Periodic Reporting for period 1 - GRAPES (learninG, pRocessing, And oPtimising shapES) Reporting period: 2019-12-01 to 2021-11-30 Summary of the context and overall objectives of the project Geometry is everywhere. Using digital technology to automate and facilitate the use of geometric shapes is the focus of GRAPES. We focus on optimising the digital representation of 3D objects, devising efficient ways for handling and visualising them, and of course making them available for a wide spectrum of everyday uses ranging from urban planning to producing digital twins, and from drug design to 3d printing. More concrete applications addressed within GRAPES include geometric design in shipping, retrieval and mining of non-rigid shapes, and reconstruction of landscapes for the prevention of natural disasters. To devise robust and general methods for the aforementioned issues, GRAPES aims at advancing the state of the art in a variety of fields ranging from Computational and Numerical Mathematics, to Geometric Modelling and CAD, up to Data Science and Machine Learning. Besides applicable research, we address the technological challenge by creating synergies between Universities, Research Centers, and private companies. Our focus is the training of 15 PhD candidates; our structure allows them to benefit from top-notch research as well as a strong innovation component through a nexus of intersectoral secondments and Network-wide workshops. Work performed from the beginning of the project to the end of the period covered by the report and main results achieved so far A considerable amount of effort has been devoted in organizing the managerial structure, concluding the Consortium and Partnership Agreements, the various committees, and the details of the implementation of our work plan. We have set up a mechanism that will efficiently design, implement and monitor all activities within the Project aiming at providing the best possible training to our fellows. This is complemented by a strong network, including industrial stakeholders as well.The main results achieved so far can be partitioned into 3 scientific domains: 1. High-order methods and alternative representations deal with nonlinear methods for accurately representing complex shapes, while different representations offer complementary advantages. We have advanced in algebraic methods supporting such methods, including some prototype software for testing the new algorithms. Our emphasis has been on methods that exploit the structure of the given equations, such as their sparseness, aiming at techniques whose runtimes depend on the intrinsic rather than nominal input complexity. 2. Algebraic & numeric tools in shape optimisation and analysis have focused on isogeometric analysis, which allows for combining geometric modeling with computational fluid mechanics (typically governed by partial differential equations). Software engineering is a core target in bridging algorithms design with industrial-strength applications. A concrete example is modeling of ship parts for simulation, such as ship hulls, and propelers. 3. Machine Learning for shapes has concentrated on supervised methods for representing shapes and for offering robust methods in similarity search and object segmentation. We have pioneered specific neural network architectures for 3d objects as well as the combination of 2d image and 3d information (typically point clouds, but also meshes, implicit functions) for handling deformable objects. A more applied series of results concerns large scale 3D modeling and reconstruction with industrial partner GeometryFactory. 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) Progress in research has been marked by new algebraic algorithms in handling geometric problems, including intuitive control mechanisms for interactively modifying shapes, by novel methods in iso-geometric computing, and by new AI approaches regarding understanding and decision-making around 3d shapes. Software development has been a horizontal activity that allows our fellows to interact with industry and to disseminate their methods through open-source repositories. Applied results that may have a direct socio-economic impact include highly sophisticated methods for re-using geometric models across different platforms (with industrial partner ITI), shape optimisation for sensitivity analysis and other maritime applications by implementing viable search spaces, and modeling of dangerous geological regions. These innovation directions are expected to bear fruit within the lifetime of the Project. A more general direction of the Project that shall lead to tangible outcomes is to allow for complex geometric operations by novel AI algorithms in an accurate, efficient and, hopefully, predictable and understandable way.