• Canada
• 2017-2021close
• 2018 close
• 2019 close

The subject of study of differential geometry are smooth manifolds, which correspond to smooth curved objects of finite dimension. In modern differential geometry, it is becoming more and more common to consider sequences (or flows) of smooth manifolds. Typically the limits of such sequences (or flows) are non smooth anymore. It is then useful to isolate a natural class of non smooth objects which generalize the classical notion of smooth manifold, and which is closed under the process of taking limits. If the sequence of manifolds satisfy a lower bound on the sectional curvatures, a natural class of non-smooth objects which is closed under (Gromov-Hausdorff) convergence is given by special metric spaces known as Alexandrov spaces; if instead the sequence of manifolds satisfy a lower bound on the Ricci curvatures, a natural class of non-smooth objects, closed under (measured Gromov-Hausdorff) convergence, is given by special metric measure spaces (i.e. metric spaces endowed with a reference volume measure) known as RCD(K,N) spaces. These are a 'Riemannian' refinement of the so called CD(K,N) spaces of Lott-Sturm-Villani, which are metric measure spaces with Ricci curvature bounded below by K and dimension bounded above by N in a synthetic sense via optimal transport. In the proposed project we aim to understand in more detail the structure, the analytic and the geometric properties of RCD(K,N) spaces. The new results will have an impact also on the classical world of smooth manifolds satisfying curvature bounds.

"Software is the most prevalent of all the instruments used in modern science" [Goble 2014]. Scientific software is not just widely used [SSI 2014] but also widely developed. Yet much of it is developed by researchers who have little understanding of even the basics of modern software development with the knock-on effects to their productivity, and the reliability, readability and reproducibility of their software [Nature Biotechnology]. Many are long-tail researchers working in small groups - even Big Science operations like the SKA are operationally undertaken by individuals collectively. Technological development in software is more like a cliff-face than a ladder - there are many routes to the top, to a solution. Further, the cliff face is dynamic - constantly and quickly changing as new technologies emerge and decline. Determining which technologies to deploy and how best to deploy them is in itself a specialist domain, with many features of traditional research. Researchers need empowerment and training to give them confidence with the available equipment and the challenges they face. This role, akin to that of an Alpine guide, involves support, guidance, and load carrying. When optimally performed it results in a researcher who knows what challenges they can attack alone, and where they need appropriate support. Guides can help decide whether to exploit well-trodden paths or explore new possibilities as they navigate through this dynamic environment. These guides are highly trained, technology-centric, research-aware individuals who have a curiosity driven nature dedicated to supporting researchers by forging a research software support career. Such Research Software Engineers (RSEs) guide researchers through the technological landscape and form a human interface between scientist and computer. A well-functioning RSE group will not just add to an organisation's effectiveness, it will have a multiplicative effect since it will make every individual researcher more effective. It has the potential to improve the quality of research done across all University departments and faculties. My work plan provides a bottom-up approach to providing RSE services that is distinctive from yet complements the top-down approach provided by the EPRSC-funded Software Sustainability Institute. The outcomes of this fellowship will be: Local and National RSE Capability: A RSE Group at Sheffield as a credible roadmap for others pump-priming a UK national research software capability; and a national Continuing Professional Development programme for RSEs. Scalable software support methods: A scalable approach based on "nudging", to providing research software support for scientific software efficiency, sustainability and reproducibility, with quality-guidelines for research software and for researchers on how best to incorporate research software engineering support within their grant proposals. HPC for long-tail researchers: 'HPC-software ramps' and a pathway for standardised integration of HPC resources into Desktop Applications fit for modern scientific computing; a network of HPC-centric RSEs based around shared resources; and a portfolio of new research software courses developed with partners. Communication and public understanding: A communication campaign to raise the profile of research software exploiting high profile social media and online resources, establishing an informal forum for research software debate. References [Goble 2014] Goble, C. "Better Software, Better Research". IEEE Internet Computing 18(5): 4-8 (2014) [SSI 2014] Hettrick, S. "It's impossible to conduct research without software, say 7 out of 10 UK researchers" http://www.software.ac.uk/blog/2014-12-04-its-impossible-conduct-research-without-software-say-7-out-10-uk-researchers (2014) [Nature 2015] Editorial "Rule rewrite aims to clean up scientific software", Nature Biotechnology 520(7547) April 2015

This project concerns computational mathematics and logic. The aim is to improve the ability of computers to perform ``Quantifier Elimination'' (QE). We say a logical statement is ``quantified'' if it is preceded by a qualification such as "for all" or "there exists". Here is an example of a quantified statement: "there exists x such that ax^2 + bx + c = 0 has two solutions for x". While the statement is mathematically precise the implications are unclear - what restrictions does this statement of existence force upon us? QE corresponds to replacing a quantified statement by an unquantified one which is equivalent. In this case we may replace the statement by: "b^2 - 4ac > 0", which is the condition for x to have two solutions. You may have recognised this equivalence from GCSE mathematics, when studying the quadratic equation. The important point here is that the latter statement can actually be derived automatically by a computer from the former, using a QE procedure. QE is not subject to the numerical rounding errors of most computations. Solutions are not in the form of a numerical answer but an algebraic description which offers insight into the structure of the problem at hand. In the example above, QE shows us not what the solutions to a particular quadratic equation are, but how in general the number of solutions depends on the coefficients a, b, and c. QE has numerous applications throughout engineering and the sciences. An example from biology is the determination of medically important values of parameters in a biological network; while another from economics is identifying which hypotheses in economic theories are compatible, and for what values of the variables. In both cases, QE can theoretically help, but in practice the size of the statements means state-of-the-art procedures run out of computer time/memory. The extensive development of QE procedures means they have many options and choices about how they are run. These decisions can greatly affect how long QE takes, rendering an intractable problem easy and vice versa. Making the right choice is a critical, but understudied problem and is the focus of this project. At the moment QE procedures make such choices either under direct supervision of a human or based on crude human-made heuristics (rules of thumb based on intuition / experience but with limited scientific basis). The purpose of this project is to replace these by machine learning techniques. Machine Learning (ML) is an overarching term for tools that allow computers to make decisions that are not explicitly programmed, usually involving the statistical analysis of large quantities of data. ML is quite at odds with the field of Symbolic Computation which studies QE, as the latter prizes exact correctness and so shuns the use of probabilistic tools making its application here very novel. We are able to combine these different worlds because the choices which we will use ML to make will all produce a correct and exact answer (but with different computational costs). The project follows pilot studies undertaken by the PI which experimented with one ML technique and found it improved upon existing heuristics for two particular decisions in a QE algorithm. We will build on this by working with the spectrum of leading ML tools to identify the optimal techniques for application in Symbolic Computation. We will demonstrate their use for both low level algorithm decisions and choices between different theories and implementations. Although focused on QE, we will also demonstrate ML as being a new route to optimisation in Computer Algebra more broadly and work encompasses Project Partners and events to maximise this. Finally, the project will deliver an improved QE procedure that makes use of ML automatically, without user input. This will be produced in the commercial Computer Algebra software Maple in collaboration with industrial Project Partner Maplesoft.