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Open consultation on Mathematics for Horizon 2020 next Work Programme

Anni HELLMAN

28 January 2016 - updated 4 years ago
179 comments
Total votes: 50

This consultation is closed. The report and its annex are now available.

________________________________________________________________________

Mathematics is recognized today as essential and indispensable for addressing the major challenges in science, technology and society. Faced with the abundance of data on social, technical, economic, ecological, and technological systems new and sophisticated mathematical tools are required for these data to help us tackle pressing societal challenges and provide us with the necessary technological advantages. The “knowledge society” is “ignorant” without mathematics!

The consultation seeked to inform the future work programmes (2018-19-20) with innovative mathematical content. Two facts motivated this request: today’s digital society depends on mathematics and algorithms; there is a vast pool of mathematical talent in Europe. The conclusion is that Europe can be first in mathematical applications for big data, computing and especially HPC, to be first in modern science and innovation.

The potential of mathematics was already explored in a very successful online consultation which provided excellent food for thought for a workshop held in November 2014. The conclusions of that workshop were collected in a report (to link).

This consultation is closed. Submitted ideas and comments made can be found here.

"There are 10 types of people in this world: those who understand binary and those who don't." Anonymous

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Comments

Gernot Beer,

28/05/2016 02:54

One of the main problems of large scale simulation is the requirement to generate a mesh. As identified 10 years ago by Tom Hughes, mesh generation is one of the most time consuming parts of a simulation and also a main source of errors.
In an ideal world, data describing the geometry of the problem should be taken directly from a Computer Aided Design (CAD) program without the requirement to approximate it with a mesh. Furthermore, most simulation programs still use volume discretization and very basic functions such as Lagrange polynomials to approximate the unknown values. Using simulation methods based on boundary discretization and more sophisticated functions such as NURBS can result in more accurate simulations with a significant reduction in the number of unknowns.
Currently there are some efforts made by small research groups around the world towards achieving the "holy grail" of simulation, i.e. one without the requirement of mesh generation. However this is a challenging task and requires a coordinated effort by engineers, software developers and -most importatly- mathematicians.

Anni HELLMAN,

25/05/2016 08:08

The list of suggested areas contains some very important topics many of which merit investment through H2020.

However there are some wider points which need to be recognised and taken into account in any proposed programme. Principal among these is the qualitative distinctions to be made between machine-based data analysis and human data analysis. Recent work has demonstrated the substantial difference in analysis outcomes achieved by machines and humans, resulting from greater human flexibility and capability in reduction as well as in data integration. This stands behind the use of citizen scientists as part of the data reduction process in Citizen Science projects.

The human mind can make the kinds of “leaps” that computers, irrespective of level of sophistication, currently cannot. Therefore it would seem appropriate to develop approaches that would address how machines might make similar inferences.

Humans are particularly good at integrating divers sources of data. For example a human can look at stock markets, satellite imaging, climatic/weather data, traffic flow in ports and on sea-lanes as well as other sources to produce predictors of future commodities prices. Such approaches often draw on learned knowledge or use inference processes drawn from similar or related fields.

This might be considered to be addressed by the inclusion of topological data analysis. However this is a field in its infancy. While it may – and this has yet to be adequately demonstrated – work well in certain fields, it is unclear whether or not it will work when integrating diverse datasources, still less in how it might handle the constraints imposed by prior knowledge/expertise in similar but different fields.

Therefore it is appropriate to argue for an extension of the coverage to include the investigation (by topological means or otherwise) of how diverse data can be integrated. Equally important, if not more so, is to understand why some cases are not amenable to the technologies suggested.

To be able to handle large-scale diverse datasets may necessitate the creation of radical new mathematical technologies, but the ability to handle large data in scenarios such as those outlined above would give Europe a world-leading position in data analysis and reductuion.

Broadly I would suggest that these be added to the list (although there is some overlap in several areas) with the proposed list:

1. Mathematics of complex dynamical networks:
We are starting to build complex dynamically changing structures whether viewed in the form of physical networks or networks of data or dataflows. At both large and small scales these structures will vary in time (for example of dataflows among a large fleet of driverless vehicles moving through an intelligent city, or other ultra-large scale integrated information systems). In such systems even local micro-reconfiguration may have a macro-impact. Within the next decade the number of nodes in the subsystems of such networks may reach substantially in excess of 10**9, yet we lack the mathematical underpinnings to handle such networks. We need exact techniques (probably the most difficult), approximation schemes, heuristics etc with appropriate domains of coverage to enable us to manage such networks. The internet offers examples of some heuristics, but not on the sorts of timescales that these systems could reconfigure. Such an approach calls for

2. Novel Mathematical Techniques for Exascale: In order to be able to address the types of engineering problems - and others - that are envisaged as being addressed by exascale computing we will require algorithms for the solution of very large scale matrix problems (>>10**6 ) that will scale over many orders of magnitude for use in areas such as FEA of order 10**12 and beyond for problems such as FEA, engineering simulations, general matrix problems. These should be implementable for exascale in standard libraries similar in style to the existing LAPACK/LINPACK.

With regard to Quantum Computing technologies; while these represent very worthwhile mathematical areas to be working in and pose a raft of new issues which FET must address, by the same token these are technologies which are neither widely applicable nor are they likely to be within the timescales of H2020. Further there are not quantum computing technologies (as distinct from quantum encryption techniques) that are sufficiently robust that they could be delivered to mainstream, or even research facility, IT departments within those timescales. Therefore, if realisability is a criterion and therefore return on investment for the European tax payer this area cannot be one of the highest priorities.

Jeffrey Johnson,

19/05/2016 11:54

Global Systems Science combines Policy Problems, Complex Systems Science , Policy Informatics, and Citizen Engagement. Mathematics is essential to Complex Systems Science and Policy Information in order to model the dynamics of complex social and physical systems, such models being the precursor to data structures and computation.

Global Systems Science is an essential way of providing real world problems for new mathematical research, situating that research in real world problems.

Jeffrey Johnson,

04/05/2016 10:53

The DYM-CS programme has identified the following loosely grouped and intersecting areas of mathematics as requiring further investment to support Europe’s R&D in ICT and other high added-value technologies:

• topology, algebraic topology, computation topology, topology of data, tensor algebra, persistent homology
• graph & network theory, complex networks, network dynamics, multigraphs, multiplex networks, hypernetworks, diffusive processes.
• statistical mechanics, multilevel Hierarchies of subsystems, self-organisation, dynamical organisation & evolution, game theory, evolutionary game theory. Cellular automata.
• dynamical systems and random dynamical system, symbolic and hypersymbolic dynamics
• intermittency and extreme events, cascades of failure. criticality, emergence, phase transitions, tipping points, multilevel dynamics. anticipatory & predictive actions of units,
• statistics, theory of statistical learning, deep learning, information geometry, information-driven emergence, information theory, persistent entropy, state space reduction, multiscale simulation
• category theory, topos theory

Jeffrey Johnson,

04/05/2016 10:53

Vassil Alexandrov,

02/05/2016 16:38

Novel mathematical methods and scalabale algorithms are key in producing robust applications that can help leverage upcoming exascale computing architectures. The justification, ideas and way forward are well described in the ETP4HPC Strategic Research Agenda -2015 update (http://www.etp4hpc.eu/). On the other hand we observe the Big Data explosion, and key aspects and the importance of mathematics are outlined in Frontiers in Massive Data Analysis, US National Research Council of the National Academies, Report 2013 . What we also observe is the increasing need to tackle and solve both Data Intensive and Compute Intensive problems. What I would like to stress also that recent experiments, for example, on the exascaling workshops in LRZ (Germany) have shown that employing novel mathematics and algorithms can lead to a substantial improvement in the performance of important applications thus clearly supporting the needs for novel mathematics for Exascale, Data Science and Digital Science overall.

Route to exascale: HPC systems continue to scale up in compute node and processor core count. These extreme-scale systems require novel mathematical methods to be developed that lead to scalable scientific algorithms to hide network and memory latency, have very high computation/communication overlap, have minimal communication, have fewer synchronization points. Scalable mathematical methods and corresponding scientific algorithms for multi-petaflop and exa-flop systems also need to be fault tolerant and fault resilient, since the probability of faults increases with scale. Resilience at the system software and at the algorithmic level is needed as a crosscutting and co-design effort. Finally, with the advent of heterogeneous compute nodes that employ standard processors as well as GPGPUs, mathematical methods developed and corresponding scientific algorithms need to match these architectures to extract the most performance. This includes different system-specific levels of parallelism as well as co-scheduling of computation. Key science as well as industrial applications require novel mathematics and mathematical models and system software that address the scalability and resilience challenges of current- and future-generation extreme-scale HPC systems. In more detail important research topics include:
- Develop novel hierarchical multilevel approaches for mathematical multiscale models for modeling variety of physical phenomena that employ that leads to design of hybrid methods enabling calculations with different precision at the different levels of hierarchy and leading to corresponding algorithms with high scalability matching the computer architectures hierarchies.
- hybrid (stochastic/deterministic and deterministic/deterministic) methods for variety of problems including Linear Algebra, Optimization etc. For example, these are Monte Carlo and quasi-Monte Carlo hybrid (stochastic/deterministic) methods as well as deterministic/deterministic ones. These are inherently parallel and are expected to achieve high scalability.
- Developing communication avoiding, falt-tolerant, resilient algorithms needed at exascale , the above mentioned approaches will greatly contribute to this.

Data Science: With the advent of Data Science in the past few years the need of such scalable mathematical methods and algorithms able to handle data and compute intensive applications at scale becomes even more important.

The need is for:
- Scalable methods and algorithms that enable to discover global properties of data ( we are talking here about network science methods, multi-objective and multi-constrained optimization, variety of classification methods, machine learning, etc.

Would like to stress the importance of achieving scalability at all level, starting from mathematical models level through algorithms level down to systems level, we call it “vertical integration and validation of mathematical methods and algorithm” in ETP4HPC.

There is certain commonality in the mathematical approaches above both in HPC and Computational Science in the route exascale and in Data Science and in particular in the case of Data and Compute Intensive science and these should be further exploited.

Marie Farge,

01/05/2016 22:38

Research is a collaborative endeavour between peers. It should not develop as a competition for ever larger and more selective grants, as already denounced by Tim Gowers, Albert Cohen and others in previous posts. Since our research topics are highly specialised and are we are very few on a given problem, we could not afford competing between us. Indeed, to improve our ideas we share them with our peers, through discussions, seminars, workshops, long-term programs and peer-reviewed papers. The European Commission should help us to maintain such a high-level of collaboration by supporting infrastructures where we meet and discuss new ideas (e.g., Newton Institute in Cambridge, MFO in Oberwolfach, IHP in Paris…). For instance the CIRM in Marseille has played an essential role for the emergence of wavelets in the mid-80s, by offering us the possibility to organise, on short notice and without administrative burden, informal meetings between researchers from different disciplines and countries. Today, thanks to publicly-funded electronic infrastructures scientists collaborate through the Web, but they still need to meet and intensively work together for one or two weeks. I think the more international research teams will develop, the more intensive meetings in dedicated places will become necessary.

We also need another kind of infrastructures to develop open access to publications. We appreciate the ‘green open access’ policy of the European Commission, asking us to deposit our papers in an open repository (e.g., arXiv). In contrast, most of us refuse the ‘gold open access’ model, where we (or our institution) pay APCs (Article Processing Charges) to publishers, because they will thus keep their control of the publication system, by still owning the journals and fixing the price of APCs (as they are doing for subscriptions). In 2012 our group 'The Cost of Knowledge' proposed an alternative model, called ‘Diamond Open Access’, where neither readers nor authors have to pay. It is based on three principles:
– authors keep their copyrights and publish papers under the Creative Commons license CC-BY,
– editorial boards own the journal (title, peer-reviewing reports and other assets) for which they are responsible of the peer-reviewing task, they can then choose the publisher or publishing platform they prefer (publishers are service providers but not journal owners),
– if a journal is recognised to be useful to the scientific community and as long as its editorial board can prove good peer-reviewing practices, it could be published for free using editorial and publishing platforms, which are publicly-owned and publicly-funded infrastructures using open source software designed to service a very large number of journals from different disciplines (as the major publishers do, e.g., Elsevier Editorial System).
For many years publishers have already asked us to use their platforms and thus all our exchanged for peer-reviewing belong to them. We think that our research papers, our scientific journals and our peer-reviewing exchanges should no more be privately-owned by publishers. The Open Science Cloud proposed by the European Commission should provide us such editorial and publishing platforms, but publicly-owned and written in open source, with commercial publishers as service providers.

Concerning the funding of researchers I fully agree with Tim’s remark: ’Instead of forcing mathematicians to fit their projects artificially into a larger whole, you should trust us to judge for ourselves what we need and support that.’ You should a priori trust us and only a posteriori check if we have adequately used what we asked for. You should also avoid supporting someone who has already several grants and who might hire more PhDs and postdocs than s/he has time to work with. Last point, you should not forget that research requires concentration and time to understand abstract ideas and produce new ones. Unfortunately the present situation is counter-productive since we are overwhelmed by financial and administrative tasks, for which most of us have no special talent. We are asked to respect strict deadlines to apply for research grants or to send reports when we peer-review them. The perverse effect is that the system imposes on us to give priority to fund raising, peer-reviewing and administration, while there are no deadlines to do research, discuss with students and colleagues, write papers and books… This situation, which is the same worldwide, is becoming critical.

Susanne Ditlevsen,

01/05/2016 12:52

On behalf of the board of the European Society of Mathematical and Theoretical Biology, we suggest research themes in Mathematical Biology, where applied mathematicians, statisticians and probabilists can contribute. Mathematics and statistics are playing increasingly important roles in the biological sciences and contribute today at all levels from design of experiments to analysis of data, and exploration of models and their properties. It is becoming clear that most complex systems are highly nonlinear and act in non-intuitive ways, and that the causal mechanisms cannot be understood without mathematical models and their analysis. There are lots of interesting research going on in Europe within Mathematical Biology and it has increased considerably in the last couple of years. Specific calls for projects within this vast and important field are needed.

Greet Bilsen,

01/05/2016 12:12

Next to information-theoretic security and privacy, there is also a continuing interest in provable security for realistic environments. There is a growing demand for security products with a certified and significant security level. The higher certification levels require mathematical proofs.

In the area of secure implementations, there is a growing recognition for the fact that even schemes that are information-theoretically secure on paper, may be insecure in practice because common computation platforms leak information in the form of radiation, execution time, etc. In order to obtain real-world security, we need to adapt the mathematical security models in order to include implementation aspects.

Furthermore, in the Internet of Things, the secure generation of true randomness turns out the be a problem: cheap small devices operating on harvested power or batteries cannot be relied upon the generate true randomness in a hostile environment. Hence there will continue to be a strong need for cryptography based on methods relying on pseudo-randomness. Fields like number theory, coding theory and other areas of discrete mathematics can be
used with succes here.

(comment of behalf of Prof. Vincent Rijmen - KU Leuven)

Jiří Rákosník,

01/05/2016 10:07

Many thanks to Anni Hellman for arranging this important discussion, mathematics is lucky to have such enlightened person in the EC structures. I endorse particularly those contributions presenting broader, fundamental views. Even though it might be out of scope of this discussion, I want to remind what has been mentioned already in the previous consultation: the need of a specific infrastructure for mathematics. The term „infrastructure“ appeared four times in the whole discussion, however, in different connections. Mathematics never becomes obsolete, what was true yesterday, remains true today. Mathematical knowledge is a magnificent ever growing edifice in which each building stone has important role. In certain sense, this is kind of big data requiring a specific attitude. We need an infrastructure which will enable not only create new knowledge but also preserve the existing one and provide a sophisticated access to it. This is currently dependant on commercial activities and on rather dispersed public initiatives like the database zbMATH, the EuDML, and number of networks and projects managing the mathematical knowledge. A reasonable support is needed to build upon these efforts and create a stable infrastructure for mathematics which will remain essentially under public control.

Susanne Ditlevsen,

01/05/2016 07:40

Mathematical modeling including stochastic effects in biology has huge potentials for improving our understanding of the studied systems. There is a long tradition of studying average behavior of the systems, but it is essential to also include the stochastic behavior, which might have huge impacts on the dynamics, especially in nonlinear systems. Noise is ubiquitous in all living systems, but our understanding is still limited.

Tanja Lange,

01/05/2016 04:59

Glad to see funding support for mathematics by the EU. I think the proposed topics are highly relevant.

Volker Bach,

30/04/2016 22:12

Being the president of the german mathematical society, DMV, I would
like to draw the attention to the "Rapport de Prospective" of the
french "Comité national de la Recherche scientifique" available
under

http://www.cnrs.fr/comitenational/doc/rapport/2014/Rapport_prospective_C...

This report gives a well-balanced and up-to-date (2014) description
of the current research trends in mathematics and their importance
for other sciences, technology or society at large, as well as, the
impact development outside of mathematics has on modern mathematical
research.

Jean Roman,

30/04/2016 20:50

For scientific and engineering computing, exascale is the next step in the long road of exponential performance increase that has continued since more than 50 years. High performance computing advances have been largely dependent on research works on numerical algorithms, software development, architecture design that allowed higher performance for computational models and applications. Today and in the future, advances will strongly depend of new research on data analysis, machine learning tools and on data centric architectures.

Very important research topics (both from applied mathematics and computer science) are identified in order to achieve scientific breakthroughs using exascale systems for challenging scientific and societal problems using extreme amount of data :
⁃ mathematical modelling including multiple criteria optimization, control and uncertainty quantification issues for multi-scale and multi-physics complex problems
⁃ scalable model discretization combining continuous and stochastic approaches leading to new algorithmic approaches including scalable code coupling, parallelization-in-time, model reduction for high-dimensional problems, adaptativity and scalable mesh generation
⁃ highly scalable solvers and numerical libraries
⁃ mathematically well-designed methods and algorithms for data analytics, detection of structures and correlation in big data, statistical approaches
⁃ high performance algorithms for machine learning
⁃ hierarchical algorithms exploiting all the levels of parallelism of the platforms (from very fine to coarse grain of parallelism)
⁃ resilience of algorithms including reproducibility, verification and validation issues
⁃ data intensive computing platforms (big data management coupled with data analytics pipelines both included in the numerical simulation process)
⁃ architecture-aware algorithms and software for exascale computing including programming methodologies, environments and tools, runtime systems allowing the dynamic execution of programs with a very large number of threads (DAG scheduling issue with low communication and synchronization overhead), minimization of data movements for energy consumption issues.

Several of these topics have been identified in the report from the « Workshop for Mathematics and Digital Sciences published in March 2015 (https://ec.europa.eu/digital-single-market/news/workshop-mathematics-and...). This report points out the importance of mathematical support on one hand to HPC development towards exascale, and on the other hand to mathematical solutions for pattern recognition and simulation on big data.
Most of these research topics at studied at Inria which is a public science and technology institution supervised by the French ministries for research and industry with the missions to produce outstanding research in the computing and mathematical fields of digital sciences and to ensure their impacts on the economy and society.

Marco Secondini,

30/04/2016 15:51

The nonlinear Fourier transform can help to develop a new generation of optical fiber systems

Keywords: nonlinear Fourier transform, inverse scattering, optical fiber communications, nonlinear channels, integrable nonlinear equations.

Optical fiber communication systems form the backbone of the world’s communication infrastructure and provide for most of the global data traffic. However, the exponential growth of network traffic is pushing current technology towards its limits. In fact, the maximum rate at which information can be reliably transmitted through the optical fiber is limited by fiber nonlinearity. Apparently, a capacity crunch is looming on the future of optical networks.

Communication engineers are striving to solve this big issue, trying to devise novel transmission techniques that are suitable to achieve reliable communications over highly nonlinear channels. Recently, the nonlinear Fourier transform (NFT), also known as the inverse scattering transform, has been recognized as a fundamental tool to solve this problem. As the (ordinary) Fourier analysis underlies the great development of communication systems occurred during the second half of the twentieth century, the NFT could be the basis to develop a whole new theory of communications over nonlinear channels.

The NFT is expected to eventually replace its linear predecessor, which has emerged from wired and wireless transmission principles and reigned in the communication culture for over a century, in fiber-optic communications. However, though the mathematical theory of the NFT has been well elaborated, the adaptation of nonlinear structural elements in optical transmission systems in place of their linear counterparts and the corresponding “change of paradigm” require serious reconsideration of the communication system's architecture at every scale. For instance, we need fast NFT algorithms for the direct and inverse transforms (something similar to the FFT algorithm), we need to develop modulation and detection techniques that operate in the nonlinear spectral domain, we need to devise optoelectronic devices that can process the nonlinear spectrum, we need to understand how noise affect the nonlinear spectrum, and we need to extend many classical results of information and communication theory (e.g., the noisy channel coding theorem) to this new class of channels.

In order to achieve such an ambitious goal, mathematicians and engineers should work together in an interdisciplinary approach, addressing fundamental theoretical problems, developing new algorithms and techniques, and eventually experimentally demonstrating a new generation of optical fiber communication systems.

Benjamin Himpel,

30/04/2016 12:34

TWT GmbH Science & Innovation develops and applies new mathematical methods towards problems in IT, Engineering and Consulting. Here are some topics, that we are interested in:

Big Data in Engineering

The automotive industry produces a vast amount of data. Either they do not use it efficiently, or they do not know what to do with it and cannot obtain any information from it. Clearly, these problems will grow and spread, if they are not addressed appropriately. Therefore, they provide the opportunity and necessity for new mathematical methods. We propose to take a close look at a variety of use cases together with research institutes and partners from the automotive, aerospace and health care industry, and develop the necessary mathematics for these cases. There are many different approaches to handle big data, but which methods you choose to pursue and combine, or if you have to develop a new one, depends on the use cases at hand. Therefore, it is necessary to connect experts and stake holders from different subfields. Mathematics is key for creating new methods which are general enough to ensure sustainability.

There are numerous new methods to study big data. Most of these methods are assume a simple data structure. However, most data in engineering is too complex for standard methods. Engineering Know-How is essential to understand different components and develop the best data mining methods. For example, we would like to test and further develop Numerical Tensor Calculation by applying it to test bays in the automotive industry and the simulation of acoustics in the interior of vehicles. Data structures (text, video, audio, etc.) fromdozens of sensors need to be analyzed and used in order to control different parts of the car.

Safety and Security for the internet of things, the connected car and autonomous vehicles: Despite impressive developments in these areas, robust and new mathematics is necessary in order to analyze existing methods, improve them or find alternate ways to ensure safety and security in these new technologies. Big Data certainly plays a big role in this, but also other areas of mathematics (e.g. encryption and data fusion). A collaboration of mathematicians, computer scientists, engineers and industrial partners will be important to make new technology safe and secure.

CAD/CAE in Engineering

CAD/CAE has a long history in design and simulation in the industry. Despite a lot of advances, the basic process has not changed much. In particular, the design and simulation still uses different structures, even though there has been a lot of effort to merge the processes and data structures.

Isogeometric Analysis takes a prominent role in the effort to combine design and simulation: This subject is well-established from a mathematical point of view. It allows for Finite Element Analysis to be integrated in conventional NURBS-based CAD tools, which makes simulations faster and more accurate. We would like to help make the paradigm shift in collaboration with research institutes as well as partners from the automotive, aerospace and health care industry.

Ruth Kaufman,

30/04/2016 01:12

1) Decision making beyond big data: most research on big data up to
now has focussed on classification (recognition, learning,
clustering,...) and inference. There is a need to go beyond these
efforts and develop efficient algorithms/approaches for decision
making and optimisation on big data sets.

This area would fit between or alongside the "Topological data
analysis" and HPC.

This is an area where the OR community could make a substantial contribution.

2) There is scope for looking at how decision making under risk and uncertainty, classical components of Operational Research, could be extended to contribute far more than at present to crime prevention, climate change, migration, and defence.

Rolf Krause,

29/04/2016 12:08

The computational power of future exascale-computers will be made available through massive parallelism. The shear amount of computing kernels (cores, nodes) opens new perspectives for the development of simulation and optimization methods for large scale problems.
In addition to classical deterministic methods, such as iterative methods for linear and non-linear systems, new classes of methods can be developed,
which combine ideas from numerics, numerical analysis, high-performance computing, probability, and inference. This approach might help not only to deal efficiently with
massive parallelism. It might also help to simplify simulation and optimization methods and can lead to a better integration of existing data into simulation runs (data driven simulation). Developing and anlysing these methods will require substantial mathematical effort.

Heleen Wortelboer,

29/04/2016 10:54

Expression of interest by Dr Heleen Wortelboer, TNO, The Netherlands

Key words: Complexity, Health, Dynamical system theory, Applied logic, Category theory, Information mining, and computer science

EU wide projects offer a chance to construct multi-country teams of researchers which can study important societal challenges from a European perspective. Providing affordable healthcare for EU citizens is a major challenge, especially due to the continuous rise of chronic complex life style diseases such as obesity, diabetes type 2, cardiovascular problems, and dementia. To solve this problem, new insights and hypotheses are urgently needed where mathematics and complexity science can be of beneficial use.

Personal health

Personal health can be identified as a complex adaptive system, in which multiple health-related nested systems (physiological, mental, societal, spiritual, cultural) are involved. Studying health from a complexity perspective across Europe via EU wide programs offers the opportunity to integrate system levels from small scale to large scale: biology, psychology, behavior, family, work environment, society, country, EU.

We foresee that a trinity of mathematics-logic-computer science is needed. Due to the interdisciplinary nature of this problem lexical ambiguity within the different semantic fields or knowledge domains can arise. Hence what is needed is a perspective on the correspondence of the concepts within the different contexts of the knowledge domains or disciplines. An area suitable to address this segment of the problem is applied logic. Next, to successfully translate the abstract concepts developed from the results of applied logic to more suitable structures computational science is a necessity to deal with processes of processes (models of models) and defining analogies and equivalences in more precise ways, and for this category theory could be a suitable candidate. Beside the more applied computer-science described above, theoretical computer science is probably also needed because of software architecture.

Community empowerment and Citizen science

As an organization of applied research, TNO has the ambition to deliver applications that are of immediate benefit to society. Besides the importance of mathematics on new insights on individual health as described above, health can only improve if people are empowered to become aware of their health, are able to monitor their health, experiment with the effects of life-style changes on health and get feedback on such experiments. To support this, mathematical algorithms and tools should use open standards, should be accessible by Health Data Cooperatives, the EU Open Science Cloud, and should be able to use existing available health data.

Mathematics could also support the development of an EU wide citizen science community. Citizen science projects have shown a huge capacity to involve citizens in research. Citizen science related to health could be a fruitful way to raise health awareness and stimulate healthy behavior changes. Furthermore, a citizen science community can accumulate huge amounts of data and interpret results from algorithms. Additionally, citizens can provide real life data that is actually relevant to the individual. Mathematical algorithms need to be developed that can deal with this real life, often heterogeneous data, captured with multiple devices such as cell phone apps, trackers and in patient records. To improve community empowerment and citizen science we would encourage the requirement of a patient or citizen organization to participate as a recognized partner in H2020 proposals.

The approach indicated could have immediate impact of mathematics and complexity science on the health of EU citizens.

Anni HELLMAN,

29/04/2016 10:52

Dear fellow mathematicians,

this consultation on mathematics for Horizon2020 has proven a great success and it also brings the message, loud and clear, that the mathematicians in Europe - and wider - have a lot to offer to science and innovation. The response has exceeded all our expectations both in number and in quality of contributions and contributors!

We will analyse the full content at the end of the consultation - we have now given a few extra days for the consultation. A report will be drafted outlining the main conclusions, and highlighting the ideas that have been included in the report. The report will be made available on this platform.
It should be used widely, to highlight the importance of collaboration across borders between sciences and between mathematics.

The report will be presented within the Commission in the context of the drafting process of the Horizon2020 Work programmes for the period 2018-2019-2020.

Due to the quality and quantity of responses to this consultation, we will have a session on mathematics in the "ICT Proposers Day" on 26-27 September in Bratislava. The aim of the Proposers Day is to inform and bring together potential proposers for the upcoming Work programme calls - more information will be available on the Futurium platform in May.
The session on mathematics seeks to bring together mathematicians and potential proposers for topics where mathematics are needed, such as those relating to simulation, modelling, requiring creation of algorithms or big data analytics. We will publish on this platform a blog with some further information regarding this also in May.

There are several areas of mathematics indicated as areas for future development, and some very interesting mathematical tools and methodologies are proposed.
Very importantly, in many contributions the working across borders is becoming key: between mathematics and other fields such as biology, chemistry, physics, finance, as well as within mathematics a collaboration between different mathematical fields will be required for future needs.

Mathematics does not cease to bring surprises: for example topology was deemed a rather theoretical subject years ago when I studied it. Now it can be key to big data analysis. There are some interesting contributions on this.

It is clear that to tackle the big data challenge and to be able to extract that wealth of information that lies within, new ways to employ existing mathematics, and even new fields of mathematics may be required. New methods to sample and extract the necessary data without losing key information are needed.

HPC and its development towards exa-scale, not to speak of quantum computing, are fields where mathematics plays a key role.

Modelling, simulation and optimisation (MSO) has proven to be a field of crucial importance. MSO is hugely important to the industrial processes for manufacturing, from aviation and automobile industry to fashion design and life sciences' industry - and it relates closely to big data and implies serious computing power requirements, hence a link to HPC.

Biomathematics is present very strongly with some excellent contributions. As I have heard a fellow mathematician say, "biology is the new physics". We believe that the role of mathematics will grow strongly in this field.

There are other, equally important fields touched upon in the contributions.

We will analyse this all. We aim to publish the report by end of June on this platform.

Thank you to all contributors!

Anni HELLMAN,

29/04/2016 08:55

Comment by Dr Luis Alvarez Esteban, University of Turku, Finland

The global financial markets have experienced an enormous increase in complexity over the past decades. Both the amount as well as the variety of market participants has more or less exploded and the range of different available financial products has increased significantly. This trend persists regardless of the recent financial crisis and its consequences. In light of this observation, it is clear that there is a need for mathematical approaches that can take into account the multiple sources of risk and uncertainty that this complexity calls for: Namely, credit and liquidity factors (credit valuation adjustment, funding valuation adjustment, counter-party risk, etc.), interest rates and inflation (multiple-curve models), and demographic factors such as longevity risk. As models incorporating these driving components become increasingly complex, their analysis becomes computationally more challenging and call for more efficient numerical techniques based on the recent developments in computational mathematics and mathematical statistics.

Anni HELLMAN,

29/04/2016 08:57

Comment by Dr Mika Hirvensalo, University of Turku, Department of mathematics and statistics

Communication security addresses most European citizens in an invisible way: Commercial banks, credit institutions, and companies such as PayPal provide electronic transactions channels whose communication security should be established in a reliable manner: It is highly undesirable that a third party could interfere with a confidential transaction and potentially obtain information or alter the messages between communicating parties. Until now, classical information and complexity theory has been used to develop protocols providing security against eavesdropping and interference. However, in the beginning of quantum computing era, those solutions may become powerless.

Theoretical investigations on quantum computing have appeared very interesting and promising: The quantum nature of the information sometimes offers computational shortcuts leading to very efficient solutions for tasks which seem very arduous for computers handling classical (non-quantum) information only.

For many purposes, an improvement of the information handling capacity is usually seen as a benefit, but this is not necessarily the case when protecting the information. In fact, improved computation power may be used to break a code used for secure communication. This is actually the case in all currently used asymmetric encryption procedures, which do not require a meeting of communicating parties prior to communication: An owner of a quantum computer could be able use efficient quantum algorithms to decipher the messages in a reasonable time, whereas such a fast deciphering procedure is not known for classical (non-quantum) computers.

However, the technological development towards a quantum computer has been a difficult task, indeed. According to Washington Post January 2014 article, NSA has invested approximately 80 million dollars to develop a quantum computer for breaking codes. Canadian company D-Wave Systems founded in 1999 has developed technical solutions to implement quantum computing, and as of August 2015, offers devices for quantum computing capable of handling more than 1000 quantum bits. On the other hand, no-one, including D-Wave Systems itself, claims that their solution is an all-purpose quantum computer. Rather on contrary, it seems that the technological solutions of D-Wave Systems are too limited to implement quantum algorithms for fast deciphering of the aforesaid communication protocols. However, existing research is not capable to properly estimate the computational power of limited quantum computing architecture used by D-Wave Systems, and hence the potential security threat caused by that type of quantum computers is not known well enough.

Quantum computing with an all-purpose instruction set is currently much more limited. The number of quantum bits technologically available seems not to exceed 15, and this limited amount of information storage is certainly not sufficient to endanger currently used asymmetric security protocols. On the other hand, any two-state quantum physical system can at least in principle serve as an implementation of quantum bit, and therefore the technological development towards a large-scale all-purpose quantum computer remains eminently unpredictable.

Nevertheless, in a due course currently used security protocols may be eventually threatened by quantum computers and need to be replaced with protocols believed to be resistant against a quantum computer attack. Currently however those protocols bear a burden of relatively large encryption key sizes. It may also be emphasized that the quantum-resistance of the proposed protocols is merely an assumption, but so is also the resistance of currently used protocols against classical computing.

Another possibility to improve the communication security is to use quantum information instead of classical already for the communication. Swiss company ID Quantique offers commercial solutions for quantum cryptography, utilizing security provided by the nature of quantum information. However, it is not very straightforward to start using these protocols everywhere, as they technically require a contiguous light fiber between communicating parties to ensure the security. Moreover, the security of the quantum protocol is known for the ideal version, but the practical resolutions are with no exception stressed with imperfections.

To ensure the communication security now and in the future, the computational power of D-Wave type quantum computers should be studied thoroughly, protocols believed to resist quantum attackers should be studied and developed further, and also the imperfect realizations of the protocols utilizing quantum information should be investigated further. The aforesaid type of research typically requires interaction of mathematicians, physicists, and computer scientists.

Mika Hirvensalo, University of Turku, Department of mathematics and statistics

Ton Jorg,

28/04/2016 16:57

Expression of Interest by Ton Jörg 27 April 2016
Complexity poses a fundamental and foundational question for science in general. Not many scholars of complexity seem aware of the fundamental and foundational consequences of complexity for science in general, with some exceptions like e.g. Helga Nowotny (2013, 2016) and Brian Arthur (2013, 2015). That’s why there is still no general theory of complexity, neither an adequate science of complexity. Too many scholars operate in their own discipline, disregarding the transdisciplinary nature of complexity. They do not even seem to miss such a general theory of complexity or a science of complexity.
In my view complexity poses the question of the need for a radically altered account of reality itself (Kauffman, 2009). The focus, then, is on replacing science-as-we-know-it into a new science: one that fully incorporates complexity into a new, unexpected worldview. The new science is about the generative, emergent nature of complexity, as being linked to a fundamental generative nature of order (Bohm & Peat, 2000).
What is needed is a new generative foundation of science that describes the dynamics of complexity, operating as generative, emergent complexity. A science that recognizes complexity as potentially self-potentiating (Rescher, 1998) and self-propagating (Arthur, 2013, 2015). For this new foundation, we need to replace the paradigm in use into a new paradigm: the paradigm of self-generative, emergent, and self-potentiating complexity. The scientific realm of study, then, is not about fixed entities but about complexly generative, dynamically evolving entities (Arthur, 2015). To study this as a subject of study, we urgently need a new verb-based science (Arthur, 2015), to become able to study the very complex dynamics of the entities we study, like economy (Arthur, 2015).
With the new science it will be possible to open the sciences (Wallerstein et al., 1996), and to open new spaces of possibility in a new ‘world of the possible’, hitherto unknown (Kauffman, 1993, p. 375).
The fundamental challenge for the so-called “Age of Complexity” is to recognize the hitherto still unknown complexity of reality and to become transdisciplinary in developing a new generative complexity approach. This demands for a radical change of thinking: of thinking in complexity about the unknown complexity of the real world as a complex world.
We do need a new language of complexity, with a new vocabulary, to describe what we’re actually studying (Mitchell, 2011): the still very unknown complexity of reality, with its unknown territories.
We urgently need to replace the notion of ‘Grip on Complexity’ into the notion of “navigating the sea of complexity” by taking the real world as a complex world of the possible. Uncertainty and complexity should not be taken as unfriendly concepts anymore but as potentialities of opening new spaces and unknown territories, all for the sake of shaping the future in the best way possible (Scheffer, 2009). This is what a practical science of complexity, about transitions and transformations in the real world, might actually be about (Vygotsky, 1978, 1987; Scheffer, 2009). We may become able to navigate these transitions and transformations within a new world of the possible. This navigating may be linked to a new, complexity-inspired design approach of Imagineering (Nijs, 2014, 2015). This new approach may open the wondrous world of the possible for education (Jörg, 2016a).
We may get aware that, living in the Age of Complexity, we are actually living in a transition stage (Nowotny, 2013), to be taken as a pre-revolutionary phase. In opening the sciences for complexity, we are at the edge of a scientific revolution indeed (Jörg, 2016b). Complexity, then, should not be taken as the problem for science but as the solution. This is what a new paradigm is really about.

Pavel Exner,

28/04/2016 11:00

I feel that I should add a few words in my position of the European Mathematical Society President before this consultation closes. To begin with, I want to thank the European Commission, and in the first place to Anni Hellman personally for opening this forum at which ideas and concerns about the European support of mathematics can be voiced.

The discussion brought in many specific topics worth to be pursued, from the expected ones like the big data to some far less obvious. I will not comment on them, however, and I focus on a couple of general issues.

One recurrent topic in the discussion was the size of grants, in particular, in connection with the European Research Council. As a member of the ERC Founding Scientific Council, and the only mathematician there, I always argued there that the half a million lower limit is too big for any mathematician. I was overruled, however, and looking now on what people ask for and get, you find that the average size of a mathematical grant is 0.8 million in the junior categories, and 1.3 in the advanced one, below the ERC average but way above the said lower limit.

If somebody comes here to the conclusion that ERC is not a proper tool to finance mathematics I can only say that such a point of view is fundamentally wrong. One of the sources of the ERC strength comes from the fact that all fields of science, from mathematics to history of arts, speak the same voice and share a common understanding what an excellent research is. The ERC has to be supported and helped to resume the dynamics it had during the Seventh Framework Programme, possibly on a different and more permanent institutional basis which would better shield it from political influences and administrative hurdles. Only is such a way it can become in a long run an equal partner to the institutions such as the NSF in the United States.

What is painfully missing is a “mycelium” from which ERC results can grow, a palette of smaller funding opportunities for mathematicians with the potential to become visible personalities, temporary positions, small- and medium-scale collaborations, research conferences, etc. A part of that was once covered by the now defunct European Science Foundation the demise of which left a palpable hole. It may happen that one day this would be covered by the ERC in a way the NSF does it, but it is a long shot; one should not forget that the latter is almost ten times older than the former.

This concerns, in particular, the Marie Curie Sklodowska Programme which some of the participants of the discussion mentioned. It would certainly fit together with the ERC and other funding schemes to create a unique system covering the whole professional life of a mathematician, from the student age to the retirement, however, one serious obstacle is the fact that they belong to the portfolio of different Brussels “ministries”.

I have lauded Anni Hellman and her colleagues for opening this consultation which I see as a nontrivial act of wisdom. Sad to say, the democratic mechanism are not automatically supportive to long term vision projects; if you need an example, recall the last year attempt to cut the ERC and MSC budgets in the name of boosting the economic growth here and now. And it is our duty to repeat that strong science, and mathematics in particular, is crucial for the future of the European society.

The point is the atmosphere in the society which in a sense could be called schizophrenic.
Many of our fellow citizens enjoy results of scientific research at every angle and at the same time they despise the tools. It would socially destructive to acknowledge publicly that you fail to understand basic rules of grammar, but we see scores of people saying with a pride: “I have never mastered a simple proportion and look how big I became”. Well, you are not surprised when it comes from sports or pop music stars, but sometimes you hear such things from people who feature themselves as political philosophers.

With all that in mind, the European mathematical community is prepared for a permanent dialogue with the European politicians, being convinced that a well devised support of science, and mathematics in particular, is the best way to the prosperous society we all wish at our continent.

Ronald Rösch,

27/04/2016 22:07

"Sparse" data:
Image data sets get larger and larger, a 3D image generated by micro computed tomography can easily be larger than 20GB. Often, the major part of the processing is spend on finding interesting regions - inhomogeneities, defects, particular structural features. Strategies for detecting such regions automatically are needed.

Space-time series:
In-situ experiments produce sequences of spatial data. While imaging and reconstruction methods are improved continuously, quantitative analysis of the resulting data lags behind. Registration methods being able to cope with large distortions/sudden structural changes are needed. Moreover, special tinme-space characteristics and methods for measuring them based on image data have to be developed.

Analysis and modelling of multiscale structures:
Geometric modelling of materials microstrcutures is a prerequisite for structural optimization based on the real materiual. For multiscale structures, strategies for integration of structural information from several scales into one geometric model are missing. Moreover, analysis methods and models for structures (e.g. biological), where the scales are not clearly separated are needed.

Real time algorithms:
Depending on the algorithm, the computing time for series of 2d or 3d images can be enormous. In many applications, however, time is limited: By the cycle time for inline inspection systems in production or by the impatience of the users. A new generation of algorithms adapting automatically to the available hardware (optimization & parallelization & new numerics) is needed. Additionally, methods for estimating the needed computing time in advance have to be developed.

Image analysis with integrated simulation of physical properties:
Numerical simulation of macroscopic physical properties based on image data (e.g. visual, xray, ...) of the microstructure and upscaling has been established during the last years. Applications are mainly in materials science. To make an impact in production, too, this prediction of physical properties should be possible with minimal human interaction, only. The quality (e.g. stiffness, damaging, ...) of each produced single component could be evaluated individually: ok or waste, and/or each individual part could then be applied, where its properties fit best.

Ruben Sanchez-Garcia,

27/04/2016 20:09

I would like to stress the need for translational research in mathematics. We have researchers thinking 'inside' and 'outside the box', but we also need mathematicians 'running between the boxes'.

This analogy goes beyond the pure/applied divide, as in fact some pure mathematicians are extremely good at making connections between different fields or applications, and some methods in applied maths can become routine and lack innovation.

I would still say that real-world applications, normally the starting point in applied research, remain a key motivation for ‘running between the boxes’. This can include tackling major challenges (‘hot topics’ if you wish), such as Big Data. I don’t think that motivation from overarching themes is necessarily a bad idea, in particular if it encourages translational research.

In my opinion, to realise translational mathematics we need a combined effort from experts to reach out and communicate effectively beyond their fields (this is a big issue in pure maths), and opportunities for interaction, including with potential end users. This is precisely the right context for large EU grants.

If I could add a final comment about Topological Data Analysis (TDA); I agree with previous comments that topology does not have all the answers, but it is an exciting new avenue and I am hoping it can be successfully integrated with statistical and machine learning techniques, and extended to more geometrical methods.

Enrico Scalas,

27/04/2016 19:39

I am glad to see that several of my concerns on the role of mathematics in H2020 have already been addressed by other mathematicians and I am particularly glad to see that some of them are from the UK.

In particular, I liked the replies by JF Toland, Pavel Drabek and Timothy Gowers.

Within the Maths community, it is commonplace that there is virtually no space for mathematics in H2020, except for a few funding programs such as ERC grants (that give large amounts of money to a very small number of mathematicians), Marie Curie (essentially the only reasonable funding scheme for mathematicians) and, potentially, FET Open (now with success rates well below 3% and close to 1%).

Even if it may sound surprising to applied scientists and politicians, mathematicians have their own research agenda and this is often "orthogonal" to the "top-down" (i.e. scientific research themes decided by politicians) agenda of H2020. The mathematics of today will help inform the applications of tomorrow and the "top-down" approach to scientific research has never been a good idea. Moreover, we do not know what kind of mathematics will be needed in the future and, by funding only a tiny subset of mathematicians, we can eventually kill many promising and potentially useful research programs.

As mentioned by other colleagues, Mathematics will not die if not funded by the EU. However, there is a simple set of actions that could be immediately taken and could become the norm in the next framework program and would help in engaging mathematicians more in European research frameworks. I try to list them below.

1. ERC: Instead of awarding huge grants to a small number of individuals, the ERC could consider awarding individual grants on the order of 100,000 up to a maximum of 500,000 euros for for periods from two to five years. With the same amount of money used now, they could help instead of 1, up to 25 mathematicians. I am also expecting that this simple action would boost the outcome of ERC-funded research in Mathematics.

2. Other H2020 funding programs: When appropriate, the participation of one or two groups in departments of mathematics and/or statistics could become an element of positive evaluation (similar to the participation of companies). For instance, groups of statisticians are particularly needed in applied proposals where large data sets will be collected (for rigorous design of experiments) and evaluated (for sound data analyses), but this is just an example among many I could give.

Enrico Scalas
Professor of Statistics and Probability
Head of Department, Mathematics
University of Sussex

Dan Kral,

29/04/2016 16:58

I fully share the views expressed in the reply of Pavel Exner to this post. A "big grant" may mean something else for each of us. When I joined the debate started by the post of Timothy Gowers, I had in mind grants in the order of many millions EUR for large consortia, which I indeed believe not to be appropriate for mathematics (but they might be appropriate in other areas). However, 100K EUR for a 2-year grant is clearly insufficient if that grant should support even a small group consisting of a single postdoc and a single PhD student. If mathematicians should have time for doing research rather than applying all the time to get funding for a postdoc, a student, etc. separately, the amounts mentioned in Pavel Exner's reply seem in the right range to support the PI together with say 2 postdocs and 2-3 PhD students for 5 years. I believe that the idea is that an ERC grant will be the main source of funding for the PI during the grant period to allow the PI to focus on research (similar for example to the RS URF scheme in the UK), so the PI does not need to apply for any additional funding during the grant period. Marie Curie is a good way of supporting young mathematicians - still, I have seen many strong mathematicians accepting an offer for a postdoc in US/Canada rather than waiting for the results of Marie Curie and risking the offer to be gone. Having postdoc funding on the ERC grants allows European scientists to compete on the global market for top postdocs.

I understand that the budget is and will always be limited. I believe it is better for mathematics (and in fact, any science) in the long term to support top research in a sufficient way (and I mean sufficient, not irrationally large, e.g., multipartner international consortia in mathematics) rather than to support everybody little (in a way not sufficient to do too much). For example, I think that most of UK-based mathematicians would agree that DTCs of roughly half the current size would be much more appropriate for mathematics but DTCs five times smaller might be too small. Similarly, I can imagine ERC grant sizes in mathematics to be 2/3 of the current sizes and still be sufficient for their purpose (but the amounts should be such that the grants can still serve their purpose). Having said all these, there should always be a large amount of diversity in the topics of research supported to keep all areas alive. Because of this, I fear that identifying key research themes is dangerous as the areas that will not be identified might lack funding. So, not surprisingly we can see posts from all different areas of mathematics in this discussion.

Luis Vega,

27/04/2016 18:23

From the point of view of BCAM – Basque Center for Applied Mathematics, the future H2020 work programmes (2018, 2019 and 2020) should address the following two key aspects.

On one hand, BCAM encourages the adoption of mathematical Modelling, Simulation and Optimization (MSO) as a transversal and rich research area. In particular, we would like to support the statements of F. Planchon, Maria J. Esteban and Volker Mehrmann, among others.

On the other hand, we would like to make a special emphasis on the growing demand to develop mathematical tools for the next industrial (Industry 4.0) and social revolution, which requires not only research on Computational Mathematics but also on Data Analytics. The growing amount of data in different fields such as Biosciences, Energy or Advanced Manufacturing is leading to a new scenario in which Mathematics will play a key role to face this “Data Revolution”.

Therefore, in line with the European Cloud Initiative, it is crucial to involve disciplines such as High Performance Computing (HPC) and innovative technologies for the management of Big Data. The grand challenge of Algorithmic Mathematics is to design the most appropriate model of computing in order to solve the current computational problems and data infrastructure limitations. A new theory needs to be built for the development and assessment of statistical methods and learning algorithms tailored to Big Data requirements.

The interaction and connection between the mathematical sciences and other fields should be part of the agenda in H2020 for the forthcoming programmes in order to contribute to the societal challenges in the indicated sectors (Advanced Manufacturing, Biosciences, Energy) and the Society in general. In fact, there are large examples of how Applied Mathematics (e.g.: Computational Fluid Dynamics, Image Analysis, Molecular Dynamics, Cloud Computing, Financial Mathematics) can contribute to the better understanding of real world problems. And that is why from BCAM we strongly support the idea that the EU could provide funding to mathematical research and its applications.

Ronald Rösch,

26/04/2016 18:18

With the PRIIPs regulation, risk and performance scenarios for all Packaged and Insurance-based Investment Products need to be calculated. The stochastic models in use are complex, especially when it comes to
specific products. We therefore suggest to review the existing approaches for scenario calculation, in particular with a focus on the persisting phase of low interest rates. Moreover, a modelling framework for different product classes should be developed. This ensures the comparability of the results and therefore poses a real benefit for the consumers.

Muhammad Amin,

26/04/2016 06:03

Big Data have new opportunities to modern world and challenges to statisticians. On one hand, Big Data hold great promises for discovering subtle population patterns and heterogeneities that are not possible with small-scale data. On the other hand, the massive sample size and high dimensionality of Big Data introduce unique computational and statistical challenges. These challenges are distinguished and require new computational and statistical paradigm. The penalization techniques to handle ultra high dimensional data are the best way to cope these problems. I just completed my PhD titled "Penalized quantile regression methods for high dimensional data and their applications".

Helena Smigoc,

25/04/2016 10:23

An abundance of questions from Matrix Theory are motivated by applications. I highlight completion and nearness problem, and matrix factorisation.

Completion problems arise in applications when a partial information on a matrix is known and a completion of this matrix satisfying some conditions is wanted. Similarly, nearness problems are looking for a matrix with some desired properties, but in this case a matrix that is as close as possible (in some sense) to a given matrix is requested. In both problems some structural and spectral constraints are imposed, for example:
Low rank, Positive semidefinitness, Stability, Prescribed inertia, Orthogonality, Zero-nonzero pattern, Sparsity, Positivity of elements.

Matrix factorization is a factorization of a matrix into a product of two (or more) matrices, and it is of interest to find matrix factorizations where the factors have some desired properties, examples of those properties can be taken form the list given above. Matrix factorizations reveal hidden structures of a matrix and are used in machine learning, signal processing, analysis of big data, textural analysis, etc.

While some aspects of the questions that are given above are classical and well researched, there is an increasing demand from applications to impose several conditions at the same time. Moreover, looking at the problems for matrices over finite fields gives another dimension of difficulty. In those cases existing tools do not apply and new ones need to be developed. Problems that demand structural as well as spectral constraints are particularly challenging and they connect several areas of mathematics from linear algebra, to combinatorics, graph theory, and number theory.

Thomas Laffey,

29/04/2016 13:55

I strongly endorse the importance of the topics listed by Helena Smigoc. Motivated by applications, a fundamental change in the study of matrix theory has been the current emphasis on patterns of the entries in the matrix or the type of entries, such as 0,1, integers, positive reals, etc.,for which the traditional linear algebras of transformations has to be supplemented by completely new techniques from geometry, algebra,analysis and probability theory. Answering the questions described by Dr. Smigoc will, in addition to having applications throughout the sciences, make a profound contributions to diverse areas of pure mathematics.

Miguel Bustamante,

25/04/2016 10:07

Mesoscale systems occur in many areas of science and are characterised by the complex interaction of a large number of degrees of freedom, at length and time scales for which the analysis is neither amenable to a direct discrete approach nor to a size-independent continuum approach. Correspondingly, the modelling of these systems cannot be based solely on analytical results or on purely numerical simulations; rather, it requires for its success the combination of analytical and numerical methods, with a rigorous mathematical basis that is still in development. More specifically, the so-called range of mesoscales of a system is that range of scales at which a direct numerical approach is not feasible due to the lack of either computer memory/speed or robust validation methods, while classical continuum analytical approaches fail too due to the presence of nontrivial dynamics involving intermittency, spatio-temporal chaos, nonlinear resonances or turbulence. Thus, the mathematical modelling of mesoscale systems is one of the greatest challenges of this century, and provides a fertile ground for multi-disciplinary collaborations, spearheaded by mathematical research and going hand-in-hand with high-resolution numerical simulations.

In terms of real-life impact, the understanding of mesoscale systems is, without exaggerating, an urgent matter in Earth sciences, materials sciences, agriculture, food science and the global economy in general. For example:

(1) In the atmosphere, the understanding of mesoscale convective systems and other mesoscale patterns provides continued improvements of numerical weather prediction models, which have strong impact on the world economy and safety. On a more topical note, climate change is the most urgent environmental challenge faced by humankind today. Scientific success on this matter requires a mesoscale approach where the most relevant degrees of freedom can be singled out, in order that accurate predictions are made. Again, an interdisciplinary approach is required involving statistics, mathematical analysis, computer science, physics and mathematical modelling.

(2) In materials science, multi-scale simulations involve the nested coupling of models at several levels of scales: from the smallest quantum mechanics scale to the molecular dynamics to kinetic theory to continuum mechanics.

(3) In fluid mechanics, and more recently in complex fluids involving the presence of active matter (such as microorganisms), the goal is to model mathematically in an adaptive way both the microscopic and the macroscopic dynamics. To validate the models, experiments are performed at all scales using the latest technology, such as high-speed cameras and particle image velocimetry. The same approach is followed in the case of nonlinear optics, where the main goal is to understand how to design optical media so that optical signals are transmitted efficiently and with minimal losses. The greatest challenge in all these areas is the understanding of the inner workings of the nonlinear interactions, which sets the way energy is transferred across scales, and the preferred directions of such transfers.

Beyond urgent issues involving immediate impact, mathematical modelling of mesoscale systems has the potential to contribute to the development of more fundamental areas of research, such as high-energy physics, cosmology and the newborn field of gravitational wave astronomy. In the latter, for example, the LIGO family of gravitational wave detectors might very soon detect events that do not have a theoretical explanation yet, and whose explanation might require the modelling of fast, highly energetic and highly nonlinear processes occurring in massive collisions, involving the interaction of matter, light and gravitation at the mesoscale.

paolo Orlandi,

25/04/2016 16:45

important field for mathematics

Michael Wilczek,

28/04/2016 22:30

Thanks, Miguel, I think your post identifies a number of important research topics! Research on these topics is urgently needed, and I believe that the combination of most recent experimental, numerical and theoretical methods puts breakthroughs within reach.

Jacek Miękisz,

24/04/2016 21:51

Living cells store, process, and transmit information. They respond to stimuli and make decisions in highly stochastic environments. Such small biological systems should obey (some yet unknown) principles of non-equilibrium physics. So we need collaboration between biologists who know real-life phenomena, physicists to construct models using concepts and techniques of statistical and other branches of physics, and mathematicians to develop new analytical tools to deal with such models and to find relations between abstract structures present in life processes. We need BIOPHYSMATH programmes to understand biological, physical, and mathematical theory and practice of information transmission. This might be useful for example in designing efficient micro-machines in synthetic biology and socio-economic regulations and mechanisms of early warnings of possible crises.

Jacek Miękisz,

24/04/2016 21:48

I agree with my predecessors that huge projects with many investigators are not optimal for mathematical research. The best applications arise when we are focused on fundamental research done in small groups. So why do we need EU programmes? I think they are indispensable to foster interdisciplinary research. National science and scientific careers are unfortunately compartmentalized. Moreover, you observe that in some places, branches of science which owe their birth to clever observations of real-life phenomena, like for example dynamical systems, are now investigated in closed self-centered groups. Their members are no longer interested in problems outside mathematics. They created enough interesting inside problems to go on in a conservative way. The other so to speak extreme consists of scientists who use buzz words such as for example chaos theory and complex systems to attract attention and funding. They try to create new paradigms (social networks) and work within them. Unfortunately very often they do not have much contacts with mathematicians and physicists who studied chaos theory and complex systems for decades. One goals of interdisciplinary EU programmes would be to make such conservative and paradigm groups to meet and work on some common tasks.

Daniel Fernandez,

22/04/2016 17:35

Excellent idea

John Toland,

22/04/2016 15:44

1. The following remarks are predicated on a profound commitment to fundamental (often highly abstract) mathematical science research at the highest possible level and a belief in the importance of communicating to the world beyond mathematics recent (and not so recent) discoveries therein. However the different expectations and time scales for pure research on the one hand and its possible impact on the other must not be conflated; rather their differences should be recognized and managed. In particular, mechanisms should be in place for dealing with impact, especially a long time after the original research.

2. From the diversity of INI programming across the whole range of mathematics (pure, applied and statistics) http://www.newton.ac.uk/science/programmes/past it is clear that mathematics has important applications in a vast variety of applications. For examples, important advances in materials science are being led by sophisticated interpretations of pure mathematical theory of equations arising when modelling materials; in biology, cell motility poses wonderful mathematical challenges, and imaging analysis, vision, medical and geophysical scanning (including land mine detection) are intrinsically mathematical.
All of these involve a great deal of mathematics well beyond big data analytics.

3. It is important for funded research programs to be open to all the mathematical sciences including those of a fundamental nature, and on a variety of scales, and not restricted a priori to particular themes.

4. There is an urgent need to support inter-disciplinary training, facilitating introductions between potential collaborators and nurturing teams who in the first instance may have little in common. This means supporting activities (scoping meetings etc) leading to research, rather than research per se. Although each such activity is small scale and short term, such activities are essential and collectively must be funded in the large for the long term.

JF Toland
Director of the Isaac Newton Institute (INI)
http://www.newton.ac.uk/
including the
Turing Gateway to Mathematics (TGM)
http://www.turing-gateway.cam.ac.uk/

Vera Pawlowsky-Glahn,

22/04/2016 12:52

Many of the previous comments address the need for novel mathematical tools in the Biosciences, in Environmental Studies, in Industrial, Political and Social Studies, in interdisciplinary fields in general. Compositional Data Analysis (CoDA) is such a new tool. Its roots go back to a paper on spurious correlation by K. Pearson, published in 1897, and it has seen an amazing expansion since J. Aitchison introduced in 1982 the logratio approach. Nowadays it builds on the fact that the sample space of compositional data, i.e. of data in proportions, percentages, or, in general, parts of a whole that carry relative information, is a constraint subset of real space that has its own algebraic-geometric structure. More precisely, it has a Euclidean vector space structure which has been extended to a Hilbert space structure for functions with finite or infinite support. Recently, the importance and necessity of these tools in microbiomics, or other "omics", has been recognised, and all the problems related to these fields, like fat data, represent new challenges.

This type of problem is close to the fundamentals of statistics, which is based on the interplay between observations and parameters. Both, observations and parameters, are in measurable spaces, the sample space and the parameter space. Measurability of these spaces is the minimal structure required, but generally richer structures are convenient, if not necessary. For instance, vector space operations have to be defined for attaining central limit results; distances are required for straightforward definitions of mean values and variability (see Fréchet, 1948). Most contributions to statistics are at present assuming that both sample and parameter spaces are subsets of the real space endowed with the sum as group operation and with Euclidean metrics. At most, in functional data analysis or stochastic processes the sample space is a Hilbert $L^p$ space with its metrics. However these structures inherited from real or $L^p$ spaces may fail at modelling the main features of observations and parameters. We can mention some examples: compositional data which sample space can be represented as a simplex with its own Euclidean structure, where the sum is not the group operation; directional data, which sample space is the sphere, which can not be equipped with its own Euclidean structure; random sets, which need special operations and metrics; random positive measures, represented by densities, which can be included in vector spaces (Bayes spaces) where the group operation is Bayes updating.

It would be a great opportunity for European science to have this topic included in the program of Horizon 2020.

Shirley Coleman,

22/04/2016 10:36

Waking up to the benefits of analysing their internal company data is a new challenge for small to medium enterprises (SMEs). They can see the enormous potential in terms of offering new products based on visualisation and insight but are not sure how to get started. Levels of confidence in handling data are worryingly low in many SMEs due to diverse factors such as loose responsibility for collecting and checking data, poor numeracy skills, secrecy and privacy fears about giving too much away upsetting customers and losing competitive advantage. SMEs are so important to our EU economy but risk being left out of the great advances in big data analytics realised by large organisations. A thorough review of the needs and barriers experienced by SMEs would be a good start to tackling the knowledge and confidence gap. In addition it is not enough to make help available, it needs to be implemented and this requires strategic intervention which can only happen when promoted and supported by evidence. Therefore further research should be carried out to explore the opportunities, fears and barriers to SMEs using their data resources fully and the interventions that would be effective. Rather than being a social policy or business study, this research needs to be grounded in knowledge of the advances in mathematics which are facilitating new ways of analysing data, for example developments in encrypted data mining.

Jean Clairambault,

22/04/2016 10:19

I fully endorse this proposal for MSO in applied mathematics. I would like to stress that a field in which optimisation should be particularly welcome is medicine. This is seen from the point of view of mathematicians, as physicians seldom even suspect that there can exist a field of mathematics called optimisation and optimal control, applicable to medical treatments. Therapeutic strategies in the clinic usually rely on sheer empiricism, for interactions between therapists and optimisers are scarce. It could be a goal of this MSO initiative to encourage their multiplication on well-defined applications by clearly mentioning therapeutics in its objectives and proposing to actively support it by budgeted lines.

Julian Stephens,

22/04/2016 10:11

To encourage interest and funding in this area there could be more focus on "how can quantum algorithms solve the World's problems?". For example are there relatively short-term applications that can optimise industry operations without needing a full Universal Quantum Computer?

Irena Ograjenšek,

22/04/2016 04:48

Predictive analytics (use of simple and / or complex models based on historical data) has become increasingly important for anticipatory business decision-making in the past few years. Decisions such as whom to target in a marketing campaign, what products to stock, possibility of fraud, or who the "best" customers are for a firm are just a few illustrative examples of predictive analytics applications. Consequently, managers in consumer packaged goods, retailing, banking, gambling, energy and healthcare industries are currently the most active users of predictive analytics.

We believe predictive analytics should also be at the core of policy making. Government agencies should harness the power of (micro) data they routinely accumulate to a much larger extent when designing policy measures to deal with the challenges of population ageing, new migratory patterns, labour market shifts, capital flows, corruption prevention, etc. Predictive analytics should facilitate faster policy making and, when deemed necessary, also faster corrective measures design and implementation - both for the benefit of the general population and its specific vulnerable subgroups.

Rainer Goeb,

27/04/2016 00:15

Business analytics enable success in the private sector. A study from the MIT Center for Digital Business shows that organizations driven most by data-based decision making had 4% higher productivity rates and 6% higher profits. Why should analogous success not be possible in policy making? The problem: even in the private sector, many companies still struggle with adopting data-based business analytics, particularly when it comes to processing and analysing big data. The public sector is much further behind in this respect. Further impeding factors: lack of interest in evidence-based decisions in ideology-driven political parties, reservations and fears in the population against data collection and monitoring, data protection and privacy issues. We need a rational discussion on all these issues in society and politics on European level, and we need sound studies on how predictive analytics can work in policy making.

Rick Quax,

21/04/2016 14:07

Seemingly very different complex systems may actually display the same kind of behavior. For instance, a phenomenon called ‘criticality’ is a useful type of instability to rapidly respond to external stimuli in neuronal networks in the brain, gene-regulatory networks, bacterial colonies, and the flocking of birds, among others. Other cross-domain phenomena include tipping points, resilience, controllability, and the emergence of scales. From this the picture emerges that the mechanistic details of complex systems may be irrelevant for generating such cross-domain phenomena. However, the young field of complex adaptive systems currently lacks a unified framework to study emergent, cross-domain phenomena in a way that ignores these irrelevant mechanistic details. The current approach is to study such phenomena in domain-specific models, such as a neuronal network, which complicates the matter and limits the rates of scientific progress on questions which span multiple domains.

Given also the fact that H2020 intends to increase its focus on interdisciplinarity, I argue for the topic of 'Complex Adaptive Systems' to be included as a research area in its own right. Foundational research should be invited which specifically crosses domain boundaries and develops theory and mathematical modeling to understand phenomena which are observed in multiple domains.

Bodo Manthey,

20/04/2016 20:33

Algorithms are at the heart of many modern systems, ranging from cars to methods for analyzing big data. Algorithms predict our behavior and influence our decisions. However, many of the algorithms used are not well understood. Understanding modern algorithms is pivotal for two reasons: First, it is the only method for humans to stay in command over what happens in a world of algorithms. Second, only a deep and rigorous understanding of the current algorithm provides us with means to design better the algorithms. Because of this, mathematical analysis of algorithms is a key topic with impact far beyond the borders of mathematics and computer science.

Although this sounds rather applied, this is more fundamental research. We still lack the mathematics tools to understand many of the most successful algorithms. There are a few concepts that provide deeper and refined insights than pure (classical) worst-case analysis (such as smoothed analysis, semi-random models, fixed-parameter tractability, or approximation stability), but these are still fields in their infancy that require a lot more research to make them widely applicable.

Following other comments, this area of research does not need a few large grants, but more smaller grants would be preferable.

Massimo Ferri,

19/04/2016 15:44

A sort of political comment.

In my opinion a central issue is the continuity of the research spectrum. OK with MSO, of course; OK also with “pure” maths, meaning research which apparently does not have any direct applications and which draws its inspirations from within mathematics itself. But it is important to have applications-oriented mathematics, which works at building theoretical frameworks like pure maths, but finding motivation in science and technology.

This builds a connecting link useful for both the applied domain - to which it offers more structered tools -and to the pure fields, to which it gives new impulse and a wider visibility. After all, it is important to make our work understandable by the politician and the tax-payer.
This is the case, e.g., of topological data analysis and topological robotics.

Unfortunately, at least in my country (Italy) this viewpoint meets some resistance in the “pure maths” circle, which praises in the media the great applicability of our discipline but punishes applications-oriented maths in the competitions.

Fabrice Planchon,

19/04/2016 13:05

Mathematics have long had natural interactions with many other sciences, as well as industry and social issues. In recent years, important mathematical developments have occurred at the following interfaces and are likely to intensify:

a) Interactions of mathematics with life sciences, biology, health and medicine have progressed and offer a broad potential for innovation. This involves modelling complex biological phenomena, by the use of more-and-more sophisticated techniques (PDEs, stochastics, probability, statistics). This also includes the development of a large range of mathematical techniques dedicated to the study of physiological signals and medical images.
b) Interactions with computer science in general, and HPC and big data in particular, is already well identified and require combined efforts. Statistical learning and deep neural networks have raised important challenges. Important developments in the possibility of dealing with high dimensional problems have emerged, for example by identifying and exploiting sparsity concepts.
c) More traditional interactions with physics at all scales as well as engineering sciences is a fruitful source of potential innovation. We can mention analytical relativity and detection algorithms based on Wilson wavelets as two outstanding examples that contributed to the recent discovery of gravitational waves. In geophysics, the modelling of the “whole-earth system” is a major issue.
d) Interaction with chemistry is still underestimated today. The potential is huge. For example in quantum chemistry and physical chemistry, where powerful numerical approaches are needed to accompany theory and applications.
e) Interaction with social sciences, in the broadest sense, is an important field for investment by mathematics. Either by participating more actively in the development of complex systems, or by seizing questions that have a strong demand for modelling or data analysis.

In most , if not all previous items, one may recognize the trilogy Mathematical Modelling, Simulation, Optimisation that has been highlighted in other posts in this open consultation. More generally, interactions between fundamental mathematics, applied mathematics, modelling and its applications are happening at a fast growing rate and prove to be mutually beneficial. Tackling any of the aforementioned challenges is likely to require combined efforts of mathematicians of several different fields collaborating with scientists from relevant domains.

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