle. It has dominated physics research in the twenty-first century.In this ERC project, we propose new technical tools to study how the Holographic Principle can be applied to our Universe and, in particular, to our Cosmology. Because of the expansion of space and time, there is a Cosmological Horizon beyond which we cannot see. It has been proposed that this horizon should also be described within the Holographic approach. In GENGEOHOL we use modern geometric and quantum mechanical techniques to put this problem on equal footing to previously better understood but less realistic models for Holography.Rather surprisingly, the same techniques that can be used to understand our Cosmology have direct implications for the physics of certain exotic materials that are studied in the lab. These are called "strongly coupled systems" and cannot be currently understood by other techniques. In the recent past new phases of matter have been discovered with new exotic symmetries. It is these symmetries that connect these systems through the holographic principle to our previous questions. Therefore, the understanding of space and time can directly lead to progress in more experimentally accessible setups.Mankind has always wondered about the nature of reality. What is time? How did it begin? How will it end? If we are to answer these questions, we need to first understand what the Holographic Principle has to say about realistic models for space-time. GENGEOHOL is a step in this direction. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far In the course of time in which GENGEOHOL has been running important progress has been made in different directions. As important examples we can point out:First, a proposal was put forward on how to apply the Holographic Principle to Cosmological space-times. The solution proposed within GENGEOHOL is novel in two ways. First, it gives an important role to the observer asking the physical questions. This allows the model to account for time dependent questions which were outside the scope of previous proposals. Second, it embeds Cosmological setups in previous well understood less realistic models. As such, there is great power in this formulation from what has been understood about the Hologrphic Principle in the last twenty years.Besides Cosmological setups, GENGEOHOL has made progress in understanding the Quantum Information aspects of Wormhole geometries.Second, it was understood how certain generalized geometries are introduced in Holographic setups. This is of great importance as these symmetries connect the study of phases of matter for real physical materials to the question about the individual building blocks of space and time within black holes. A nice application of these ideas was obtained in GENGEOHOL to the physics of superfluids.Third, it was understood what type of generalized symmetries are responsible for the emergence of General Relativity in Holographic theories. This result is the most concise proof of this phenomenon to this date and provides some perspective on why General Relativity is a preffered theory of gravity. Progress beyond the state of the art and expected potential impact (including the socio-economic impact and the wider societal implications of the project so far) By the end of GENGEOHOL we hope to be able to fully understand what generalized symmetries are responsible for the geometry of space and time that we observer in our Universe and how to connect this physics to that of exotic phases of matter.