Traditional relational data management systems are challenged by the abundance of highly interconnected heterogeneous data. This led to a surge in the popularity of graph databases in many industry and academic areas. For example, graph datasets with world-wide multi-omics data for genomic analyses and contact tracing were collaboratively curated during the pandemic (EU Datathon, CovidGraph, Covid-19 Knowledge Graph). Other critical societal graph databases use-cases include finance, telecommunications, journalism, and intelligent transportation. However, when processing geo-distributed graph data at scale, custom distribution models are needed, whose support still poses practical challenges. These are: replication, to mitigate slow and unreliable networks; sharding, to horizontally partition large graphs; and partial replication, to favor access locality, replicating data close to clients. Moreover, local-first models provide high availability, combining sharding and replication for read and write access to a relevant data subset, even when disconnected. While distribution mechanisms, such as Replicated Data Types (RDTs), have become well-established for local-first key-value stores, their usage in graph databases is largely unexplored. This is a more complex setting, due to its stronger demands to compositionally maintain connectivity invariants. VERDI proposes a novel interdisciplinary methodology to reliably devise such foundational distribution devices, cross-cutting the areas of databases, distributed systems, formal methods, and programming languages. It comprises four work packages (WPs). WP1 will build a unified formal foundation for prototyping and extracting correct-by-construction graph RDTs (GRDTs). WP2 will tailor these to provably enforce complex invariants under weak consistency models. WP3 will extend GRTDs with parametricity and transactional support and WP4 will evaluate their performance on a decentralized graph-based ledger industrial use-case.

Spintronics is the area of research dedicated to the study of how 'spin'--the quantum mechanical currency of magnetism--can be used realize new types of information transport, storage, and processing system which surpass the capabilities of those currently found in our computers and other electronic devices. Over the quarter century since its conception this field has branched in many directions. Some have already delivered spectacular real-world impact: spintronic magnetic field sensors are, for example, the bedrock of contemporary hard-disc technology and key players in the choreography of the information age. Others hold exciting promise as the basis for technologies of the future: a host of spintronic sub-disciplines feature among the most active and innovative areas of contemporary solid-state physics research. Today, with many spintronic sub-fields now well-developed, there is mounting interest in unlocking the rich and subtle physics which connects them. Against this background, this proposal is about recognizing and investigating a new and exciting frontier: the interface between magnonics and magnon spintronics, and quantum information. The field of magnonics is the area of magnetics dedicated to the science of quasi-particles known as magnons. In certain magnetic systems, magnons are able to play the role of microscopic spin-carrying tokens which can be generated and transmitted over relatively long distances (up to centimetres) and at high speed (many tens of kilometres per second). Magnon spintronics, magnonics' emerging sister discipline, is concerned with structures and devices which involve the passing of spin-information between magnons and electrons, the familiar workhorses of conventional electronics. As appreciation of the interplay between magnonic and electronic spin-transport deepens, so excitement surrounding its possible contribution to next-generation information technology heightens. To date however, work in magnonics and magnon spintronics has focused on the study of room-temperature magnon and magnon/electron systems in the classical limit. As a result, the field of experimental quantum measurement and information processing has yet to explore what the magnonic theatre has to offer. This project will develop the first experimental systems dedicated to a broad and systematic investigation of magnonics and magnon spintronics at the quantum level. Building on this, it will take the maiden steps towards accessing the new physical insight and potential technological opportunity at the interface between the rich physics of magnonic and magnon spintronic systems, and the techniques of contemporary quantum measurement and information processing. If successful, it will define and open up an entirely new field of research: quantum magnon spintronics.

Strings in programming languages are sequences of characters that represent any kind of text. They are a fundamental aspect of information representation: user names, passwords, or indeed any kind of text are handled as strings. The manipulation of strings, however, can also lead to subtle programming errors, that can have consequences for program correctness as well as information security. For example, a malicious user may enter computer code as their username. If the system is not sufficiently secure, this code can allow the user to hack the system. The Open Web Application Security Project lists this kind of attack among the top 10 application security risks. Despite this kind of attack being well-known, it has proved surprisingly difficult to avoid due to the complex nature of computer programs. Ideally, programming mistakes will be caught during testing. However, manual testing is a tedious and time-consuming process which requires the developer to imagine every possible user input. Automatic test-case generation can take this burden away from the developer and allow more complete testing to be done more efficiently. However, this relies on the tools used for test-case generation to be able to accurately reason about how software will run. This project will focus on how software deals with strings. Typically "regular expressions" are used for this purpose. However, current research takes an idealised view of regular expressions that omits many important features of the regular expressions provided by modern programming languages. We will address this shortcoming both in the theory of computer science and in practice. In particular, we will create a test-case generation tool-chain that will provide better test-case generation for software dealing with strings. These tools will be tested on real-world industrial code provided by Prodo.ai.

We are interested in geometrical shapes given as sets of solutions of many polynomial equations in many variables. More than the abstract study of shapes, it is writing down and solving explicit equations that lends our work its special flavour and power in applications. A first aim of our program is to vastly improve the understanding of the structures that control how we write these equations. All our shapes can be classified into three different kinds: Fano (positively curved), Calabi-Yau (flat) and General Type (negatively curved). Fano and Calabi-Yau shapes play a special role in geometry as "atomic pieces" when breaking complex shapes down to simpler ones, and in physics as backgrounds for string theory. The second key aim of this research is to classify and write down the equations of these Fano and Calabi-Yau atomic pieces in 3, 4, and 5 dimensions, thereby producing a vast "periodic table" of all possible shapes. String theory is a leading candidate for a "theory of everything". It postulates that the fundamental objects in physics are not point-like particles but strings. These strings move in a background that, in addition to space and time, has extra hidden dimensions curled up in (depending on the version of the theory) either a 3-dimensional Fano or Calabi-Yau shape (3 complex dimensions = 6 real dimensions). Thus, our classification of shapes is also an encyclopaedia of all the possible background geometries of string theory. To tackle our classification, we will show how to associate a Fano shape to every reflexive polytope. A reflexive polytope is a geometric object closely related to the "Platonic solids" that we learn about in school. To give an idea of the size and complexity of the problem, the complete list of 4-dimensional reflexive polytopes is known and there are nearly half a billion of them. There is an operation on polytopes, called a mutation, and it is possible to mutate a polytope into another polytope. If two polytopes are related by mutation then they give rise to the same Fano shape, so we need to sort all polytopes up to mutation. In order to do this, we need to perform parallel computations on a massive scale distributed over a High Performance Computing cluster. In string theory, it can happen that two mathematically very different background geometries produce the same physics. When this happens, the two geometries are said to be "mirror symmetric" to each other. Mirror symmetry is a most fascinating aspect of string theory, and our third key aim is to give the first truly transparent explanation of this phenomenon. Our project requires us to integrate the scientific expertise of our three institutions, and enlist the specialist collaboration of many mathematicians nationally and internationally. In turn, the results and methods of our work will have significant applications in many areas of mathematics, science, and scientific computation. In a very exciting spin-off project, we will study some avatars of Fano shapes in the world of congruences between coefficients of Fourier expansions of L-functions. This is a profound area of number theory, started by Ramanujan, that lies at the heart of Wiles' proof of Fermat's last theorem. As scientists, we are inspired when we see the same structures arise independently and for separate reasons in different parts of mathematics and science - this reveals deep connections between seemingly unrelated scientific disciplines. An example was the discovery of group theory (the theory of symmetries) in mathematics and in quantum physics at the turn of the 20th century. There are several examples in our own work: Fano shapes correspond to lattice polytopes and to congruences of coefficients of L-functions; Calabi-Yau shapes arise independently in geometry and in string theory; cluster algebras are found at the same time in algebra and in the geometry of mirror symmetry.

Endomyxa are a very poorly known but large and diverse group of organisms in the protozoan phylum Cercozoa. Endomyxa includes the commercially important plasmodiophorid plant parasites, and haplosporidia and relatives, which include parasites of a wide range of invertebrates, most famously MSX disease of oysters. The evolutionary relationships of both of these groups were for a long time unknown (plasmodiophorids were considered fungi for a long time), but good ribosomal DNA trees show that they are related to each other (a relationship reinforced by morphological synapomorphies) and several lines of molecular evidence show that they belong within the eukaryotic supergroup Rhizaria, specifically as a subphylum within Cercozoa, but separately from the so-called 'core' Cercozoa. More recently, free-living relatives of these parasites have been found through culturing/cell isolation and rDNA sequencing. These include the large testate marine filose amoeba Gromia, and large reticulose, naked amoebae: the bacterivorous Filoreta and the predatory Arachnula and Platyreta. Such organisms represent very distinct and poorly studied protozoan morphotypes whose ecological roles are almost completely unknown. Other research has revealed more endomyxan parasites, for example the spot prawn parasite, and Paradinium, a parasite of copepods. Culture-independent environmental rDNA libraries show that there are a large diversity of other endomyxan clades and lineages which remain uncharacterised - the only information we have for them is the provenance of the samples in which the sequences were detected. The environmental libraries also show that there is very strong ecological structuring - many endomyxan clades have so far only been found in quite specific habitats, for example deep-sea samples, anaerobic marine, anaerobic freshwater, or phylloplane communities. These patterns suggest high levels of ecological specialization, perhaps involving interactions with other organisms. This project aims to identify as many of these novel lineages and clades as possible, whether they are free-living, symbiotic, or parasitic, using a combination of intensive selective culturing using a diversity of food sources and culture conditions informed by the results of the environmental libraries, and fluorescent in situ hybridization (FISH) using fluorescent probes that will specifically detect chosen uncharacterized lineages. FISH will also show the relative abundance of endomyxan lineages in different habitats, and whether they are associated with, or are found inside, other eukaryote cells, and if so, which organisms they co-occur with (and are therefore perhaps parasites of). Multiple and diverse SSU rDNA environmental libraries will be constructed, using a set of overlapping PCR primers covering all Endomyxa, to show much more clearly and realistically than previously possible the true diversity of Endomyxa, and their ecological nature by analyzing the sequence data in the context of other biotic and abiotic variables co-measured at each sampling site. The libraries will be made using both rDNA and crDNA reverse transcribed from rRNA: the latter is generally taken as a surrogate for the level of activity of the cells (more ribosomes will be present in more active cells), rather than just presence/absence and biomass as indicated by rDNA. This approach will provide more informative about the ecological characteristics than rDNA libraries alone. The environmental sequences will be used to improve phylogenetic reconstructions of Endomyxa and their relationships to the rest of Rhizaria both by increasing taxon sampling of this part of the tree, and by providing sequence data that can be used to obtain LSU rDNA sequences for representative genotypes directly from environmental DNA samples. Where cultures are available, other genes can be targeted by PCR, to be combined with existing data in multigene phylogenies.