CoGNATE

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.