Periodic Reporting for period 3 - NewtonStrat (Newton strata - geometry and representations) Reporting period: 2021-06-01 to 2022-01-31 Summary of the context and overall objectives of the project The Langlands programme is a far-reaching web of conjectural or proven correspondences joining the fields of representation theory and of number theory. It is one of the centerpieces of arithmetic geometry, and has in the past decades produced many spectacular breakthroughs, for example the proof of Fermat’s Last Theorem by Taylor and Wiles. The most successful approach to prove instances of Langlands’ conjectures is via algebraic geometry, by studying suitable moduli (or parameter) spaces. Their cohomology carries actions both of a linear algebraic group (such as GLn) and a Galois group associated with the number field one is studying. A central tool in the study of the arithmetic properties of these moduli spaces is the Newton stratification, a natural decomposition based on the moduli description of the space. Recently the theory of Newton strata has seen two major new developments: Representation-theoretic methods and results have been successfully established to describe their geometry and cohomology. Furthermore, an adic version of the Newton stratification has been defined and is already of prime importance in new approaches within the Langlands programme.This project aims at uniting these two novel developments to obtain new results in both contexts with direct applications to the Langlands programme, as well as a close relationship and dictionary between the classical and the adic stratifications. We investigate the geometry of Newton strata in loop groups and Shimura varieties, and representations in their cohomology. Secondly, we study corresponding geometric and cohomological properties of adic Newton strata. Finally, we establish closer ties between the two contexts. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far For the first part of the project we studied affine Deligne-Lusztig varieties. These arise naturally as a description in terms of linear algebra of Newton strata in the moduli spaces that arise in the Langlands program. We found new ways to describe which of these affine Deligne-Lusztig varieties are non-empty, to compute their dimensions and the closure relations between the associated Newton strata, and various other geometric properties. Further, we applied geometric results to show that Langlands correspondences are realized in the cohomology of certain unitary Shimura varieties.For the second part of the project, we used new methods introduced by Fargues and Fontaine in p-adic Hodge theory to study the recently defined adic Newton strata. We determined which strata lie in the closure of a given one, and considered intersections with another interesting space, the so-called weakly admissible locus. 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) The main question left for the second half of the project is the thirs set of questions regarding the relation between classical and adic Newton strata, where we want to study analogies and direct relations. Furthermore, we want to finish our study of geometric properties of affine Deligne-Lusztig varieties in affine flag varieties, extend our recently begun theory of the geometry of adic Newton strata, and apply all this to obtain new insight into the cohomology of these spaces.