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We construct an equivariant version of discrete Morse theory for simplicial complexes endowed with group actions. The key ingredient is a 2-categorical criterion for making acyclic partial matchings on the quotient space compatible with an overlaid complex of groups. We use the discrete flow category of any such compatible matching to build the corresponding Morse complex of groups. Our main result establishes that the development of the Morse complex of groups recovers the original simplicial complex up to equivariant homotopy equivalence., Comment: 31 pages, 6 figures

International audience; Precision medicine allows the extraction of information from complex datasets to facilitate clinical decision-making at the individual level. Topological Data Analysis (TDA) offers promising tools that complement current analytical methods in precision medicine studies. We introduce the fundamental concepts of the TDA corpus (the simplicial complex, the Mapper graph, the persistence diagram and persistence landscape). We show how these can be used to enhance the prediction of clinical outcomes and to identify novel subpopulations of interest, particularly applied to understand remission of depression in data from the GENDEP clinical trial.

Understanding the biological function of knots in proteins and their folding process is an open and challenging question in biology. Recent studies classify the topology and geometry of knotted proteins by analysing the distribution of a protein's planar projections using topological objects called knotoids. We approach the analysis of proteins with the same topology by introducing a topologically inspired statistical metric between their knotoid distributions. We detect geometric differences between trefoil proteins by characterising their entanglement and we recover a clustering by sequence similarity. By looking directly at the geometry and topology of their native states, we are able to probe different folding pathways for proteins forming open-ended trefoil knots. Interestingly, our pipeline reveals that the folding pathway of shallow knotted Carbonic Anhydrases involves the creation of a double-looped structure, differently from what was previously observed for deeply knotted trefoil proteins. We validate this with Molecular Dynamics simulations., 5 figures, Supplementary information contained as an appendix

We explore pseudometrics for directed graphs in order to better understand their topological properties. The directed flag complex associated to a directed graph provides a useful bridge between network science and topology. Indeed, it has often been observed that phenomena exhibited by real-world networks reflect the topology of their flag complexes, as measured, for example, by Betti numbers or simplex counts. As it is often computationally expensive (or even unfeasible) to determine such topological features exactly, it would be extremely valuable to have pseudometrics on the set of directed graphs that can both detect the topological differences and be computed efficiently. To facilitate work in this direction, we introduce methods to measure how well a graph pseudometric captures the topology of a directed graph. We then use these methods to evaluate some well-established pseudometrics, using test data drawn from several families of random graphs., Comment: 27 pages, 7 figures

To any finite simplicial complex X, we associate a natural filtration starting from Chari and Joswig's discrete Morse complex and abutting to the matching complex of X. This construction leads to the definition of several homology theories, which we compute in a number of examples. We also completely determine the graded object associated to this filtration in terms of the homology of simpler complexes. This last result provides some connections to the number of vertex-disjoint cycles of a graph., 15 pages, 8 figures, comments welcome. V2 accepted by the Australasian Journal of Combinatorics

This report presents the findings of a global survey on AI in Financial Services jointly conducted by the Cambridge Centre for Alternative Finance (CCAF) at the University of Cambridge Judge Business School and the World Economic Forum in Q2-Q3 2019. Representing one of the largest global empirical studies on AI in Financial Services, a total of 151 respondents from 33 countries participated in the survey, including both FinTechs (54% of the sample) and incumbent financial institutions (46% of the sample). The study was supported by EY and Invesco. The study’s objective was to analyse and understand the current state of AI adoption in Financial Services, as well as its subsequent implications. This was done through the comparative analysis of empirical data collected via a web-based questionnaire. This research provides a comprehensive picture of how AI is currently being applied in Financial Services by both FinTechs and Incumbents; driving different business models; underpinning new products and services; and playing a strategic role in digital transformation. The findings also reveal how financial service providers across the globe are meeting the challenges of AI adoption with its emerging risks and regulatory implications, as well as the impact of AI on the competitive landscape and employment levels. The overarching findings of the study suggest that AI is expected to transform a number of different paradigms within the Financial Services industry. These anticipated changes include how data is utilised to generate more actionable insights; business model innovation (e.g., selling AI as a service); changes to the competitive environment with the entrance of ‘Big Tech’ and consolidation; various impacts on jobs and regulation; impacts on risks and biases; and the further development and adoption of game-changing technologies.

Technology-enabled innovation in financial services (FinTech) can extend the benefits of financial inclusion to millions of unbanked and underbanked people around the world. However, the rise of FinTech presents many regulatory challenges - for emerging and developing economies in particular. Consequently a number of regulators in advanced, emerging, and developing economies have responded to such challenges by innovating on their own. These innovative regulatory initiatives include innovation offices, regulatory sandboxes, and RegTech for regulators. Against this backdrop, the Early Lessons on Regulatory Innovations to Enable Inclusive FinTech Report aims to provide readers with key insights on how regulators are innovating to better respond to financial innovation. This empirical study provides independent and comprehensive insights on the inter-relationships between financial and regulatory innovation and financial inclusion, together with lessons learned from early examples of regulatory innovation. The report also provides practical considerations for those regulators seeking to develop their own initiatives. This report has been produced by the Cambridge Centre for Alternative Finance at Cambridge Judge Business School in partnership with the FinTech Working Group of the United Nations Secretary-General's Special Advocate for Inclusive Finance for Development (UNSGSA). It would not have been possible without the support of the Monetary Authority of Singapore (MAS) and the supervision and coordination of the Office of the UNSGSA. Global interest in innovative regulatory initiatives is strong, with regulatory sandboxes now live or planned in over 50 jurisdictions. There are also innovation offices in operation in over 30 jurisdictions, while 20 jurisdictions are pioneering RegTech models. Some jurisdictions have established innovation offices as a first step in the regulatory innovation journey. They are a particularly compelling initial option for capacity-constrained regulators in emerging and developing economies. They require no protracted changes to legislation or regulation, or related resource implications. Key lessons learned from innovation offices underscore the importance of early, and close, engagement with innovators. Executive buy-in and inter-agency coordination can ensure their effective functioning, and they can be a powerful support to regulators with a financial inclusion remit. Innovation offices are also only as good as the quality of their resources, such as the technical capacity of their staff. Regulatory sandboxes have been widely adopted as an innovative regulatory initiative, including several with a specific financial inclusion focus. Sandboxes can help regulators gain a better understanding of FinTech and develop evidence-based regulations that promote inclusive FinTech. Key lessons learned from early regulatory sandboxes highlight that they are neither necessary nor sufficient for promoting financial inclusion. Sandboxes do offer benefits but are complex to set up and costly to run. Experience shows that most regulatory questions raised in connection with sandbox tests can be effectively resolved without a live testing environment. Similar results may be more affordably achieved through innovation offices and other tools. RegTech (regulatory technology) is a distinct innovative regulatory initiative, which focuses on how to monitor and enforce compliance against relevant regulations. RegTech can support a more responsible delivery of innovative financial services, which may directly impact financial inclusion. It also allows regulators to swiftly respond to market developments, better protect consumers, and enhance institutional supervision. There are currently limited examples of RegTech solutions being deployed in emerging and developing economies. Lessons learned from initiatives that do exist highlight the merits of beginning with a problem that will likely gain broad support and has a high likelihood of successful resolution. Crowdsourcing potential solutions to regulatory problems, paired with strong executive sponsorship, has also proven rewarding. Finally, multi-disciplinary teams with complementary skill sets can help foster long-term regulatory transformation. It is clear that no single initiative is a "silver bullet" for effective regulation. Many are resource intensive and require careful cost-benefit analysis. However there are a number of considerations for regulators who do seek to implement their own innovative regulatory initiatives. An internal mindset and culture which is supportive of innovation within the regulator is crucial, while direct engagement between regulators and innovators has also proven to be mutually beneficial, regardless of the form which this takes. Regulators have also benefited from close inter-agency collaboration and opportunities for mutual learning. Regulators in emerging and developing economies need to remain agile and open as they innovate and create regulatory initiatives. When effectively carried out, innovative regulatory initiatives enable inclusive FinTech and support regulators in long-term capacity building.

The first global benchmark study of the RegTech sector by the Cambridge Centre for Alternative Finance, conducted with the support of EY Japan, is based on a survey of 111 firms as well as regulators and industry experts. We estimate that the global RegTech industry generated $5 billion in revenue in 2018, and had raised $9.7 billion in external funding as of early 2019. The financial services sector dominates demand for RegTech services but most vendors now target non-financial sectors, and this share is set to grow. The RegTech market can be broken down into five distinct segments: 1. Profiling and due diligence 2. Dynamic compliance 3. Reporting and dashboards 4. Risk analytics 5. Market monitoring. The sector underwent a period of rapid growth between 2014 and 2018, but is now growing mostly through incumbent expansion rather than market entry. Long sales cycles, complex IT planning within client institutions, difficulties in establishing trust and high levels of competition have left some vendors struggling to gain traction. A handful of larger vendors thus dominate most funding and commercial activity.

The 3rd Annual European Alternative Finance Report presents the most comprehensive analysis of the status of alternative finance industry in Europe, covering more countries, alternative finance models, as well as industry trends and developments than was available in its predecessors. Much effort has been placed on data quality verifications and clarifications in an ongoing productive and collaborative dialogue with all platform informants and research partners. Here, we wish to acknowledge their invaluable contributions, without which this report could not have been written. Overall, the data collected shows that 2016 saw European alternative finance doubling its volumes from 2015, and continuing an impressive growth. When placed into its short historical perspective, an impression emerges of an industry progressing from an introduction stage catering to innovators, into the growth stage catering to a growing number of early adopters. This growth phase is characterized by entry of new platforms and service providers, overall increasing competition, emergence of first cases of consolidation, diversification of products, and investments in process effectivization and streamlining. Despite exhibiting continuous healthy growth, Europe still punches well below its potential in most countries. Challenges remain both in terms of market education, and amendments of regulation. In Europe regulatory challenges are faced with two barriers. First, most countries have relatively mature and well-established regulatory systems. And, second, these vary significantly across countries. In this respect, our study provides initial evidence that more favorable regulation at national level is associated with higher volumes per capita, as well as with higher share of funding reaching businesses. Accordingly, one can expect that if regulatory conditions will continue to improve at both national and international levels, the industry will be able to maintain and enhance its growth. Nevertheless, one should acknowledge that the story of European alternative finance sector growth, more importantly, is the story of the democratization of access to finance and investment opportunities for people and ventures in Europe. When we present that the industry has generated EUR 2 Billion in 2016, it implies that close to 95% of this sum was actually invested in European venturing reaching businesses, entrepreneurs, artists, and social activists. All aim at creating value to consumers, investors and society via ventures made possible thanks to the support of the crowd. Accordingly, and by extension, more accommodating regulation for the industry, also implies greater access to finance for multiple stakeholders in the economy and more opportunities for value creation. Here, our findings leave much room for optimism for the coming years. Not only are volumes growing strongly, but platforms are investing actively in innovative solutions towards improving performance and customer service. Moreover, evidence suggests that a greater share of platforms engage in cross-border transactions, although, in most cases, these still represent a relatively modest share of overall volumes (thanks to limitations mentioned above). Finally, we hope this report, and the insights emerging form it, will be of help to all stakeholders interested in the development of alternative finance in Europe. And we look forward to a continued cooperation with all our partners in monitoring industry developments in the future.

This report presents the research findings from the 3rd Annual European Alternative Finance Industry Survey, which systematically records the development of this industry during 2016. This research was carried out by the Cambridge Center for Alternative Finance at the University of Cambridge Judge Business School, with the support of 29 major European industry associations and research partners, in partnership with BBVA.

The 4th UK Alternative Finance Report from the Cambridge Centre for Alternative Finance (CCAF), the University of Cambridge Judge Business School, captures the size and growth of UK crowdfunding and peer-to-peer lending markets in 2016. This research is based on surveys of more than 8,300 investors and lenders of online alternative finance as well as 77 crowdfunding and peer-to-peer lending platforms in the United Kingdom. Based on these findings, the CCAF has provided input to the Financial Conduct Authority (FCA), which is currently reviewing its regulatory regime for crowdfunding in the UK.

We obtain a simple and complete characterisation of which matchings on the Tait graph of a knot diagram induce a discrete Morse function (dMf) on $S^2$, extending a construction due to Cohen. We show these dMfs are in bijection with certain rooted spanning forests in the Tait graph. We use this to count the number of such dMfs with a closed formula involving the graph Laplacian. We then simultaneously generalise Kauffman's Clock Theorem and Kenyon-Propp-Wilson's correspondence in two different directions; we first prove that the image of the correspondence induces a bijection on perfect dMfs, then we show that all perfect matchings, subject to an admissibility condition, are related by a finite sequence of click and clock moves. Finally, we study and compare the matching and discrete Morse complexes associated to the Tait graph, in terms of partial Kauffman states, and provide some computations., Comment: 31 pages, 44 figures, comments are welcome! This version accepted by the Electronic Journal of Combinatorics