Most attempts to generalize the Jacobian Conjecture failed. The only such an attempt which is still standing is a conjecture due to Olivier Mathieu from 1995, concerning integrals over compact connected Lie groups. However this conjecture is extremely hard. The proof of the Abelian case, due to Duistermaat and van der Kallen, is already a tour de force. In 2009, Wenhua Zhao came up with a new set of fascinating conjectures. One of them, the Image Conjecture, implies the Jacobian Conjecture. Both Mathieus conjecture and the Image Conjecture are embedded in the context of Mathieu subspaces, a concept introduced by Zhao which generalizes the notion of an ideal. Now for the first time in the long history of the Jacobian Conjecture there is a beautiful general context in which this conjecture can be studied. Relations with orthogonal polynomials and differential operators give new oppertunities. In this proposal we intend to study examples of Mathieu subspaces, including the ones directly related to the Jacobian conjecture and, starting from these examples, try to lay foundations for the development of a theory of Mathieu subspaces.

This monograph provides an introduction to noncommutative geometry and presents a number of its recent applications to particle physics. In the first part, we introduce the main concepts and techniques by studying finite noncommutative spaces, providing a “light” approach to noncommutative geometry. We then proceed with the general framework by defining and analyzing noncommutative spin manifolds and deriving some main results on them, such as the local index formula. In the second part, we show how noncommutative spin manifolds naturally give rise to gauge theories, applying this principle to specific examples. We subsequently geometrically derive abelian and non-abelian Yang-Mills gauge theories, and eventually the full Standard Model of particle physics, and conclude by explaining how noncommutative geometry might indicate how to proceed beyond the Standard Model. The second edition of the book contains numerous additional sections and updates. More examples of noncommutative manifolds are added to the first part, in order to better illustrate the concept of a noncommutative spin manifold, as well as to illustrate some of the key results in the field (such as the local index formula). The second part now includes the full noncommutative geometric description of particle physics models beyond the Standard Model of particle physics. This is very desirable given the developments and discoveries at the Large Hadron Collider at CERN during the last few years. We have also included a Chapter on the recent progress on the formulation of the noncommutative quantum theory.

Over the past decade, the possibility of quantum computers and the reality of quantum information theory have led to a remarkable cross-fertilization between computer science, mathematics, logic, and physics. This proposal lies in the interface of these fields, as reflected by a team of applicants consisting of a computer scientist, a mathematical physicist specializing in quantum theory, and a pure mathematician with a strong background in logic. Each of the applicants is the author of a monograph on a research topic of immediate relevance to this project: - B. Jacobs, Categorical Logic and Type Theory (Elsevier, Amsterdam, 1999). - N. P. Landsman, Mathematical Topics Between Classical and Quantum Mechanics (Springer, New York, 1998). - S. Mac Lane and I. Moerdijk, Sheaves in Geometry and Logic: A First Introduction to Topos Theory (Springer, New York, 1992). Subsequently, each was the recipient of an NWO ?Pionier? grant that was used to further develop and innovate these topics; the present project may be seen as a fusion of the three research lines in question. Our aim is, briefly, to combine two different branches of category theory (viz. tensor categories and topos theory) and to apply this combination to the logical analysis of composite quantum systems and quantum computation. We apply for €750k, to be spent on a postdoc (senior enough to relate the following topics to each other) and on three PhD students, working on three concrete subthemes: - Modularity in quantum computation; - Construction of examples of toposes with a nontrivial tensor structure; - Development of internal logic of ?tensor toposes?. The three applicants will be in charge of individual themes as well as of the overall coherence of the project, enhanced by a PhD training programme and frequent joint activities.

The recent direct detections of gravitational waves and multi-messenger discovery of a binary neutron star merger opened new exciting opportunities for probing fundamental physics in unexplored regimes. Key scientific targets in this field are the rich phenomena of neutron star binary systems, whose unique science is imprinted in the gravitational waves generated during their inspiral and merger, as well as the accompanying electromagnetic counterparts. Each messenger conveys complementary information about these violent events, with a joint analysis being essential for tests gravity and cosmology, for probing the microphysics of matter at supra-nuclear densities, and for gaining deeper insights into neutron star and black hole formation. The aim of this project is to jointly analyze information from gravitational waves and electromagnetic counterparts to probe the rich physics of neutron stars. Neutron stars contain matter compressed by gravity to up to several times the density of an atomic nucleus and represent exceptional environments where all four fundamental forces are simultaneously important. Despite a decades-long effort in theory, experiments, and astrophysical observations to probe neutron star physics, we still have only a diverse set of hypotheses about the composition and properties of matter in such extreme conditions. The proposal is organized around three interrelated projects. We will first develop a systematic analysis strategy for the joint interpretation of gravitational waves and electromagnetic counterparts to determine the nature of the compact objects in a merging binary system. We will then address several current challenges in extracting the fundamental science from gravitational-wave signals, which relies on robust theoretical models, by advancing models to include more realistic physics and developing efficient descriptions for practical use. Finally, we will assess the prospects for probing black hole formation and measuring new physics with gravitational wave experiments such as LIGO, Virgo and the Einstein telescope. This project will provide key inputs for using the new field of GWs to (i) probe the fundamental physics of matter in unexplored regimes, (ii) distinguish double neutron star binaries from those involving a black hole or exotic object, (iii) measure the microphysics and energetics driving the merger, tidal disruption, and black hole formation, and (iv) elucidate the full cosmological context of these cataclysmic events. The proposal’s timely and urgent approach leverages experimental opportunities opening up with new facilities (including NWO funded projects) that, in return, will rely on the deliverables of such research. Given that this is an emerging new field, the current proposal may also achieve unexpected discoveries.

Extremes in the Universe, such as black holes (BHs) and strongly-curved spacetimes, are key areas for astrophysics this century. Within this context, a defining question is the formation of BHs themselves. The birth of black holes (BHs) provides unique astrophysical conditions where extreme gravity, densities and magnetic fields come together. This proposal aims at directly measuring these extreme astrophysical processes by tracking the mergers of pairs of neutron stars (NSs) or a NS-BH that ultimately form BHs. Although rare and short-lived, NS mergers produce copious amounts of electromagnetic (EM) and gravitational wave (GW) radiation within short timescales. Our goal is to measure, for the first time, the physics driving the mergers through observations of their EM and GW radiation. It leverages the opportunity today to observe these events thanks to new time-domain telescopes and GW detectors (LIGO and Virgo). GW observations will measure the fundamental parameters of NS and BHs, such as their masses, spins and equation of state. EM observations, sensitive to the composition and thermodynamic state of matter, will be complementary to the GW signals that constrain the NS and BH masses and spins. My objectives are to: i) interpret combined EM and GW measurements, and hence ii) determine the astrophysics driving the NS/BH mergers leading to the BH formation. To achieve them, I have designed a multi-disciplinary program, bridging time-domain astronomy, general relativity and statistical and computational astrophysics. I will provide a real-time software necessary to characterise the first EM and GW mergers. We will work in close collaboration with the LIGO-Virgo detectors, the iPTF/ZTF and BlackGEM optical telescopes and radio MEERKAT and LOFAR observatories, to apply the new tools and directly witness the first BH births within the next five years.