In this proposal semi-integral points refer to notions of rational points on algebraic varieties that satisfy an integrality condition with respect to a weighted boundary divisor. They were first introduced by Campana and by Darmon. Campana points have recently risen to the attention of the number theory community thanks to a Manin type conjecture in the recent work of Pieropan, Smeets, Tanimoto and Várilly-Alvarado. Semi-integral points provide both an intermediate notion and a generalisation of the notions of rational and integral points, thereby unifying the two theories. This proposal concerns the existence of semi-integral points and the density of orbifold pairs in general families having semi-integral points. The aims of this proposal are to determine good upper bounds for the density of orbifold pairs in a general family that have semi-integral points (WP1) and to compute obstructions to the existence of semi-integral points (and hence to integral points) in key examples corresponding to long-lasting questions in number theory (WP2). The approach will combine a variety of techniques from analytic number theory, algebraic geometry and arithmetic statistics. For (WP1), the experienced researcher and the supervisor will develop a criterion to detect local semi-integral points together with a sieve method to estimate the number of everywhere locally soluble varieties in the family. For (WP2), the research team will develop a Brauer-Manin obstruction theory for semi-integral points to compute failures of the integral Hasse principle in fundamental examples and handle classical Diophantine problems such as the existence of integral points on diagonal cubic surfaces and the non-existence of consecutive powerful numbers.

The project contextually sets up a novel framework to study the spectral-theoretical properties of classes of non-selfadjoint (NSA) operators related to Markov processes (MP) via their intertwining to a continuous path selfadjoint (SA) MP. Conceptually, this means that the jumps of each class of NSA MP can be considered a perturbation of one SA MP realized by an intertwining kernel. This approach can have far-reaching consequences for understanding classes of MP as the reduction to SA MP leads to well-studied objects whereas the spectral theory of NSA operators is far from understood. The price of that is the non-invertability of the intertwining kernels. This framework is explored and crystallized by a challenging, detailed spectral-theoretical study of an enormous class of NSA operators directly arising from the key phenomenon of self-similarity and in duality from branching. This is achieved by a synergy of research fields complementing each other to obtain the spectral properties of those operators culminating in the derivation of spectral expansions of the generated semigroups. As a result of this synergy, a number of tools and techniques with impact, including applications to fields beyond the scope of the project, are derived. A particular development in the area of recurrent equations and special functions will be unexpectedly exploited to the effect of a comprehensive theoretical and applied study, including numerical schemes, of key quantities in financial and insurance mathematics such as Asian options and perpetuities. A training-through-research in line with the fellow’s affiliation to the host institution and the proposed secondment will critically contribute to the optimal completion of the proposal in terms of time, scope and quality.

The goal of the project is to present a long term vision for a new Centre of Excellence (CoE) for Mathematical Modeling and Advanced Computing in Science and Engineering and a business plan for its establishment. The founders of the center are the two leading scientific institutions in Bulgaria in the area of Mathematical Modeling and Advanced Computing – the Institute of Information and Communication Technologies (IICT) – coordinator, and the Institute of Mathematics and Informatics (IMI), both from the Bulgarian Academy of Sciences, jointly with the advanced partner Vienna University of Technology (TU Wien). The CoE will concentrate its activities in four carefully selected key areas with high potential for first-rate scientific achievements and innovations: (i) Mathematical Modeling; (ii) Advanced Computing; (iii) Content Technologies, Intelligent Interfaces and Knowledge Processing; (iv) Flagship Applications in Science and Engineering. They are primarily focused on scientific excellence and technology breakthroughs in priority directions aligned with the challenges of the Programme Horizon 2020 and the Bulgarian Innovation Strategy for Smart Specialization 2014-2020, especially the priorities "Informatics" and "ICT and Mechatronics and Green Technologies". The CoE will be supported by the e-infrastructure "National Centre for High Performance and Distributed Computing", which combines hardware, software, middleware, and services, coordinated by IICT. This infrastructure is a part of the Bulgarian National Road Map for Research Infrastructures. The CoE is aimed at developing new mathematical approaches and advanced computing tools for efficient solutions of problems with high scientific and social impact. The strong research and development potential and mutually complementary expertize of the two Bulgarian institutions, IICT and IMI, and the advanced teaming partner from TU Wien, fully meet the challenging requirements of the planned new truly joint center.

Basing on the collaboration between academic and non-academic partners from Italy, Spain, Bulgaria, and Norway ISPAS’ overall objective is to develop a new joint curricula of PhD courses that will take place in both academic and non-academic surroundings to equip PhD candidates of the STEM Sciences, Medical Sciences, Social Sciences, Arts and Humanities with competences and skills in the areas of open innovation and open science to facilitate their employment after graduation in and outside of academia. The new joint curricula of PhD skills courses will address the recommendation of reinforcing tools, resources and guidance in education, training, employment and other learning settings to support people in managing their lifelong learning pathways. The development of the courses will be organized based on the quadruple helix model of collaboration where consortium’s academic partners collaborate with consortium’s non-academic partners, representatives of industry and public sector, and involve governmental representatives and citizens. Further, ISPAS will include courses for training data stewards. ISPAS' activities will produce an impact on the improvement of the innovation potential of future PhD candidates. ISPAS' activities also aim to produce an impact on the joint collaboration between academia and stakeholders in the regions of Catalonia in Spain, Piemonte in Italy, Rogaland in Norway, and Bulgaria by providing opportunities for improving skills intelligence, skills visibility and comparability for better career choices, learning about future Open Science and entrepreneurship skills needs and employment potential of scientists of the STEM Sciences, Medical Sciences, Social Sciences, Arts and Humanities in various interdisciplinary and intersectoral fields.