The impact of applied and industrial mathematics on innovation.

Maths can help a lot in industry for two main reasons: first, as lingua franca between engineers, physicians, mathematicians and executives (even with a mathematician as "translator") that is one of the fundamental premisses to have a good model for the scientific representation of real world processes.
Second, in a very competitive world almost all the SME's need to optimize their processes in order to become more competitive. The role of hand-made models (and corresponding software) is crucial, as the comercial products/software in the market usually don't take into account the particularities of their own companies. This fact, together with a precise definition of validity domains for each applied domain is crucial to a good and sustainable development.

Great point about mathematics being the common language and vehicle to carry ideas across international and discipline borders. As for SME's it is important to say that using mathematics even a single person can make advances showing that smaller groups can contribute to the digital revolution - it does not have to be the big data companies (although once you have an idea they will surely make an offer and so it may appear that only the bigger companies are making the innovative advances). Steven Bishop, UCL

In the UK, we celebrate the work of Alan Turing whose mathematics was critical in advancing computing science. The UK Government have announced that a major Turing Centre will be set up with something like £42m set aside for this. In weather forecasting (surely one of the big users of HPC) mathematics has lead to jumps in improvement of our predictive power (e.g. for hurricane paths) that would not have been achieved by increased computer power alone. Steven Bishop, UCL
PS a new film is due to be released soon with Benedict Cumberbatch which will only increase the interest in this area

Mathematics can play an important role in digital science distinct from its contribution to the big data revolution. The new applications of graph theory to networks is advancing our understanding of how patterns can be detected, understood and managed (ok controlled if you like) and this may yet provoke mathematicians to devise new mathematical techniques.

Meanwhile, some of the same methods can help sort and organise big data which may require HPC in order to manage distributed sources and analysis.

Mathematics, and in particular Operations Research and Mathematical Optimization, may be of great help in all these topics. For instance:

- Big data: structure detection in big social networks and information retrieval from "big data" databases can be formulated as a large mathematical optimization problem. Mathematics are also needed in big-data confidentiality issues: how to guarantee that individual confidential information is not disclosed?

- HPC: mathematical optimization problems are among the most CPU time consuming today algorithms, and they may require HPC infrastructures for its efficient solution. This includes exploitation of parallelism capabilities. At the same time, optimization problems are a source of difficult instances to test new HPC paradigms.

- Applied and industrial mathematics: all industrial problems, in any field, involve the solution of an optimization problem, either to minimize costs or to maximize benefits.

I would like to comment on the role of e-Infrastructures for Mathematics. European e-Infrastructures can support the computational needs of mathematicians by providing the capacity and capability in terms of storage and computing that may not be available locally to the scientist, as part of the ICT services offered by the home research institute.

The European Grid Infrastructure (EGI, http://www.egi.eu/) provided to user communities active in the Mathematics domain 30 Million CPU hours of computation since 2004, where the largest amount of computing resources was offered by the Dutch national infrastructure, followed by CERN, France and Greece (http://accounting.egi.eu/vodis.php?query=sumelap&startYear=2004&startMon...).

Being EGI a federation of national infrastructures, it can help mathematicians by offering the ICT services that may not be available locally/nationally. So far, EGI has offered services for Authentication, Authorization and Identity Management, computing and data management.

To date, we (EGI) lack a Virtual Research Community active in the Mathematics disciplinary area, so it is difficult to approach Mathematicians at a European level, most of these are being supported by the national e-Infrastructures as part of the long-tail of science.The EC could help EGI in approaching European networks of Mathematicians with dedicated workshops. Such events would help European e-infrastructures to better network with this European research community and to understand the needs both in terms of ICT capacity and capability.

The impact of applied and industrial mathematics on innovation.

The development of new products or new production processes is at the heart of European industry. With growing complexity and shorter innovation cycles, mathematical modeling, simulation and optimization (MSO) becomes an indespensible prerequisite for the development of European Industry and technologies. The virtual product that accompanies the real product and allows product verification, risk analysis and product optimization is the future. For this we have to strengthen interdisciplinary cooperation and to include mathematical MSO into all major technology funding streams. European research groups have been very active to support this development but there is a strong imbalance within the Euurpean research area that needs to be balanced. The role of mathematical methods in big data and HPC is typically underestimated. Too much emphasis is paid on new hardware developments, here new developments require the incoporation of mathematical MSO.

The progress in performance in the methods is larger than in the hardware and this has to be continued.

Emphasis should be given to new mathematical methods and also to the creation of e-infrastructures that allow the whole European science community to make use of new methods.

To maximize the impact of MSO on innovation can be inparticular successful in the development of infrastructure networks, such as gas, electricity, communication, etc where the developments of the last years have been tremendous and where a major European research initiative may allow in the near future to take a holistic point of view and to simulate, analyze, and also control and optimize a whole national or supranational network in its planning and in its operation. For this direction a FET initiative, bringing together the experts from the applications and from mathematical MSO, would be the right framework and could have a major impact. On the national level in some countries such initiatives have started, involving industries and academic institutions, but more national researchinitiatives should get involved from different countries.

Economic impact of an e-infrastructure in industrial mathematics

Mathematical tools are fundamental for the simulation processes that are necessary in the design of new technologies and procedures. Mathematical modelling carries with it robustness, stability and safety, in the sense that mathematicians can deliver solutions with error estimates and the knowledge that the proposed method does what it is supposed to do, which is not necessarily the case of any simulationMathematicians, allied with other scientists, engineers and computer scientists can make a big difference for companies involved in innovation and new technologies. But solving complex problems often needs to involve groups with different expertise, groups that do not necessarily come from the same place or country. In that sense, disposing of databases of expertise, past experiences and success stories in solving real life problems, etc, is of fundamental importance. Also having databases of successfully used algorithms and numerical or simulation methods to tackle particular problems. A European e-infrastructure devoted to the interaction of mathematics with new technologies and innovation would provide a prefect solution to share this kind of information and being able to build the ad-hoc experts' groups at best. Companies would be able to find the right people to help them with their needs.Another possible important aspect of an e-infrastructure in industrial mathematics would be to offer services to companies like a pan-European job market of experts or young people. Engineers and scientists working in companies could also be offered continuous education courses in the latest mathematical technologies in numerical computation, control, optimization, etc. Science advances quickly and it is difficult to keep in touch with the new tools that offer much better solutions. Such an e-infrastructure could offer such information and possibilities to companies which want to upgrade the scientific level of their engineers and scientists.Europe and the European economical competitiveness would certainly take a big advantage of the existence of such an e-infrastructure, providing the best existing services and solutions for companies aiming at building the most innovative solutions for all aspects of our European society.

From the contributions already received, I am convinced we will have a very interesting workshop, alive with active discussions about mathematics and what it has to offer to tackle modern digital challenges. Many thanks for everything that has been posted already!

I have had today discussions with some mathematicians, both academians and those working with industry, all showing enthousiasm in getting together to discuss and identify mathematics as one of the powers behind the digital revolution. This will be a very interesting workshop, I am convinced!

The potential for mathematicians to contribute, at various levels, to the development of new digital tools and methods, is clear. That development potential and need is not only growing at high speed but at an exponentially growing speed. The more complex our environment becomes - the more data, the more powerful computers, the more linked everything is with everything else - the more demand there is for optimisation algorithms, pattern recognition, modelling, in general finding new, innovative solutions to tackle the new problems encountered due to new magnitudes. It will be good to identify what this means or can mean in practical terms - perhaps we can start that reflection at the workshop!

There has been discussion around a Virtual Research Environment for mathematics. This linked to a specific thematic field of mathematics, bringing together SMEs and researchers as well as research centres, mathematicians and technicians, to tackle and generalise mathematical challenges to serve the European research community, continues to be an excellent idea. Our Workprogramme for Research Infrastructures - one of the background documents - contains a topic inviting these interests to come together: Topic EINFRA-9-2015 – e-Infrastructures for virtual research environments (VRE).

As collaborators for industry, mathematicians have a lot of innovation potential - and can be solution oriented. Several areas are already mentioned in the comments: Mathematical modelling, simulation and optimization amongst others are areas which are key to both HPC development and Big Data, and on the other hand, can produce fantastic results from the vast amounts of data that we are struggling to find ways to tackle in Europe.

I thank you for all contributions so far and encourage new contributions!

Please key to your calendars the preliminary dates: at present we are hoping to organise the Workshop in Brussels 6 or 7 November. To be confirmed...

and ps: thanks for the tip on the Turing movie...! another Beautiful Mind -style I suspect!

The science of data needs Mathematical Statistics methodology

The science of data needs a lot of Mathematical Statistics research to develop new robust procedures to deal with this (good) monster of big data. The support should be this and can not came from smoking sellers, whcich is a real threat out there with this topic.

Connecting mathematics with industry and the society

No doubt that mathematics is a major source of innovations for industry and the whole society. However, very often new mathematical ideas do not immediately benefit to industry but take some time to percolate through other (more applied) scientific fields, mostly engineering, before reaching their "final" target. I believe it is an outmost important challenge to directly connect mathematicians to industry. It may be already the case in some instances for large companies which have their own research departments, well connected to mathematicians (among others). But it is clearly not the case for most small or medium companies. It is obvious that Europe could help a lot, in this respect, by organizing these connections on the model of national institutions like Franhaufer in Germany or AMIES in France, just to quote a few.

Napoleon said "The advancement and perfection of mathematics are intimately connected with the prosperity of the state".

It has never been more true than today. With the challenges we are facing today, paradigmal shifts in economy, data, health care, the role of mathematics will only grow. In fact, large scale pan-European intiatives to update and build modern mathemtatical infrastructure should have been initiated yesterday. We are already slightly behind. However, not all is lost, with the a few strategically chosen programs in applied mathematics, optimization, operations research, we can prepare Europe for the changes ahead.

I have seen the development of computer simulations and more generally the growing number of scientific riddles at the interface between computer and mathematics and my experience tells me that it is just the beginning!

Computers have been invented by two outstanding mathematicians and seventy years later HPC is still piloted to and for applied mathematicians. Although supercomputers are more important to theoretical chemists, astrophysicists, engineers etc than to mathematicians, the code-libraries that they use are the fruit of a collaboration between them, mathematicians and computer scientists.

Such collaborations are essential to science now a days and seems that the future of science belong to centers capable to unite actively mathematics, computer science and application fields.

Better Methods Allow More Appropriate Incorporation of Data

Here are a couple of comments:

Re: The role of mathematics in Big Data: (1) We need to distinguish "big data" from "good data", as those are not remotely the same. (2) The availability of richer and more finely-grained data necessitates the development of methods that allow one to throw away less of the data. Throwing away data can lead to misleading or even simply wrong answers (e.g. losing causal relationships for disease or idea transfer when going from a temporal network to some sort of "effective" static network based on averaged contacts between people). One recent develop, in which I have been deeply involved, is generalizing ordinary network theory to "multilayer networks", in which one can incorporate time-dependence, multiple types of edges, interdependent networks (as occurs in e.g. infrastructure), and so on.

Impact of industrial and applied mathematics on innovation: It is worth keeping in mind that many branches of mathematics previously 'confined' to theoretical studies are now seeing an increasing number of practical applications --- including harmonic analysis, algebraic geometry, algebraic topology, tensor analysis, and so on. There are powerful tools available that may not be traditionally "applied", and it's really important that we take advantage of them and create ways to use them with data (which is active in many of these areas, but not necessarily applied to industry very much).

FET programs: I think it would be nice to have a call on taking traditionally "pure" areas of mathematics and bringing them closer to applications through concerted collaboration of both mathematicians (and other scientists) and end users.

Mathematics is much more widely applicable than just to HPC and Big data and making a useful mathematical model of any process will almost always lead to greater understanding of the process. It also requires collaboration
between mathematicians and scientists and we need infrastructure to make this happen and to make sure the mathematical expertise available in Europe is used effectively.

Mathematics is much more widely applicable than just to HPC and Big data and making a useful mathematical model of any process will almost always lead to greater understanding of the process. It also requires collaboration
between mathematicians and scientists and we need infrastructure to make this happen and to make sure the mathematical expertise available in Europe is used effectively.

I certainly agree that mathematics can be used for much more than HPC and Big data. As a mathematician and programmer I worked in several (including international) projects on diverse subjects: map creation using satellite data, simulation of phone booths, simulation of a mobile phone company, heart section analysis, glass cut optimization, lossy image compression, surface triangulation for glass/metal buildings, route optimization, numerical solution of differential equations, database analysis and visualization, databases for WWW pages for internal and external use (including energetics, wireless service optimization, traffic lights, and more). The first problem is always to persuade the company that they do need a mathematician to model the problem and find ways to optimize the handling of that. That is why I strongly back up the ideas above and suggest more. Namely:

1. Create a forum, where companies and mathematicians can find each other.

2. Create a forum, where one can publish and find mathematical ideas, algorithms for industrial problems.

3. Create a forum, where companies can publish their problems requiring mathematics to solve.

These forums already exist: the European Consortium for Mathematics in Industry (http://www.ecmi-indmath.org/). The problem is that the membership is not free for any of the participants and since none of those participants know exactly what are the benefits of the membership, it cannot be widely accepted.

About the Big data. One of the main problems is that everybody generates data, but not all is useful for anything. So before doing anything with the "Big data", one needs to clear that data from the garbage, remove duplicates, etc. It is well possible, that after clearing the data is not at all so big. Also (from my own experience) I know that people make mistakes in entering data in databases, so they need to be cleared not only syntactically, but also semantically. Another problem is that for different reasons (business interests, personal rights) some of the data will never will be available for analysis, and without those the aquired results will never be complete or usable.

About HPC. I did some work in this area too, and I know that there is no standard for HPC. I myself used at least least 3 different kinds of parallel computing, and I know that they need quite different way of thinking when it comes to programming. So our goal in this direction is not clear at all, we would need much more (practical) information to do anything useful in this direction.

Some of the problems summarised in quotes:

"Weinberg's Second Law: If builders built buildings the way programmers wrote programs, then the first woodpecker that came along would destroy civilization."

"Andrew S. Tanenbaum: The nice thing about standards is that you have so many to choose from."

"William E. Vaughan: To err is human, to really foul things up requires a computer."

FYI: Weinberg's Second Law is applicale to this WWW page too. None of the select form elements work correctly in Firefox 29.0.1, I could not set some of the subscription preferences, there are dangling internal links on this page, and the server responds very slowly to everything.

We're improving the website and communities continuously, so thanks for your comment: we'll look into this and fix the html issue asap. We're also working with our providers to improve the performance, gradually - we try to keep costs down as well.

For anyone interested in the mathematics needed for modeling, simulation, and optimization of complex systems, I recommend to look at the the Exa-math workshop that was held a year ago in Washington. See https://collab.mcs.anl.gov/display/examath/Agenda
From there a number of interesting position papers can be accessed that identify many mathematical research topics that need to be addressed to better use the upcoming generations of supercomputers.

Topics include (among others)

- ultra-parallel algorithms that can exploit beyond a million parallel threads

- the need for a mathematical analysis of algorithms that can be used to understand computational cost quantitatively. This is in contrast to the usual asymptotic convergence results that are rigorously proven, but that are unable to distinguish between fast and slow methods since they involve unspecified constants.

- algorithms with inherent fault tolerance on systems that are not perfectly reliable

- communication and synchronization avoiding algorithms, realizing that the dominating cost of computing is caused by moving data (and not number crunching)

There are 3 "hot areas" of intensive research on Digital Science, strictly related to Maths.

1. Exascale
2. Quantum Technologies
3. Neurosynaptics and Neuroprocessors

In particular the area of algorithms and programming techniques were, with these new contexts would required new ideas, creativity and differents approaches . It 's high time to consider Mathematics from a wider perspective dedicating a single EC Directorate General for the Nature's Language. You should be able to better coordinate and focus the investments, research and policies.
Without forgetting to put in the same context the Complexy Research field.

I am writing this on behalf of ETP4HPC (www.etp4hpc.eu) - the European Technology Platform for High-Performance Computing. Our members will log their comments in this consultation process on an individual basis. Also, in relation to this item:

• The preparation of the FET Proactive (HPC) and/or the e-Infrastructure Work Programmes 2016-17 under the Excellent Science pillar of Horizon 2020. Do you have a concrete proposal for a topic linked to this discussion to be included in the next work programmes?

ETP4HPC has a Strategic Research Agenda in place (SRA - available at the website above) and we are working on another document including proposals for the next round of the FETHPC calls. Our next SRA will include a reference to Mathematics and Digital Sciences and we will issues a separate document to express our position on this topic (target date - end of Sept 2014). We are also interested in participating in any events/workshops in relation to this consultation process. Thank you.

The WP2016-2017 is now in its preparatory phases which has included several consultations. Also this consultation will feed into the WP drafting. The consultation has certainly been very succesful with lots of high level participation and comments of great relevance, and we see a clear need for a follow-up! This is why we aim to organise a workshop (which we hope we will very soon be able to announce) and in that workshop get more concrete input for the Workprogrammes and topics supporting mathematics' development.

Hello,
I think that e-infrastructures can be a valid instrument for mathematicians in resolving problems, to search counterexamples and expand the knowledge in several fields of Math, specially in Algebra, Non Commutative Geometry, Number theory. The challenges are from my point of view the following:
- expand and enrich the european e-infrastructure with the software and tools needed by the mathematicians to work easly.
- promote the pervasive use of distributed computing tecnologies (Cloud or other paradimgs) in the math community by improving the Universitary courses and with specific traning activities to help scientists in discover the benefits and the high impact of e-infrastrucutre in math
- Promote the creation of technologists able to work toghether with math scientists in order to help them to generate optimized software for their math problems and to approach the porting the code in Cloud or distributed computing infrastructure. These experitese are popular in physics and chemistry but not so popular in the fileld of math.
In 2012 we used the European Grid Infrastructure to accellerate the ongoing Glodbach Conjecture verification started 10 years ago. In few months we was able to complete the 20% of the total computation, reaching the current world record of computation, thanks to the large resource availability offered by distributed e-infrastructure. (see http://home.web.cern.ch/about/updates/2012/09/lhc-grid-tackles-270-year-...).
In addition in 2014 Still using e-infrastructure we was able to improve some interesting results in the field of non commutative Geometry.
Thank you for attention.

Maths is Underpinning for Analytics which creates value

The Mathematical Underpinnings of Analytics

In almost every sector of commercial and public endeavour there has been or there is about to be a data deluge. The innovation and exploitation, and also the hype, are driven by (a) the availability of data from emerging and converging digital platforms, (b) the increased amount of online and off-line traffic, data collection and surveillance, (c) the commercial imperatives to create greater value from existing customers and distilled knowledge; and (d) growing open data initiatives. Companies have become more aware of their own data resources, and see the future exploitation of these resources as a strategic path to growth.

When the data is very large, or continuously arrives very rapidly, doing anything smart is heavy lifting, and doing anything smart in real time is a challenge. Data has volume and velocity. Given enough processing time and capacity anybody might achieve anything. Yet companies and institutes have in many cases already invested their money in their infrastructure, architecture, enterprise, platforms, and access. Now they need to see some value return and value growth.

However, the data itself is merely the raw material. It is nothing without analytics: the concepts, methods and practices the can conjure valuable and actionable insights and radical knowledge from the large volumes of data. Such smart analytics goes well beyond “dice and slice”, alerts, reporting and querying, and it is far beyond the provision of infrastructure, computing architectures, and data handling resources. The latter are all necessary but are simply not sufficient for success. It is the analytics that will provide distinctiveness and unique capabilities, and allow us to see “further than others”. Such analytics needs to be founded on rigorous mathematical concepts, ideas, and methods, that may be deployed so as to underpin and innovate new concepts, methods and algorithms. In turn this could inspire new products and services and expand what companies and public institutes could achieve. It could even give rise to new business models. Com- panies can and will differentiate themselves on their exploitation of such analytics, and there clearly is an opportunity for them to obtain more insight and business growth from their data resources, going beyond current operations.

As different forms of data become ever more pervasive and more available, so the next generation of businesses and services within significant sectors (including commercial services, digital media, communications, domestic energy, security, environment, mar- keting, targeting, customer relationship management), and engagement across almost all public sectors will be developed by those companies, small and large, best placed to innovate. Our digital platforms will evolve rapidly year on year, converging and becoming pervasive with 24/7 operations, and data will become more open, as a currency or commodity.

Yet what is modern data science? What will data scientists need to achieve? This is a rather a crowded space. Other voices and opinion formers may argue from their own experience that it must subsume some or all of the following issues and activities: Foundations, Infrastructure, Management, Database Management, Security and Privacy, Search and Mining, and so on

True analytics is about none of these things, but any successful analytics strategies, operations and methods will surely rely on elements of them. In this book we shall focus on the mathematics that underpins world class analytics, and thus creates some differentiation and value.

It is arguable that "game changing" capability must come from creative applications of analytics to novel data sources, since access to digital platforms and computational power (in house or in the cloud) is now ubiquitous. In any case most companies have made (or will make) decisions on a long term basis about their operations (compute resources, data architectures, enterprise systems, and so on), and they, together with their suppliers, should be ever more focused on getting a return from those investments and plans. Hence their need for creative analytics and novel theoretical concepts that may in time become applicable, deliver actionable insights, and, thus, some agility and an ability to act.

Big data is presently on an trajectory resembling the Gartner hype curve. It is confusingly disorganised because it should really be split into at least four different challenges within data science. This way we prevent groups of people contributing very different things, with distinct aims, all claiming to be doing the essential element of vast data science.

1. There is vast data in the sense that we know a little bit about a very large number of things/objects/people. For example, in customer facing industries such as mobile tel- cos, retailers, ISPs, retail banks, smart metering in energy, digital marketing, and social media, etc. We call these customer- or consumer-facing applications, where the data is often proprietary.

2. There is vast data in the sense that it is constantly arising (has velocity,) and so real time analytics are required. For example, analysis of social media, or monitoring peer- to-peer conversations, emails, messaging, financial market data, or real time response to e-commerce, etc. Again this data is often proprietary.

3. There is vast data in the sense that we know a very large amount about a relatively small number complex objects, such as images for transmission and compression, sci- entific applications from crystallography, proteomics, fMRI scans, or spectroscopy, etc. We call these scientific or engineering applications, where the high throughput of novel scientific equipment spews out extremely high resolution data.

4. There is vast data in the sense that we know medium amounts about each of a number of distinct types of data objects belonging to individuals, which we will join or match together. For example, by joining ’omic data and clinical data (unstructured). We call these multi-source applications. The data is big through the consolidation of the distinct data resources, for example, in health initiatives, public sector, security,...

Primarily we shall be dealing with the first two of these challenges, and less so with the issues raised by the latter, since this is where commercial interests lie, and we are perhaps working over proprietary data. Occasionally we shall digress into the other fields where this is helpful to do so and there is some common cause.

The key challenge for those of us researching, working with, and exploiting analytics is to produce insights that are data-driven, that are usually hidden at first sight, and reflect some knowledge that is novel or private to the data owner, and thus advantageous. Even seasoned professionals are not able to guess what the consequent best action to take is, or which are the best opportunities to exploit. They seek business growth within a competitive environment. So our outputs should be quantitative (what might happen?) and actionable (what are the priorities, most likely scenarios and possible tactics?).

A common problem is that analytics teams often do things just because they can. They make the mistake of producing multiple outputs covering all possible applications of a given method without priority. A key word here is distillation, a lovely word. We must distil the outputs from the data. Many of us have sat through exhaustive presentations where diligent analysts have turned some handle and converted a kilogram of data into a kilogram of powerpoint, and they expected our gratitude. Decisions have to be both evidence-based and justified. Yet they should be “smart” because they are data-driven.

“This insight is data-driven, we did not hypothesise this, we found it”, whereas the IT team’s favourite hypothesis-driven analytics is often merely dice and slice, and so it “just reports what they think that you need to know.” We wish to say “There are no more compelling options available”, and so “one can rest assured that there is no better hypothesis or option that one could have checked”. “These are options/tactics listed in priority order” so these investments can be ranked and justified.

Access to a range of mathematical ideas and methods is essential for success. Within commercial environments, when challenged by client-companies or business stakeholders, it is often necessary to get a team of analytics folk together to argue over alternative approaches and to sense-check the usefulness of the outputs that different approaches might provide. Two heads are always better than one when planning analytics. We need both effectiveness and efficiency, and these are not to be confused.

As the data gets very large, or it arrives at a high rate, computational efficiencies be- come very important. The good news is that computational power and resource is ever increasing. The bad news is that the data deluge never ceases. So our key asset must be an understanding of what various methods may and may not achieve. As mathe- maticians we habitually abstract and translate analytics concepts and methods into new domains to solve new problems, but we cannot do so innovatively and securely without understanding the rigorous mathematical underpinnings of analytics. This is why considerations such as those set out in this book are foundations for a career in analytics and not simply “how to apply” notes for a set of pre-cooked algorithms.

Nowadays mathematics is evolving rapidly with respect to its external fields of application, and thus impact, whilst internally it is grinding on. Against this backdrop we all travel on our own journey. The fundamentals of analysis, algebra, geometry and calculus provide us with rigorous frameworks, concepts, vocabulary, methods and notation. Indeed the last of these can often be the most important with regard to communication, abstraction and simplification. The applications (the external ramifications of mathe- matics) are changing though, and changing at a faster pace than many professionals may care to concede.

For many hundreds of years since the Renaissance, mathematics was the key to an un- derstanding of the natural - mostly physical - sciences. It reached its zenith in providing insights into physical phenomena on the scales of time and space most easily explorable, and thus most easily comprehensible and exploitable, by mankind. The observations from celestial mechanics, previously a mystery of the ancients, gave way to the understating of gravity and many other natural physical phenomena. Broadly speaking, all such applications have a foundation of conservation laws at small scales: conversation of mass, momenta, energy. They are integrals for the fundamental equations of motion, reflecting symmetries at work. (We shall seek our own symmetries later in this text.) They usually give rise to continuum models for linear and nonlinear rates of change which have become phenomenally successful, expressed in terms of partial differential equations.

Just over one hundred years ago the science of the very large and the very small (scales, velocities, forces) provided a novel set of challenges: not just in explicating newly observ- able phenomena, but in providing theoretical support to as yet unobserved possibilities. Physics thus shifted from justification and explanation to inference and hypothesis cre- ation. Statistical mechanics and stochastic fluctuations provided further challenges in passing from the small to the large scales. Meanwhile dynamical systems, even of deterministic systems, emerged as a major source of uncertainty, sensitive dependence, unpredictability, structure, form and pattern.

By the middle of the 20th century, in the UK at least, the continuum mechanics paradigm was dominant in applied mathematics. From the Navier-Stokes equations in fluid mechanics to Maxwell’s equations in electromagnetics (and their plethora of approximations and simplifications), there was both willing momentum from within the science and a strong desire from within the applications (industry, military, communications). Weak solutions (shocks), boundary layers and moving boundary problems of all kinds stretched the knowledge further.

Then the thrilling emergence of modern computing changed the way that mathemati- cians could work for ever. The birth of numerical analysis not only responded to a desire to calculate, simulate, and predict, but also yielded novel topics in its own right: optimisation in all its forms; finite differences and elements; and glorious numerical linear algebra. The growth in computational resources also heralded the return of some older mathematical ideas that had remained largely parked and impotent for almost two hundred years, the most important of which was Bayesian reasoning. The previous difficulties in its application were merely practical. Only simple or contrived examples that were reliant on useful trickery, such a conjugate priors, were amenable to analysis: otherwise posterior distributions became easy to think about yet largely inaccessible. The consequence was that probability theory lost its meaning. The rise of frequentist “cooking” methods was welcomed by many practitioners within social and scientific ap- plications, yet was fundamentally and intellectually bankrupt at birth. Computational resource reversed this calumny not least with sampling methods. Today, a week or two spent reading Jaynes’ book can be a life changing experience.

From the 1970s onwards these and other mathematical ideas found two new outlets: the mathematics of finance and risk, and the mathematics of the life-sciences. These fields provided two distinct types of challenge: the former being the need to qualify uncertainty, risk, and, thus, value; the latter being about form and function (both normal and aberrant). By the end of the 1980s almost all universities had undergraduate courses on some elements of mathematical finance and mathematical biology, not least because these fields are large employers of graduates, and also because they were both about to explode. Ultimately one was driven by the global information and communication avalanche enabling electronic trading, and the other was driven by the informatics from high throughput genomics. A data deluge drove these subjects.

Though disruptive to the old school, these transitions were very comfortable for most mathematicians, relying largely on continuum concepts and models. The words changed but the song remained the same.

However, somewhere in the past forty years, a schism crept in. There is nowadays a denominational division between methods-based applied mathematicians, whom we shall call pragmatists, and the theory-based applied mathematicians, whom we shall call theorists. The pragmatists prize methods, learning through doing, and analytical approaches that exploit the particular structure of equations and problems. They have a great eye for problems, they zoom in and chase details, and their applied mathematics is a plumbers’ bag of exotic tools. Meanwhile the theorists integrate the applications within a wider phenomenology and they use them to challenge and expand both the theory and the applied activity. They generalise and abstract to ever bigger problem classes.

The biggest losers from a restricted pragmatist’s diet are students. They take the short term gains of being able to answer some standard problems (oft rather special cases, oft leaned by wrote), but giving up, ignoring, and not challenging themselves with any mention of the larger problems. The normal response to this would be to have a foot in both camps, but this is often not achieved, especially given the polarisation of the interests of lecturers.

One should always resist over-reliance on methods and pragmatics. One might spot oneself behaving as a man with a hammer who is obsessively searching for nails. At best, if successful this leads to a virtuoso’s skill. At worst, one’s appetite and horizons become restricted and impact, esteem and influence will forever be limited to a smallish community of similarly obsessed cognoscenti. The antidote is adventure, the launch-pad for which is almost always found at the intersection between some fundamental mathematical idea and the allure of some fresh field of application.

Mathematics is subject to a further malaise called the “King Herod” principle. Estab- lished fields, championed by their participants, seek to weaken and extinguish novel and radical topics in mathematics whilst they are embryonic or infantile, and before they can grow up and compete for research funding, staffing, and esteem. This is natural in many ways, but it is also destructive and murderous. As the funding situation gets tight in straightened times all sorts of arguments based on critical mass, retention of key players, and continuity of perceived critical competencies are all trotted out to justify the massacre of innocent ideas.

It is against these background forces that a new field of applied mathematics has staked its claim: the mathematics of behavioural analytics. This was driven by the data deluge from novel digital platforms, and could equally be termed the mathematics of the digital society.

Yet even before academic mathematicians became aware of this the genie was already out of the bottle. Many companies and institutions simply could not wait for the mathemat- ical research community to catch up with the applications. The solutions to disruptive challenges and the novel opportunities so created were simply too valuable. The result was that many of the pioneering ideas in “analytics” (the mathematical and quantita- tive analysis of data resources) were produced by small research groups working within industry, and especially within start-up companies. Even the vocabulary was that of business competitiveness, and, for many analytics practitioners like the author, it was essential that analytics was seen as the provider of a competitive edge, and an activity championed in business schools and adopted in boardrooms long before it crossed over into academic research with the mathematical sciences. In particular Davenport was a highly influential figure in arguing for this culture change. The challenges and benefits that were set out in such work, drawn against the backdrop of the explosive growth in the digital society and commerce, certainly made it straightforward to grow analytics practices. The demand for creativity in analytics, and thus the insights to drive highly differentiating and competitive actions/options, is still accelerating.

In turn, this requires both sure-footedness and good decision making skills of analytics professionals, and these can only come from a firm foundation for models, methods, and algorithms within mathematics and some experience (exemplars). Over the next few years there will be a deluge of analytics taught in universities as part of graduate-level courses and professional development courses on Data Science, Big Data and Analytics, and Discrete Modelling for Commercial Sectors. The mathematical sciences community urgently requires some scholarship and exemplars to define and drive the leading edges of theory and practice. This book is a contribution to that effort.

The birth of any new field of mathematical applications is never entirely straightforward, yet it has happened. And now, after the physical and life sciences, it is the turn of the social sciences to become transformed from being a retrospective and narrative-led theory (telling us what may have happened) into a speculative, insightful and forward-looking activity (telling us what might happen).

Of course in any new field of mathematical application there is also risk to the mathe- maticians who are involved. The provision of leadership and scholarship itself requires people to commit to these new fields that, at best, colleagues and others simply may not value, understand, nor accept. At worst they may be derided or despised. This should not happen, but we know it does.

Happily one’s adherence to the new discipline of mathematics for behavioural analytics may be validated every time one speaks to the potential exploiters, especially within service sector companies. For example, those within retail, consumer goods, telecoms, online businesses, energy, finance, betting, leisure, health, IT and communications (including software and services). All of these sectors’ operational and research groups will recognize the value of analysis that rises to these digital challenges. They are intrinsic to the future success of our companies, our economy and our international competitive- ness, so we can take comfort from this very strong interest from our potential exploiters. These are companies that work in hugely competitive sectors, that are internationally excellent, and where every decision for investment and activity is tensioned. It can be argued that we are already living through a “boiling up” phase prior to the setting out of a formal foundation and scope. Perhaps in ten years time, the mathematics of behavioural analytics will be common place: every mathematics department will be doing some of it. Industry and commerce need it, the government and public regulators will require it, and our students will be attracted towards it. Economic growth, careers, entrepreneurial opportunities, and research challenges will be the drivers.

Necessarily, novel ideas and methods have arrived piecemeal, by trial and error, so it is essential now to avoid the pitfalls of the pragmatists. The comparison to earlier applications of applied mathematics, and the harmful schism with its ignorance of theoretical foundations, is compelling. Let us avoid this from today. We shall deal with discrete mathematics (graphs and networks), probability and inference (forecasting and unsupervised discrimination), optimisation (calibration by discrete non derivative search as well as continuous gradient methods), and system dynamics (stability, response and structure).

Should not the key skill of the analytics professional be the perspective to see their challenges within the bigger picture? We need a balanced mixture between pragmatics (method – what works?) with the theoretical underpinnings (why and how might anything work?).

An official position paper of the European Service Network of Mathematics for Industry and Innovation (EU-MATHS-IN http://www.eu-maths-in.eu/)

Where is mathematical modelling, simulation and optimisation?
A call for reason to capitalize on European Mathematical Expertise
for industrial innovation and European competitiveness

It has become widely recognized that the approach of modelling, simulation and optimisation (MSO) builds the third pillar for scientific progress and innovation, besides experiments and theory building. In the various Working Programs of Horizon 2020 of the European Commission, however, the use of mathematics/mathematical modelling simulation and optimisation is mentioned fewer than a handful of times (one positive exception is the FET paper). While in the current version of the Work Programs the terms modelling and simulation are used in a somewhat inflationary way, the connection to a sound mathematical basis is rarely made. But, in our experience and expertise, future challenges for innovations in industry and society exhibit increasing complexity and at the same time have to obey ever-shorter innovation cycles. For this it takes more than just trying out all possible parameter variations on a computer. The real-world challenges to be dealt with on our way towards innovations in industry and society exhibit properties that make MSO a far more non-trivial task. In fact, many of the current (and future) problems require the development of mathematical methodologies, such as the areas of:

• Multi-physics and multi-scale systems,
• Combined discrete and continuous non-linear systems,
• Account for non-deterministic stochastic influences,
• Treating systems too large to be dealt with in all details requires adequate model reduction schemes,
• Dealing with inhomogeneous uncertainties among the various parts of the model
• Coupling of various models
• Multi-criterial optimization
• Etc.

All these aspects require ongoing and advanced multidisciplinary application-driven basic mathematical research and the development of advanced and new mathematical tools for MSO to meet the expectations that are raised in the present version of the Working Programs.

We therefore strongly suggest and call for the establishment of mathematical MSO as transversal Key Enabling Technology (KET). We need continuous multidisciplinary research in application-oriented basic mathematical methods to provide the tools needed. Such mathematical tools will be intrinsically suited or can be easily adapted for a wide variety of real-world problems actually covering the whole range of identified KETs (nanotechnology, micro- and nano-electronics including semi-conductors, photonics, advanced materials, biotechnology).

Establishing mathematical MSO as transversal KET enables to capitalize on the European leadership in application-driven mathematics, to strengthen European competitiveness in industrial innovations and to meet societal challenges ahead of us.

An official position paper of the European Service Network of Mathematics for Industry and Innovation (EU-MATHS-IN http://www.eu-maths-in.eu/)

Value creation through mathematical modelling, simulation and optimization:
A proposal for new Research Infrastructures and dedicated FET topics

Future challenges for innovation in industry and society exhibit increasing complexity and at the same time have to obey to ever-shorter innovation cycles. One of the key technologies in this permanent fight is the use of computers at peak performance in an appropriate way, i.e. in the integrated modelling, simulation and optimization (MSO) frame. In the competitive industry and in the top scientific research projects a full holistic approach is to be applied (e.g. to use MSO on a complete vehicle, a full digital factory, the human heart or the complete vascular system). To develop such a holistic approach one needs a mathematical model that allows to simulate and optimize the real product on virtual product via the use of high performance computing (HPC) tools.

Although there are many success stories of the use of MSO (see “European success stories in Industrial Mathematics”, Springer, 2011, ISBN 978-3-642-23848-2), the full potential of MSO as an integrated discipline is not yet realized and hence the potential in the creation of value is still severely overlooked. Often, MSO is taken on board only in the final stages of a project. In order to create real value from MSO, it must be an essential part of every innovation project, and urges also developments in new mathematical methodologies, e.g. in:

• Multi-physics and multi-scale systems,
• Combined discrete and continuous non-linear systems,
• Non-deterministic stochastic influences,
• Approximating systems too large to be dealt with via adequate model reduction schemes,
• Dealing with inhomogeneous uncertainties among the various parts of the model,
• Management of big data.

All these fields (and many others) are of high importance in order to deal with problems arising in areas like nanotechnology, medicine and health, the environment, energy production and transport image and data processing, etc. Moreover, the MSO technology itself faces challenges, e.g. sensitivity analysis of and adjoint methods for the full, integrated model.

In order to create real value, many disciplines need to be involved. Clearly, mathematics is the basic discipline here, integrated to computer science, especially associated with HPC, and all disciplines of engineering and natural sciences.

In Europe several research groups with excellent researchers work on the scientific research and application of MSO in industry and innovations. Numerous of them have formed the international network EU-MATHS-IN, the European Service Network of Mathematics for Industry and Innovations (see http://www.eu-maths-in.eu/). This international network currently consists of a dozen of national networks representing a large number of research groups that follow the goal to boost mathematics for industry in Europe.

Many recent top scientific and innovation projects make use of the MSO technology. An example of a success case is the virtual paint shop: initiated by Volvo Car Corporation in 2006, the Fraunhofer-Chalmers Research Centre for Industrial Mathematics developed mathematical software for virtual spray painting in 2009, which reduced the environmental impact and increased the product quality.

Highlighting MSO as a Research Infrastructure and FET would provide both the scientific and industrial research communities with an advanced way of using newest mathematical technology combined with high performance computer resources and give them a tool to systematically achieve new results of high impact in their fields. Establishing MSO as a future emerging technology will enable Europe to capitalize on the current European leadership in application-driven MSO, to strengthen European competitiveness in industrial innovation in providing industry with tools of higher precision within the same time scale, and to meet important future societal challenges. Moreover, many significant research projects could be brought into the breakthrough level.

It has become widely recognized that the approach of mathematical modelling, simulation and optimization (MSO) is the third, and indispensable, pillar for scientific progress and technological innovation, besides experiments and theory building. When full-scale simulation is no more possible, mathematical modelling and simulation and optimization can do it!

This viewpoint is supported by the Forward Look on Mathematics and Industry of the European Science Foundation (http://www.esf.org/index.php?id=6264) and in further reports:
• Mathematical sciences work is becoming an increasingly integral and essential component of a growing array of areas of investigation in biology, medicine, social sciences, business, advanced design, climate, finance, advanced materials, and many more – crucial to economic growth and societal well-being (“The Mathematical Sciences in 2025”, report of the National Research Council in the USA, 2013);
• The GVA attributable to the direct application and generation of mathematical science research in the UK in 2010 was around 16 % of total UK GVA (Deloitte report on “Measuring the Economic Benefits of Mathematical Science Research in the UK”, November 2012 and a similar report issued in The Netherlands in 2014).
• German Science Council Report : Bedeutung und Weiterentwicklung von Simulation in der Wissenschaft 2014

Dear Theo,
I should not put Mathematics as a FET, I think the FETs are strictly linked to boost technology into Industry and they are quite limited about the broader field of Maths. It's is a long long term strategy what we are talking about here. IMHO we have now a big chance and a challenge on imaging how put the Nature's Language, again, as top priority for achieving new science results, theories, horizons, etc…
I will make an example of what I'm talking about:
The 20th century had 2 major scientific revolutions which started almost in the same period, the Relativity theory and the Quantum theory. Is impossible not recognize the role of the Gottingen university, in particular, the German quantum generation grew up there. Other than this the relativity theory was a titanic fight between Einstein and Hilbert (when Hilbert was in Gottingen). Looking back in time, the Gottingen reach the state of the art on mathematics during the end of 19th by merit of the Erlangen program father vision to re-establish Gottingen as the "world's leading mathematics research centre". Sooner or later all European quantum Generation attend or delivery courses or conferences there.
This is a clear result: Our information society would not exist without those two scientific revolutions and both of them were boosted by mathematical research.
A Mathematics Directorate General is an idea that should be explored, may be is too much, I don't know…, still, I think there are too much Mathematics on our daily life, and in the near future should be even more… but nobody is taking care of them. Forget that I'm proposing to control or regulate or something like that… is all about to help and promote and teach in a better way and have a common approach to this. I believe Mathematics is pure creativity… pure innovation depending how you define innovation.
Antonio Puertas Gallardo

I'm writing on behalf of the EESI (European Exascale Software Initiative), a FP7 project which aims to identify challenges, strengths and weaknesses, potential international collaborations, training needs of Europe in the field of scientific applications and system software towards Exascale (the next generation of supercomputers which aims to provide around 100x times the power of current systems).
Based on the assessment made by 200 worldwide experts, we are publishing every year some recommandations towards EC and funding agencies for funding R&D activities in order to maintain Europe in the forefront of the domain (Europe owns the development of a big part of scientific applications used in the world).
Of course applied maths are a big part of the challenges and issues raised by Ulli in a previous post about ultra scalable solvers, resilience, communication avoiding algorithms, new meshing algorithms, optimisation and UQ, reproducibility, coupling between stochastic and determinist methods, ... are clearly identified and we started to proposed last year some recommandations : http://www.eesi-project.eu/modules/download_pictures/dlc.php?file=345&id...

We are currently working on new recommandations for 2014 and we hope that some of them will be taken into account by EC into the next WP

there is also a big issue regarding the fact that we are missing in Europe the critical mass for industrializing and providing long term support to maths open source libraries.
Most of us are using BLAS, LAPACK, PETSc, Hypre and other libraires which are developed by international teams but managed and hosted by US teams with funding from DoE and NSF.
We have few exemples in Europe (like MUMPS for exemple) but its clear that if EC could fund an infrastructure and manpower for industrializing spare development into real open source librairies, clearly disseminated, with long term support it could be very important in order to avoid to our scientists to contribute into US libraires ...
This could be done through a CoE and promoted toward scientists by research infrastructure like PRACE in Europe.

I agree with the comments that several colleagues have posted. Mathematics -and in particular MSO i.e. Modelling, Simulation and Optimization- is a key technology that is absolutely crucial for innovation in Sciences as in Industry.
Based on my long experience of cooperation with industries (both large companies and SME's), I can say that there are two main obstacles that hinder the impact of mathematics on industrial innovation:
(i) the insufficient awareness (from the industrial side) that mathematical tools are evolving very quickly: something that was unthinkable with the tools that were available when the engineers and/or scientists working in the company were trained is possibly attainable now with little effort;
(ii) the fragmentation of scientific offer from the side of mathematicians: what industry needs is a "single-stop-shop" where all the information on availability of competencies can be found. This is exactly what can be provided by an e-infrastructure at European level that collects all the relevant information, organizes it and makes it accessible to end-users.

Removing these two obstacles would be an important step forward to channel "Excellent Science" to the needs of economic competitiveness of our continent!

## 54 Comments

## The impact of applied and industrial mathematics on innovation.

Maths can help a lot in industry for two main reasons: first, as lingua franca between engineers, physicians, mathematicians and executives (even with a mathematician as "translator") that is one of the fundamental premisses to have a good model for the scientific representation of real world processes.

Second, in a very competitive world almost all the SME's need to optimize their processes in order to become more competitive. The role of hand-made models (and corresponding software) is crucial, as the comercial products/software in the market usually don't take into account the particularities of their own companies. This fact, together with a precise definition of validity domains for each applied domain is crucial to a good and sustainable development.

## Great point about mathematics

Great point about mathematics being the common language and vehicle to carry ideas across international and discipline borders. As for SME's it is important to say that using mathematics even a single person can make advances showing that smaller groups can contribute to the digital revolution - it does not have to be the big data companies (although once you have an idea they will surely make an offer and so it may appear that only the bigger companies are making the innovative advances). Steven Bishop, UCL

## In the UK, we celebrate the

In the UK, we celebrate the work of Alan Turing whose mathematics was critical in advancing computing science. The UK Government have announced that a major Turing Centre will be set up with something like £42m set aside for this. In weather forecasting (surely one of the big users of HPC) mathematics has lead to jumps in improvement of our predictive power (e.g. for hurricane paths) that would not have been achieved by increased computer power alone. Steven Bishop, UCL

PS a new film is due to be released soon with Benedict Cumberbatch which will only increase the interest in this area

## Mathematics can play an

Mathematics can play an important role in digital science distinct from its contribution to the big data revolution. The new applications of graph theory to networks is advancing our understanding of how patterns can be detected, understood and managed (ok controlled if you like) and this may yet provoke mathematicians to devise new mathematical techniques.

Meanwhile, some of the same methods can help sort and organise big data which may require HPC in order to manage distributed sources and analysis.

## Mathematics, and in

Mathematics, and in particular Operations Research and Mathematical Optimization, may be of great help in all these topics. For instance:

- Big data: structure detection in big social networks and information retrieval from "big data" databases can be formulated as a large mathematical optimization problem. Mathematics are also needed in big-data confidentiality issues: how to guarantee that individual confidential information is not disclosed?

- HPC: mathematical optimization problems are among the most CPU time consuming today algorithms, and they may require HPC infrastructures for its efficient solution. This includes exploitation of parallelism capabilities. At the same time, optimization problems are a source of difficult instances to test new HPC paradigms.

- Applied and industrial mathematics: all industrial problems, in any field, involve the solution of an optimization problem, either to minimize costs or to maximize benefits.

## I would like to comment on

I would like to comment on the role of e-Infrastructures for Mathematics. European e-Infrastructures can support the computational needs of mathematicians by providing the capacity and capability in terms of storage and computing that may not be available locally to the scientist, as part of the ICT services offered by the home research institute.

The European Grid Infrastructure (EGI, http://www.egi.eu/) provided to user communities active in the Mathematics domain 30 Million CPU hours of computation since 2004, where the largest amount of computing resources was offered by the Dutch national infrastructure, followed by CERN, France and Greece (http://accounting.egi.eu/vodis.php?query=sumelap&startYear=2004&startMon...).

Being EGI a federation of national infrastructures, it can help mathematicians by offering the ICT services that may not be available locally/nationally. So far, EGI has offered services for Authentication, Authorization and Identity Management, computing and data management.

To date, we (EGI) lack a Virtual Research Community active in the Mathematics disciplinary area, so it is difficult to approach Mathematicians at a European level, most of these are being supported by the national e-Infrastructures as part of the long-tail of science.The EC could help EGI in approaching European networks of Mathematicians with dedicated workshops. Such events would help European e-infrastructures to better network with this European research community and to understand the needs both in terms of ICT capacity and capability.

## The impact of applied and industrial mathematics on innovation.

The development of new products or new production processes is at the heart of European industry. With growing complexity and shorter innovation cycles, mathematical modeling, simulation and optimization (MSO) becomes an indespensible prerequisite for the development of European Industry and technologies. The virtual product that accompanies the real product and allows product verification, risk analysis and product optimization is the future. For this we have to strengthen interdisciplinary cooperation and to include mathematical MSO into all major technology funding streams. European research groups have been very active to support this development but there is a strong imbalance within the Euurpean research area that needs to be balanced. The role of mathematical methods in big data and HPC is typically underestimated. Too much emphasis is paid on new hardware developments, here new developments require the incoporation of mathematical MSO.

The progress in performance in the methods is larger than in the hardware and this has to be continued.

Emphasis should be given to new mathematical methods and also to the creation of e-infrastructures that allow the whole European science community to make use of new methods.

To maximize the impact of MSO on innovation can be inparticular successful in the development of infrastructure networks, such as gas, electricity, communication, etc where the developments of the last years have been tremendous and where a major European research initiative may allow in the near future to take a holistic point of view and to simulate, analyze, and also control and optimize a whole national or supranational network in its planning and in its operation. For this direction a FET initiative, bringing together the experts from the applications and from mathematical MSO, would be the right framework and could have a major impact. On the national level in some countries such initiatives have started, involving industries and academic institutions, but more national researchinitiatives should get involved from different countries.

## Economic impact of an e-infrastructure in industrial mathematics

Mathematical tools are fundamental for the simulation processes that are necessary in the design of new technologies and procedures. Mathematical modelling carries with it robustness, stability and safety, in the sense that mathematicians can deliver solutions with error estimates and the knowledge that the proposed method does what it is supposed to do, which is not necessarily the case of any simulationMathematicians, allied with other scientists, engineers and computer scientists can make a big difference for companies involved in innovation and new technologies. But solving complex problems often needs to involve groups with different expertise, groups that do not necessarily come from the same place or country. In that sense, disposing of databases of expertise, past experiences and success stories in solving real life problems, etc, is of fundamental importance. Also having databases of successfully used algorithms and numerical or simulation methods to tackle particular problems. A European e-infrastructure devoted to the interaction of mathematics with new technologies and innovation would provide a prefect solution to share this kind of information and being able to build the ad-hoc experts' groups at best. Companies would be able to find the right people to help them with their needs.Another possible important aspect of an e-infrastructure in industrial mathematics would be to offer services to companies like a pan-European job market of experts or young people. Engineers and scientists working in companies could also be offered continuous education courses in the latest mathematical technologies in numerical computation, control, optimization, etc. Science advances quickly and it is difficult to keep in touch with the new tools that offer much better solutions. Such an e-infrastructure could offer such information and possibilities to companies which want to upgrade the scientific level of their engineers and scientists.Europe and the European economical competitiveness would certainly take a big advantage of the existence of such an e-infrastructure, providing the best existing services and solutions for companies aiming at building the most innovative solutions for all aspects of our European society.

## Interesting contributions already - thanks!

From the contributions already received, I am convinced we will have a very interesting workshop, alive with active discussions about mathematics and what it has to offer to tackle modern digital challenges. Many thanks for everything that has been posted already!

I have had today discussions with some mathematicians, both academians and those working with industry, all showing enthousiasm in getting together to discuss and identify mathematics as one of the powers behind the digital revolution. This will be a very interesting workshop, I am convinced!

The potential for mathematicians to contribute, at various levels, to the development of new digital tools and methods, is clear. That development potential and need is not only growing at high speed but at an exponentially growing speed. The more complex our environment becomes - the more data, the more powerful computers, the more linked everything is with everything else - the more demand there is for optimisation algorithms, pattern recognition, modelling, in general finding new, innovative solutions to tackle the new problems encountered due to new magnitudes. It will be good to identify what this means or can mean in practical terms - perhaps we can start that reflection at the workshop!

There has been discussion around a Virtual Research Environment for mathematics. This linked to a specific thematic field of mathematics, bringing together SMEs and researchers as well as research centres, mathematicians and technicians, to tackle and generalise mathematical challenges to serve the European research community, continues to be an excellent idea. Our Workprogramme for Research Infrastructures - one of the background documents - contains a topic inviting these interests to come together: Topic

EINFRA-9-2015 – e-Infrastructures for virtual research environments (VRE).As collaborators for industry, mathematicians have a lot of innovation potential - and can be solution oriented. Several areas are already mentioned in the comments: Mathematical modelling, simulation and optimization amongst others are areas which are key to both HPC development and Big Data, and on the other hand, can produce fantastic results from the vast amounts of data that we are struggling to find ways to tackle in Europe.

I thank you for all contributions so far and encourage new contributions!

Please key to your calendars the preliminary dates: at present we are hoping to organise the Workshop in Brussels 6 or 7 November. To be confirmed...

and ps: thanks for the tip on the Turing movie...! another Beautiful Mind -style I suspect!

## The science of data needs Mathematical Statistics methodology

The science of data needs a lot of Mathematical Statistics research to develop new robust procedures to deal with this (good) monster of big data. The support should be this and can not came from smoking sellers, whcich is a real threat out there with this topic.

## Connecting mathematics with industry and the society

No doubt that mathematics is a major source of innovations for industry and the whole society. However, very often new mathematical ideas do not immediately benefit to industry but take some time to percolate through other (more applied) scientific fields, mostly engineering, before reaching their "final" target. I believe it is an outmost important challenge to directly connect mathematicians to industry. It may be already the case in some instances for large companies which have their own research departments, well connected to mathematicians (among others). But it is clearly not the case for most small or medium companies. It is obvious that Europe could help a lot, in this respect, by organizing these connections on the model of national institutions like Franhaufer in Germany or AMIES in France, just to quote a few.

## Role of Mathematics

## Long term view based on personal career

I have seen the development of computer simulations and more generally the growing number of scientific riddles at the interface between computer and mathematics and my experience tells me that it is just the beginning!

Computers have been invented by two outstanding mathematicians and seventy years later HPC is still piloted to and for applied mathematicians. Although supercomputers are more important to theoretical chemists, astrophysicists, engineers etc than to mathematicians, the code-libraries that they use are the fruit of a collaboration between them, mathematicians and computer scientists.

Such collaborations are essential to science now a days and seems that the future of science belong to centers capable to unite actively mathematics, computer science and application fields.

## Better Methods Allow More Appropriate Incorporation of Data

Here are a couple of comments:

Re: The role of mathematics in Big Data: (1) We need to distinguish "big data" from "good data", as those are not remotely the same. (2) The availability of richer and more finely-grained data necessitates the development of methods that allow one to throw away less of the data. Throwing away data can lead to misleading or even simply wrong answers (e.g. losing causal relationships for disease or idea transfer when going from a temporal network to some sort of "effective" static network based on averaged contacts between people). One recent develop, in which I have been deeply involved, is generalizing ordinary network theory to "multilayer networks", in which one can incorporate time-dependence, multiple types of edges, interdependent networks (as occurs in e.g. infrastructure), and so on.

Impact of industrial and applied mathematics on innovation: It is worth keeping in mind that many branches of mathematics previously 'confined' to theoretical studies are now seeing an increasing number of practical applications --- including harmonic analysis, algebraic geometry, algebraic topology, tensor analysis, and so on. There are powerful tools available that may not be traditionally "applied", and it's really important that we take advantage of them and create ways to use them with data (which is active in many of these areas, but not necessarily applied to industry very much).

FET programs: I think it would be nice to have a call on taking traditionally "pure" areas of mathematics and bringing them closer to applications through concerted collaboration of both mathematicians (and other scientists) and end users.

## Impact of mathematics on innovation

Mathematics is much more widely applicable than just to HPC and Big data and making a useful mathematical model of any process will almost always lead to greater understanding of the process. It also requires collaboration

between mathematicians and scientists and we need infrastructure to make this happen and to make sure the mathematical expertise available in Europe is used effectively.

## Impact of mathematics on innovation

Mathematics is much more widely applicable than just to HPC and Big data and making a useful mathematical model of any process will almost always lead to greater understanding of the process. It also requires collaboration

between mathematicians and scientists and we need infrastructure to make this happen and to make sure the mathematical expertise available in Europe is used effectively.

## Impact of mathematics on innovation

I certainly agree that mathematics can be used for much more than HPC and Big data. As a mathematician and programmer I worked in several (including international) projects on diverse subjects: map creation using satellite data, simulation of phone booths, simulation of a mobile phone company, heart section analysis, glass cut optimization, lossy image compression, surface triangulation for glass/metal buildings, route optimization, numerical solution of differential equations, database analysis and visualization, databases for WWW pages for internal and external use (including energetics, wireless service optimization, traffic lights, and more). The first problem is always to persuade the company that they do need a mathematician to model the problem and find ways to optimize the handling of that. That is why I strongly back up the ideas above and suggest more. Namely:

1. Create a forum, where companies and mathematicians can find each other.

2. Create a forum, where one can publish and find mathematical ideas, algorithms for industrial problems.

3. Create a forum, where companies can publish their problems requiring mathematics to solve.

These forums already exist: the European Consortium for Mathematics in Industry (http://www.ecmi-indmath.org/). The problem is that the membership is not free for any of the participants and since none of those participants know exactly what are the benefits of the membership, it cannot be widely accepted.

About the Big data. One of the main problems is that everybody generates data, but not all is useful for anything. So before doing anything with the "Big data", one needs to clear that data from the garbage, remove duplicates, etc. It is well possible, that after clearing the data is not at all so big. Also (from my own experience) I know that people make mistakes in entering data in databases, so they need to be cleared not only syntactically, but also semantically. Another problem is that for different reasons (business interests, personal rights) some of the data will never will be available for analysis, and without those the aquired results will never be complete or usable.

About HPC. I did some work in this area too, and I know that there is no standard for HPC. I myself used at least least 3 different kinds of parallel computing, and I know that they need quite different way of thinking when it comes to programming. So our goal in this direction is not clear at all, we would need much more (practical) information to do anything useful in this direction.

Some of the problems summarised in quotes:

"Weinberg's Second Law: If builders built buildings the way programmers wrote programs, then the first woodpecker that came along would destroy civilization."

"Andrew S. Tanenbaum: The nice thing about standards is that you have so many to choose from."

"William E. Vaughan: To err is human, to really foul things up requires a computer."

## Weinberg's Second Law

FYI: Weinberg's Second Law is applicale to this WWW page too. None of the select form elements work correctly in Firefox 29.0.1, I could not set some of the subscription preferences, there are dangling internal links on this page, and the server responds very slowly to everything.

## We're improving continuously

Hi Geza,

We're improving the website and communities continuously, so thanks for your comment: we'll look into this and fix the html issue asap. We're also working with our providers to improve the performance, gradually - we try to keep costs down as well.

## Exa Scale Math Position Papers

For anyone interested in the mathematics needed for modeling, simulation, and optimization of complex systems, I recommend to look at the the Exa-math workshop that was held a year ago in Washington. See https://collab.mcs.anl.gov/display/examath/Agenda

From there a number of interesting position papers can be accessed that identify many mathematical research topics that need to be addressed to better use the upcoming generations of supercomputers.

Topics include (among others)

- ultra-parallel algorithms that can exploit beyond a million parallel threads

- the need for a mathematical analysis of algorithms that can be used to understand computational cost quantitatively. This is in contrast to the usual asymptotic convergence results that are rigorously proven, but that are unable to distinguish between fast and slow methods since they involve unspecified constants.

- algorithms with inherent fault tolerance on systems that are not perfectly reliable

- communication and synchronization avoiding algorithms, realizing that the dominating cost of computing is caused by moving data (and not number crunching)

## Mathematics Directorate General

There are 3 "hot areas" of intensive research on Digital Science, strictly related to Maths.

1. Exascale

2. Quantum Technologies

3. Neurosynaptics and Neuroprocessors

In particular the area of algorithms and programming techniques were, with these new contexts would required new ideas, creativity and differents approaches . It 's high time to consider Mathematics from a wider perspective dedicating a single EC Directorate General for the Nature's Language. You should be able to better coordinate and focus the investments, research and policies.

Without forgetting to put in the same context the Complexy Research field.

## Input from ETP4HPC

I am writing this on behalf of ETP4HPC (www.etp4hpc.eu) - the European Technology Platform for High-Performance Computing. Our members will log their comments in this consultation process on an individual basis. Also, in relation to this item:

• The preparation of the FET Proactive (HPC) and/or the e-Infrastructure Work Programmes 2016-17 under the Excellent Science pillar of Horizon 2020. Do you have a concrete proposal for a topic linked to this discussion to be included in the next work programmes?

ETP4HPC has a Strategic Research Agenda in place (SRA - available at the website above) and we are working on another document including proposals for the next round of the FETHPC calls. Our next SRA will include a reference to Mathematics and Digital Sciences and we will issues a separate document to express our position on this topic (target date - end of Sept 2014). We are also interested in participating in any events/workshops in relation to this consultation process. Thank you.

## Topics linked to the discussion

The WP2016-2017 is now in its preparatory phases which has included several consultations. Also this consultation will feed into the WP drafting. The consultation has certainly been very succesful with lots of high level participation and comments of great relevance, and we see a clear need for a follow-up! This is why we aim to organise a workshop (which we hope we will very soon be able to announce) and in that workshop get more concrete input for the Workprogrammes and topics supporting mathematics' development.

## About "The role of e-infrastructures in maths."

Hello,

I think that e-infrastructures can be a valid instrument for mathematicians in resolving problems, to search counterexamples and expand the knowledge in several fields of Math, specially in Algebra, Non Commutative Geometry, Number theory. The challenges are from my point of view the following:

- expand and enrich the european e-infrastructure with the software and tools needed by the mathematicians to work easly.

- promote the pervasive use of distributed computing tecnologies (Cloud or other paradimgs) in the math community by improving the Universitary courses and with specific traning activities to help scientists in discover the benefits and the high impact of e-infrastrucutre in math

- Promote the creation of technologists able to work toghether with math scientists in order to help them to generate optimized software for their math problems and to approach the porting the code in Cloud or distributed computing infrastructure. These experitese are popular in physics and chemistry but not so popular in the fileld of math.

In 2012 we used the European Grid Infrastructure to accellerate the ongoing Glodbach Conjecture verification started 10 years ago. In few months we was able to complete the 20% of the total computation, reaching the current world record of computation, thanks to the large resource availability offered by distributed e-infrastructure. (see http://home.web.cern.ch/about/updates/2012/09/lhc-grid-tackles-270-year-...).

In addition in 2014 Still using e-infrastructure we was able to improve some interesting results in the field of non commutative Geometry.

Thank you for attention.

## Maths is Underpinning for Analytics which creates value

The Mathematical Underpinnings of Analytics

In almost every sector of commercial and public endeavour there has been or there is about to be a data deluge. The innovation and exploitation, and also the hype, are driven by (a) the availability of data from emerging and converging digital platforms, (b) the increased amount of online and off-line traffic, data collection and surveillance, (c) the commercial imperatives to create greater value from existing customers and distilled knowledge; and (d) growing open data initiatives. Companies have become more aware of their own data resources, and see the future exploitation of these resources as a strategic path to growth.

When the data is very large, or continuously arrives very rapidly, doing anything smart is heavy lifting, and doing anything smart in real time is a challenge. Data has volume and velocity. Given enough processing time and capacity anybody might achieve anything. Yet companies and institutes have in many cases already invested their money in their infrastructure, architecture, enterprise, platforms, and access. Now they need to see some value return and value growth.

However, the data itself is merely the raw material. It is nothing without analytics: the concepts, methods and practices the can conjure valuable and actionable insights and radical knowledge from the large volumes of data. Such smart analytics goes well beyond “dice and slice”, alerts, reporting and querying, and it is far beyond the provision of infrastructure, computing architectures, and data handling resources. The latter are all necessary but are simply not sufficient for success. It is the analytics that will provide distinctiveness and unique capabilities, and allow us to see “further than others”. Such analytics needs to be founded on rigorous mathematical concepts, ideas, and methods, that may be deployed so as to underpin and innovate new concepts, methods and algorithms. In turn this could inspire new products and services and expand what companies and public institutes could achieve. It could even give rise to new business models. Com- panies can and will differentiate themselves on their exploitation of such analytics, and there clearly is an opportunity for them to obtain more insight and business growth from their data resources, going beyond current operations.

As different forms of data become ever more pervasive and more available, so the next generation of businesses and services within significant sectors (including commercial services, digital media, communications, domestic energy, security, environment, mar- keting, targeting, customer relationship management), and engagement across almost all public sectors will be developed by those companies, small and large, best placed to innovate. Our digital platforms will evolve rapidly year on year, converging and becoming pervasive with 24/7 operations, and data will become more open, as a currency or commodity.

Yet what is modern data science? What will data scientists need to achieve? This is a rather a crowded space. Other voices and opinion formers may argue from their own experience that it must subsume some or all of the following issues and activities: Foundations, Infrastructure, Management, Database Management, Security and Privacy, Search and Mining, and so on

True analytics is about none of these things, but any successful analytics strategies, operations and methods will surely rely on elements of them. In this book we shall focus on the mathematics that underpins world class analytics, and thus creates some differentiation and value.

It is arguable that "game changing" capability must come from creative applications of analytics to novel data sources, since access to digital platforms and computational power (in house or in the cloud) is now ubiquitous. In any case most companies have made (or will make) decisions on a long term basis about their operations (compute resources, data architectures, enterprise systems, and so on), and they, together with their suppliers, should be ever more focused on getting a return from those investments and plans. Hence their need for creative analytics and novel theoretical concepts that may in time become applicable, deliver actionable insights, and, thus, some agility and an ability to act.

Big data is presently on an trajectory resembling the Gartner hype curve. It is confusingly disorganised because it should really be split into at least four different challenges within data science. This way we prevent groups of people contributing very different things, with distinct aims, all claiming to be doing the essential element of vast data science.

1. There is vast data in the sense that we know a little bit about a very large number of things/objects/people. For example, in customer facing industries such as mobile tel- cos, retailers, ISPs, retail banks, smart metering in energy, digital marketing, and social media, etc. We call these customer- or consumer-facing applications, where the data is often proprietary.

2. There is vast data in the sense that it is constantly arising (has velocity,) and so real time analytics are required. For example, analysis of social media, or monitoring peer- to-peer conversations, emails, messaging, financial market data, or real time response to e-commerce, etc. Again this data is often proprietary.

3. There is vast data in the sense that we know a very large amount about a relatively small number complex objects, such as images for transmission and compression, sci- entific applications from crystallography, proteomics, fMRI scans, or spectroscopy, etc. We call these scientific or engineering applications, where the high throughput of novel scientific equipment spews out extremely high resolution data.

4. There is vast data in the sense that we know medium amounts about each of a number of distinct types of data objects belonging to individuals, which we will join or match together. For example, by joining ’omic data and clinical data (unstructured). We call these multi-source applications. The data is big through the consolidation of the distinct data resources, for example, in health initiatives, public sector, security,...

Primarily we shall be dealing with the first two of these challenges, and less so with the issues raised by the latter, since this is where commercial interests lie, and we are perhaps working over proprietary data. Occasionally we shall digress into the other fields where this is helpful to do so and there is some common cause.

The key challenge for those of us researching, working with, and exploiting analytics is to produce insights that are data-driven, that are usually hidden at first sight, and reflect some knowledge that is novel or private to the data owner, and thus advantageous. Even seasoned professionals are not able to guess what the consequent best action to take is, or which are the best opportunities to exploit. They seek business growth within a competitive environment. So our outputs should be quantitative (what might happen?) and actionable (what are the priorities, most likely scenarios and possible tactics?).

A common problem is that analytics teams often do things just because they can. They make the mistake of producing multiple outputs covering all possible applications of a given method without priority. A key word here is distillation, a lovely word. We must distil the outputs from the data. Many of us have sat through exhaustive presentations where diligent analysts have turned some handle and converted a kilogram of data into a kilogram of powerpoint, and they expected our gratitude. Decisions have to be both evidence-based and justified. Yet they should be “smart” because they are data-driven.

“This insight is data-driven, we did not hypothesise this, we found it”, whereas the IT team’s favourite hypothesis-driven analytics is often merely dice and slice, and so it “just reports what they think that you need to know.” We wish to say “There are no more compelling options available”, and so “one can rest assured that there is no better hypothesis or option that one could have checked”. “These are options/tactics listed in priority order” so these investments can be ranked and justified.

Access to a range of mathematical ideas and methods is essential for success. Within commercial environments, when challenged by client-companies or business stakeholders, it is often necessary to get a team of analytics folk together to argue over alternative approaches and to sense-check the usefulness of the outputs that different approaches might provide. Two heads are always better than one when planning analytics. We need both effectiveness and efficiency, and these are not to be confused.

As the data gets very large, or it arrives at a high rate, computational efficiencies be- come very important. The good news is that computational power and resource is ever increasing. The bad news is that the data deluge never ceases. So our key asset must be an understanding of what various methods may and may not achieve. As mathe- maticians we habitually abstract and translate analytics concepts and methods into new domains to solve new problems, but we cannot do so innovatively and securely without understanding the rigorous mathematical underpinnings of analytics. This is why considerations such as those set out in this book are foundations for a career in analytics and not simply “how to apply” notes for a set of pre-cooked algorithms.

Nowadays mathematics is evolving rapidly with respect to its external fields of application, and thus impact, whilst internally it is grinding on. Against this backdrop we all travel on our own journey. The fundamentals of analysis, algebra, geometry and calculus provide us with rigorous frameworks, concepts, vocabulary, methods and notation. Indeed the last of these can often be the most important with regard to communication, abstraction and simplification. The applications (the external ramifications of mathe- matics) are changing though, and changing at a faster pace than many professionals may care to concede.

For many hundreds of years since the Renaissance, mathematics was the key to an un- derstanding of the natural - mostly physical - sciences. It reached its zenith in providing insights into physical phenomena on the scales of time and space most easily explorable, and thus most easily comprehensible and exploitable, by mankind. The observations from celestial mechanics, previously a mystery of the ancients, gave way to the understating of gravity and many other natural physical phenomena. Broadly speaking, all such applications have a foundation of conservation laws at small scales: conversation of mass, momenta, energy. They are integrals for the fundamental equations of motion, reflecting symmetries at work. (We shall seek our own symmetries later in this text.) They usually give rise to continuum models for linear and nonlinear rates of change which have become phenomenally successful, expressed in terms of partial differential equations.

Just over one hundred years ago the science of the very large and the very small (scales, velocities, forces) provided a novel set of challenges: not just in explicating newly observ- able phenomena, but in providing theoretical support to as yet unobserved possibilities. Physics thus shifted from justification and explanation to inference and hypothesis cre- ation. Statistical mechanics and stochastic fluctuations provided further challenges in passing from the small to the large scales. Meanwhile dynamical systems, even of deterministic systems, emerged as a major source of uncertainty, sensitive dependence, unpredictability, structure, form and pattern.

By the middle of the 20th century, in the UK at least, the continuum mechanics paradigm was dominant in applied mathematics. From the Navier-Stokes equations in fluid mechanics to Maxwell’s equations in electromagnetics (and their plethora of approximations and simplifications), there was both willing momentum from within the science and a strong desire from within the applications (industry, military, communications). Weak solutions (shocks), boundary layers and moving boundary problems of all kinds stretched the knowledge further.

Then the thrilling emergence of modern computing changed the way that mathemati- cians could work for ever. The birth of numerical analysis not only responded to a desire to calculate, simulate, and predict, but also yielded novel topics in its own right: optimisation in all its forms; finite differences and elements; and glorious numerical linear algebra. The growth in computational resources also heralded the return of some older mathematical ideas that had remained largely parked and impotent for almost two hundred years, the most important of which was Bayesian reasoning. The previous difficulties in its application were merely practical. Only simple or contrived examples that were reliant on useful trickery, such a conjugate priors, were amenable to analysis: otherwise posterior distributions became easy to think about yet largely inaccessible. The consequence was that probability theory lost its meaning. The rise of frequentist “cooking” methods was welcomed by many practitioners within social and scientific ap- plications, yet was fundamentally and intellectually bankrupt at birth. Computational resource reversed this calumny not least with sampling methods. Today, a week or two spent reading Jaynes’ book can be a life changing experience.

From the 1970s onwards these and other mathematical ideas found two new outlets: the mathematics of finance and risk, and the mathematics of the life-sciences. These fields provided two distinct types of challenge: the former being the need to qualify uncertainty, risk, and, thus, value; the latter being about form and function (both normal and aberrant). By the end of the 1980s almost all universities had undergraduate courses on some elements of mathematical finance and mathematical biology, not least because these fields are large employers of graduates, and also because they were both about to explode. Ultimately one was driven by the global information and communication avalanche enabling electronic trading, and the other was driven by the informatics from high throughput genomics. A data deluge drove these subjects.

Though disruptive to the old school, these transitions were very comfortable for most mathematicians, relying largely on continuum concepts and models. The words changed but the song remained the same.

However, somewhere in the past forty years, a schism crept in. There is nowadays a denominational division between methods-based applied mathematicians, whom we shall call pragmatists, and the theory-based applied mathematicians, whom we shall call theorists. The pragmatists prize methods, learning through doing, and analytical approaches that exploit the particular structure of equations and problems. They have a great eye for problems, they zoom in and chase details, and their applied mathematics is a plumbers’ bag of exotic tools. Meanwhile the theorists integrate the applications within a wider phenomenology and they use them to challenge and expand both the theory and the applied activity. They generalise and abstract to ever bigger problem classes.

The biggest losers from a restricted pragmatist’s diet are students. They take the short term gains of being able to answer some standard problems (oft rather special cases, oft leaned by wrote), but giving up, ignoring, and not challenging themselves with any mention of the larger problems. The normal response to this would be to have a foot in both camps, but this is often not achieved, especially given the polarisation of the interests of lecturers.

One should always resist over-reliance on methods and pragmatics. One might spot oneself behaving as a man with a hammer who is obsessively searching for nails. At best, if successful this leads to a virtuoso’s skill. At worst, one’s appetite and horizons become restricted and impact, esteem and influence will forever be limited to a smallish community of similarly obsessed cognoscenti. The antidote is adventure, the launch-pad for which is almost always found at the intersection between some fundamental mathematical idea and the allure of some fresh field of application.

Mathematics is subject to a further malaise called the “King Herod” principle. Estab- lished fields, championed by their participants, seek to weaken and extinguish novel and radical topics in mathematics whilst they are embryonic or infantile, and before they can grow up and compete for research funding, staffing, and esteem. This is natural in many ways, but it is also destructive and murderous. As the funding situation gets tight in straightened times all sorts of arguments based on critical mass, retention of key players, and continuity of perceived critical competencies are all trotted out to justify the massacre of innocent ideas.

It is against these background forces that a new field of applied mathematics has staked its claim: the mathematics of behavioural analytics. This was driven by the data deluge from novel digital platforms, and could equally be termed the mathematics of the digital society.

Yet even before academic mathematicians became aware of this the genie was already out of the bottle. Many companies and institutions simply could not wait for the mathemat- ical research community to catch up with the applications. The solutions to disruptive challenges and the novel opportunities so created were simply too valuable. The result was that many of the pioneering ideas in “analytics” (the mathematical and quantita- tive analysis of data resources) were produced by small research groups working within industry, and especially within start-up companies. Even the vocabulary was that of business competitiveness, and, for many analytics practitioners like the author, it was essential that analytics was seen as the provider of a competitive edge, and an activity championed in business schools and adopted in boardrooms long before it crossed over into academic research with the mathematical sciences. In particular Davenport was a highly influential figure in arguing for this culture change. The challenges and benefits that were set out in such work, drawn against the backdrop of the explosive growth in the digital society and commerce, certainly made it straightforward to grow analytics practices. The demand for creativity in analytics, and thus the insights to drive highly differentiating and competitive actions/options, is still accelerating.

In turn, this requires both sure-footedness and good decision making skills of analytics professionals, and these can only come from a firm foundation for models, methods, and algorithms within mathematics and some experience (exemplars). Over the next few years there will be a deluge of analytics taught in universities as part of graduate-level courses and professional development courses on Data Science, Big Data and Analytics, and Discrete Modelling for Commercial Sectors. The mathematical sciences community urgently requires some scholarship and exemplars to define and drive the leading edges of theory and practice. This book is a contribution to that effort.

The birth of any new field of mathematical applications is never entirely straightforward, yet it has happened. And now, after the physical and life sciences, it is the turn of the social sciences to become transformed from being a retrospective and narrative-led theory (telling us what may have happened) into a speculative, insightful and forward-looking activity (telling us what might happen).

Of course in any new field of mathematical application there is also risk to the mathe- maticians who are involved. The provision of leadership and scholarship itself requires people to commit to these new fields that, at best, colleagues and others simply may not value, understand, nor accept. At worst they may be derided or despised. This should not happen, but we know it does.

Happily one’s adherence to the new discipline of mathematics for behavioural analytics may be validated every time one speaks to the potential exploiters, especially within service sector companies. For example, those within retail, consumer goods, telecoms, online businesses, energy, finance, betting, leisure, health, IT and communications (including software and services). All of these sectors’ operational and research groups will recognize the value of analysis that rises to these digital challenges. They are intrinsic to the future success of our companies, our economy and our international competitive- ness, so we can take comfort from this very strong interest from our potential exploiters. These are companies that work in hugely competitive sectors, that are internationally excellent, and where every decision for investment and activity is tensioned. It can be argued that we are already living through a “boiling up” phase prior to the setting out of a formal foundation and scope. Perhaps in ten years time, the mathematics of behavioural analytics will be common place: every mathematics department will be doing some of it. Industry and commerce need it, the government and public regulators will require it, and our students will be attracted towards it. Economic growth, careers, entrepreneurial opportunities, and research challenges will be the drivers.

Necessarily, novel ideas and methods have arrived piecemeal, by trial and error, so it is essential now to avoid the pitfalls of the pragmatists. The comparison to earlier applications of applied mathematics, and the harmful schism with its ignorance of theoretical foundations, is compelling. Let us avoid this from today. We shall deal with discrete mathematics (graphs and networks), probability and inference (forecasting and unsupervised discrimination), optimisation (calibration by discrete non derivative search as well as continuous gradient methods), and system dynamics (stability, response and structure).

Should not the key skill of the analytics professional be the perspective to see their challenges within the bigger picture? We need a balanced mixture between pragmatics (method – what works?) with the theoretical underpinnings (why and how might anything work?).

## Establishing mathematical MSO as transversal KET

An official position paper of the European Service Network of Mathematics for Industry and Innovation (EU-MATHS-IN http://www.eu-maths-in.eu/)

Where is mathematical modelling, simulation and optimisation?

A call for reason to capitalize on European Mathematical Expertise

for industrial innovation and European competitiveness

It has become widely recognized that the approach of modelling, simulation and optimisation (MSO) builds the third pillar for scientific progress and innovation, besides experiments and theory building. In the various Working Programs of Horizon 2020 of the European Commission, however, the use of mathematics/mathematical modelling simulation and optimisation is mentioned fewer than a handful of times (one positive exception is the FET paper). While in the current version of the Work Programs the terms modelling and simulation are used in a somewhat inflationary way, the connection to a sound mathematical basis is rarely made. But, in our experience and expertise, future challenges for innovations in industry and society exhibit increasing complexity and at the same time have to obey ever-shorter innovation cycles. For this it takes more than just trying out all possible parameter variations on a computer. The real-world challenges to be dealt with on our way towards innovations in industry and society exhibit properties that make MSO a far more non-trivial task. In fact, many of the current (and future) problems require the development of mathematical methodologies, such as the areas of:

• Multi-physics and multi-scale systems,

• Combined discrete and continuous non-linear systems,

• Account for non-deterministic stochastic influences,

• Treating systems too large to be dealt with in all details requires adequate model reduction schemes,

• Dealing with inhomogeneous uncertainties among the various parts of the model

• Coupling of various models

• Multi-criterial optimization

• Etc.

All these aspects require ongoing and advanced multidisciplinary application-driven basic mathematical research and the development of advanced and new mathematical tools for MSO to meet the expectations that are raised in the present version of the Working Programs.

We therefore strongly suggest and call for the establishment of mathematical MSO as transversal Key Enabling Technology (KET). We need continuous multidisciplinary research in application-oriented basic mathematical methods to provide the tools needed. Such mathematical tools will be intrinsically suited or can be easily adapted for a wide variety of real-world problems actually covering the whole range of identified KETs (nanotechnology, micro- and nano-electronics including semi-conductors, photonics, advanced materials, biotechnology).

Establishing mathematical MSO as transversal KET enables to capitalize on the European leadership in application-driven mathematics, to strengthen European competitiveness in industrial innovations and to meet societal challenges ahead of us.

## Establishing mathematical MSO as FET area

An official position paper of the European Service Network of Mathematics for Industry and Innovation (EU-MATHS-IN http://www.eu-maths-in.eu/)

Value creation through mathematical modelling, simulation and optimization:

A proposal for new Research Infrastructures and dedicated FET topics

Future challenges for innovation in industry and society exhibit increasing complexity and at the same time have to obey to ever-shorter innovation cycles. One of the key technologies in this permanent fight is the use of computers at peak performance in an appropriate way, i.e. in the integrated modelling, simulation and optimization (MSO) frame. In the competitive industry and in the top scientific research projects a full holistic approach is to be applied (e.g. to use MSO on a complete vehicle, a full digital factory, the human heart or the complete vascular system). To develop such a holistic approach one needs a mathematical model that allows to simulate and optimize the real product on virtual product via the use of high performance computing (HPC) tools.

Although there are many success stories of the use of MSO (see “European success stories in Industrial Mathematics”, Springer, 2011, ISBN 978-3-642-23848-2), the full potential of MSO as an integrated discipline is not yet realized and hence the potential in the creation of value is still severely overlooked. Often, MSO is taken on board only in the final stages of a project. In order to create real value from MSO, it must be an essential part of every innovation project, and urges also developments in new mathematical methodologies, e.g. in:

• Multi-physics and multi-scale systems,

• Combined discrete and continuous non-linear systems,

• Non-deterministic stochastic influences,

• Approximating systems too large to be dealt with via adequate model reduction schemes,

• Dealing with inhomogeneous uncertainties among the various parts of the model,

• Management of big data.

All these fields (and many others) are of high importance in order to deal with problems arising in areas like nanotechnology, medicine and health, the environment, energy production and transport image and data processing, etc. Moreover, the MSO technology itself faces challenges, e.g. sensitivity analysis of and adjoint methods for the full, integrated model.

In order to create real value, many disciplines need to be involved. Clearly, mathematics is the basic discipline here, integrated to computer science, especially associated with HPC, and all disciplines of engineering and natural sciences.

In Europe several research groups with excellent researchers work on the scientific research and application of MSO in industry and innovations. Numerous of them have formed the international network EU-MATHS-IN, the European Service Network of Mathematics for Industry and Innovations (see http://www.eu-maths-in.eu/). This international network currently consists of a dozen of national networks representing a large number of research groups that follow the goal to boost mathematics for industry in Europe.

Many recent top scientific and innovation projects make use of the MSO technology. An example of a success case is the virtual paint shop: initiated by Volvo Car Corporation in 2006, the Fraunhofer-Chalmers Research Centre for Industrial Mathematics developed mathematical software for virtual spray painting in 2009, which reduced the environmental impact and increased the product quality.

Highlighting MSO as a Research Infrastructure and FET would provide both the scientific and industrial research communities with an advanced way of using newest mathematical technology combined with high performance computer resources and give them a tool to systematically achieve new results of high impact in their fields. Establishing MSO as a future emerging technology will enable Europe to capitalize on the current European leadership in application-driven MSO, to strengthen European competitiveness in industrial innovation in providing industry with tools of higher precision within the same time scale, and to meet important future societal challenges. Moreover, many significant research projects could be brought into the breakthrough level.

It has become widely recognized that the approach of mathematical modelling, simulation and optimization (MSO) is the third, and indispensable, pillar for scientific progress and technological innovation, besides experiments and theory building. When full-scale simulation is no more possible, mathematical modelling and simulation and optimization can do it!

This viewpoint is supported by the Forward Look on Mathematics and Industry of the European Science Foundation (http://www.esf.org/index.php?id=6264) and in further reports:

• Mathematical sciences work is becoming an increasingly integral and essential component of a growing array of areas of investigation in biology, medicine, social sciences, business, advanced design, climate, finance, advanced materials, and many more – crucial to economic growth and societal well-being (“The Mathematical Sciences in 2025”, report of the National Research Council in the USA, 2013);

• The GVA attributable to the direct application and generation of mathematical science research in the UK in 2010 was around 16 % of total UK GVA (Deloitte report on “Measuring the Economic Benefits of Mathematical Science Research in the UK”, November 2012 and a similar report issued in The Netherlands in 2014).

• German Science Council Report : Bedeutung und Weiterentwicklung von Simulation in der Wissenschaft 2014

## FET and Maths....

Dear Theo,

I should not put Mathematics as a FET, I think the FETs are strictly linked to boost technology into Industry and they are quite limited about the broader field of Maths. It's is a long long term strategy what we are talking about here. IMHO we have now a big chance and a challenge on imaging how put the Nature's Language, again, as top priority for achieving new science results, theories, horizons, etc…

I will make an example of what I'm talking about:

The 20th century had 2 major scientific revolutions which started almost in the same period, the Relativity theory and the Quantum theory. Is impossible not recognize the role of the Gottingen university, in particular, the German quantum generation grew up there. Other than this the relativity theory was a titanic fight between Einstein and Hilbert (when Hilbert was in Gottingen). Looking back in time, the Gottingen reach the state of the art on mathematics during the end of 19th by merit of the Erlangen program father vision to re-establish Gottingen as the "world's leading mathematics research centre". Sooner or later all European quantum Generation attend or delivery courses or conferences there.

This is a clear result: Our information society would not exist without those two scientific revolutions and both of them were boosted by mathematical research.

A Mathematics Directorate General is an idea that should be explored, may be is too much, I don't know…, still, I think there are too much Mathematics on our daily life, and in the near future should be even more… but nobody is taking care of them. Forget that I'm proposing to control or regulate or something like that… is all about to help and promote and teach in a better way and have a common approach to this. I believe Mathematics is pure creativity… pure innovation depending how you define innovation.

Antonio Puertas Gallardo

## EESI Recommandation on Applied Mathematics

Dear colleagues

I'm writing on behalf of the EESI (European Exascale Software Initiative), a FP7 project which aims to identify challenges, strengths and weaknesses, potential international collaborations, training needs of Europe in the field of scientific applications and system software towards Exascale (the next generation of supercomputers which aims to provide around 100x times the power of current systems).

Based on the assessment made by 200 worldwide experts, we are publishing every year some recommandations towards EC and funding agencies for funding R&D activities in order to maintain Europe in the forefront of the domain (Europe owns the development of a big part of scientific applications used in the world).

Of course applied maths are a big part of the challenges and issues raised by Ulli in a previous post about ultra scalable solvers, resilience, communication avoiding algorithms, new meshing algorithms, optimisation and UQ, reproducibility, coupling between stochastic and determinist methods, ... are clearly identified and we started to proposed last year some recommandations :

http://www.eesi-project.eu/modules/download_pictures/dlc.php?file=345&id...

We are currently working on new recommandations for 2014 and we hope that some of them will be taken into account by EC into the next WP

there is also a big issue regarding the fact that we are missing in Europe the critical mass for industrializing and providing long term support to maths open source libraries.

Most of us are using BLAS, LAPACK, PETSc, Hypre and other libraires which are developed by international teams but managed and hosted by US teams with funding from DoE and NSF.

We have few exemples in Europe (like MUMPS for exemple) but its clear that if EC could fund an infrastructure and manpower for industrializing spare development into real open source librairies, clearly disseminated, with long term support it could be very important in order to avoid to our scientists to contribute into US libraires ...

This could be done through a CoE and promoted toward scientists by research infrastructure like PRACE in Europe.

This was my contribution to the debate ;-)

Stéphane

## "Innovate or perish": the role of mathematics

I agree with the comments that several colleagues have posted. Mathematics -and in particular MSO i.e. Modelling, Simulation and Optimization- is a key technology that is absolutely crucial for innovation in Sciences as in Industry.

Based on my long experience of cooperation with industries (both large companies and SME's), I can say that there are two main obstacles that hinder the impact of mathematics on industrial innovation:

(i) the insufficient awareness (from the industrial side) that mathematical tools are evolving very quickly: something that was unthinkable with the tools that were available when the engineers and/or scientists working in the company were trained is possibly attainable now with little effort;

(ii) the fragmentation of scientific offer from the side of mathematicians: what industry needs is a "single-stop-shop" where all the information on availability of competencies can be found. This is exactly what can be provided by an e-infrastructure at European level that collects all the relevant information, organizes it and makes it accessible to end-users.

Removing these two obstacles would be an important step forward to channel "Excellent Science" to the needs of economic competitiveness of our continent!

## Pages