The cohomology of arithmetic groups sits squarely at the intersection of several fields of mathematics. For example, it connects to number theory and arithmetic geometry via Galois representations and Hecke operators, and to representation theory, via its relationship to automorphic forms and automorphic representations. It also has deep connections with geometry, topology, and algebra, through its connections with algebraic K-theory, locally symmetric spaces, reduction theory, and lattices. Explicit calculations have played an increasingly important role in the theoretical development of the subject and its applications. For example, explicitly computing the cohomology gives tools to formulate conjectures about automorphic forms and special values of L-functions, and to try to understand the increasing influence in number theory of the torsion in cohomology. As the scale and complexity of the calculations have increased it has become more and more common for such computations to be performed with the aid of computers. This ITN Project will bring together international experts with diverse and complementary skill sets, and expertise in computational techniques relevant to such calculations and their applications to cohomology of groups, algebraic K-theory, arithmetic geometry, and lattices. This project aims to provide the computational ``cogs'' needed for the efficient application of the cohomological machinery of Hecke operators to various arithmetic groups and Artin groups which are at the cornerstone of several main conjectures. The main goal is to form a new generation of young researchers with a unique expertise at the crossroad of these topics via innovative boot camps through new research collaborations and to broaden our theoretical knowledge with a view to extending the scope of computer aided calculations in this area with potential applications to industry and quantum computing. The proposed network will: - develop computational methods in algebra, geometry, topology and number theory directed towards specific cognate open problems in mathematics and theoretical computer science (relevant to automorphic forms, cohomology of arithmetic groups and algebraic K-theory) with potential applications to industry in areas such as coding, cryptography and topological data analysis; - contribute new blood and software to EU funded symbolic computation projects such a GAP and PARI/GP; - provide training for 15 graduate students in interdisciplinary areas of mathematics, computer science and software engineering; - strengthen an existing informal interdisciplinary network of academics based at 10 EU universities and research institutes across 6 EU countries and 1 USA partner; - contribute to an open database of geometric models developed in the project, and publish a collective monograph on the mathematical and computational methods involved in the project which will contribute to the dissemination of the ESRs works.

Membranes can be used in a plethora of applications. As permeation-selective barrier they may serve as a separation membrane, for e.g., ultrafiltration, dialysis, water purification, or gas separation. In a more general context, they can also be used as a chemical, physical, or electrical barrier, e.g., in protective films, capacitors, or as sensors. For all these applications it is advantageous to make the membrane as thin as possible and at the same time as mechanically and chemically robust as possible. With respect to these criteria, graphene seems to represent the ideal material due to its mechanical strength and infinitesimal thickness of only 3 Å. Perfect graphene is impermeable for all gases and liquids and perforated graphene promises an unprecedented level of transport rates in filtering applications as the quasi two-dimensional selective membrane would exhibit negligible wall interactions. Thus, for membrane technologies novel composites based on graphene can offer significant improvements unachievable by conventional materials. The main goals of this proposal are the following: We aim to develop a process for the manufacturing of robust composites consisting of graphene and a polymer film, which will be processed further to produce ultrafiltration (UF) or nanofiltration (NF) membranes with relevance for technical separations, where the selective element is a single, artificially perforated layer of graphene. The performance of these UF and NF membranes will be assessed and the underlying mechanisms of manufacturing and separation processes will be elucidated. The perforation of the composite will be achieved by an established technology, i.e. the irradiation with swift heavy ions, enabling to control pore density and size in graphene. These pores will have a very narrow size distribution (isoporous) and their size can be selected in the range from 5 to 50 nm^2 thus offering a high degree of selectivity. By using another established technology known as ?track-etching?, the selective barrier pores will be connected to larger pores in the supporting polymer film, yielding a unique composite UF or NF membrane. Functionalization of the pore entrance, in particular with charged groups, can be used to further increase selectivity so that even desalination of water may be feasible. As the transport through the 2D barrier layer is not hindered by wall interactions very low pressures are needed. It is therefore expected that the targeted membrane prototypes for UF or NF will outperform current materials by a factor of ~100 (in terms of higher fluxes at same selectivity) which would enable substantial energy savings. However, appropriate concepts for integration of such high flux membranes into modules are also absolutely necessary and therefore in this project micro-/nanofluidic separation systems based on graphene will be designed and investigated as first steps towards implementation of such radically novel membranes.