The main goal of the proposed project is to find a practical algorithm to compute the p-adic height of a point on an abelian variety with split purely toric reduction. There are two possible approaches, which we will both attempt. On the one hand, Werner has related the p-adic heights to rigid analytic uniformization. We will make this explicit and algorithmically computable. On the other hand, when the abelian variety is the Jacobian of a curve, Coleman and Gross have shown that the p-adic height can be expressed in terms of Coleman integration.This has led to practical algorithms due to Balakrishnan, Besser and the main applicant in the good ordinary reduction case. We will use recent work in progress due to Besser and Zerbes to extend this to split purely toric reduction. The ability to compute p-adic heights explicitly will allow us to formulate a p-adic BSD conjecture for modular abelian varieties with split purely toric reduction, extending the celebrated conjecture of Mazur, Tate and Teitelbaum for elliptic curves, and to gather numerical evidence for this conjecture. In the case of good ordinary reduction this was already done by Balakrishnan, Stein and the main applicant. Moreover, we expect that the findings of the proposed project can be used to extend recent work on non-abelian Chabauty to the setting of higher curves of genus whose Jacobian have split purely toric reduction, with the goal of explicitly computing the rational or the integral points in previously inaccessible examples.

This proposal concerns solutions to systems of polynomial equations or, equivalently, rational points on algebraic varieties. This area of mathematics has seen exciting developments in recent years, and the Netherlands is home to a group of energetic young researchers at the forefront of these developments. This proposal will kickstart a strong and durable research community around these individual scientists. The 21st century is witnessing a revolution in our understanding of rational points on surfaces and higher-dimensional varieties, and it is in this field that our proposal lies. We have identified three interrelated research themes comprising three projects each. The first theme, “from curves to surfaces and beyond”, consists of projects which take established techniques from the study of rational points on curves and extend them to the substantially more difficult setting of higher-dimensional varieties. Chabauty’s method has proved very fruitful in the algorithmic study of rational points on curves, and we will investigate its application to certain classes of surfaces of general type. Algebraic geometry codes arising from curves are well established; we will build on recent constructions in the geometry of surfaces to produce good locally recoverable codes on surfaces. The third project is to investigate jumping of Mordell–Weil ranks in families of abelian varieties, generalising existing results on families of elliptic curves. The second theme, “from characteristic zero to characteristic p, and back”, looks at various settings in which geometry in characteristic p is related to arithmetic. The Brauer–Manin obstruction is an important tool for understanding rational points on a variety; we will deepen our understanding of it by relating it to the geometry of the variety when reduced modulo primes. Abelian varieties in characteristic p have consistently received a lot of attention for their theoretical relevance and real-world applications; we will study the reduction and lifting properties of abelian varieties of dimension at least two. The third project under this theme is to study the density of rational points within the p-adic or even adelic points on del Pezzo and K3 surfaces. Our third theme is “from rational points to Campana points”. Campana points are an emerging area of research in Diophantine geometry, linking rational and integral points on varieties. Manin’s conjecture was originally conceived for rational points and recently extended to Campana points; we will develop a toolbox to test the conjecture in fundamental examples. Secondly, we will develop a theory of local-global principles and Brauer–Manin obstructions for Campana points, bringing together the theories for rational and integral points. Finally, we will investigate the Hilbert property for Campana points, building on the latest covering techniques. These proposed projects are to be undertaken by six PhD students and three postdocs. While the projects themselves are independent, they share mathematical background and technical tools and will therefore benefit from extensive interaction and collaboration. To further stimulate co-operation we will hold monthly meetings for learning, discussion and collaboration. We will also organise an instructional workshop early in the programme, and an international conference towards the end of the programme.

For decades the main goal of the power system was to deliver power from generation plants to consumers through an extensive high-voltage transmission system and a medium to low voltage distribution system. As such the existing power system was designed in a centralized tree like fashion in order to connect a relatively small number of large plants to a large number of consumers. In addition, the gas infrastructure is developed separately, with an infrastructure consisting of high, medium and low pressure pipelines, more or less similar to the power grid structure. Both grids are regulated separately, even though they are coupled. So far, the analysis and design of new smart energy systems in the energy infrastructure have been mainly focused on only one particular grid such as the power grid, gas grid or heat grid. In particular, the power grid has received by far the most attention. Much of the current research evolves towards embedding of renewables such as photovoltaic cells and wind power into the power grid, where it is important to take into account the cloud and wind conditions, which are less predictable than the power production of a classical power plant. In addition, there is a fine meshed gas network in the Netherlands, where embedding of biogases and/or non-natural gases is developed. Currently, this network is coupled to the power grid on a limited basis, i.e., some Combined Heat Power (CHP) systems that produce both heat and power and run on gas are embedded. Furthermore, some larger power plants operate on gas. In the future, the embedding of many more micro-CHP systems in households is foreseen, and storage of surplus electricity in the form of power-to-gas (hydrogen) is also foreseen. Within this research proposal we aim at developing new/tailored methods that integrate hierarchical and distributed optimal control methods for large-scale networked systems necessary for the embedding of new energy systems in the coupled power and gas grids. By a large-scale networked system, we mean a collection of interconnected heterogeneous dynamical systems (agents) sharing/collecting information. Thus, the research of this proposal is concerned with designing integrated distributed controllers (at the level of agents) based on the available limited local (asynchronous) information exchange in order to achieve a global (at the level of the entire network) hierarchical optimality criterion. To the best of our knowledge, there do not yet exist control schemes with strong theoretical foundations to ensure stability and optimality for such complex structures.

This project aims at creating open educational resources (OERs) for university teachers involved in STEM courses. The final product is a combination of new and existing technologies consisting of the following components: A) a library of interactive knowledge clips, B) an adaptive question generator providing students with a personalized learning experience, and C) an analytics tool for monitoring student performance and giving early warning signs. These tools are to be used and tested in courses on Calculus which is one of the most common university courses offered in STEM programmes.

The study of rational points on algebraic curves is an important part of the classical field of diophantine equations. Such curves are classified by their genus; by Faltings’s theorem an algebraic curve over the rationals of genus g>1 has only finitely many rational points. This project will study rational points on such curves, including both algorithmic/explicit as well as theoretical/structural aspects. At present there is no general algorithm for computing the rational points on a given curve X over the rationals of genus g>1. When the Mordell-Weil rank r of the Jacobian of X is less than g, the method of Chabauty-Coleman can often be used for this purpose. The idea is to embed the rational points into a larger space, the p-adic points for a suitable prime p, on which we can do analysis. Linear relations between integrals yield a locally analytic function which has finitely many zeros and vanishes on the rational points. A recent extension of this technique, called quadratic Chabauty is based ideas of Kim and uses bilinear relations; it is applicable when r=g and some further conditions are satisfied. In this project, we will combine this method in an innovative way with other techniques, extend it further based on experimental methods, and apply it to solve several interesting moduli problems. From a theoretical point of view, it is widely believed that there should be a uniform upper bound for the number of rational points on a curve of given genus, but such bounds are only known in very restrictive situations, based on a modification of Chabauty-Coleman. We will extend quadratic Chabauty in such a way that it can be combined with existing results to extend the range of curves for which uniform bounds are known.