# MAPLE: MAthematics PLatform Engagement activity

Lead Research Organisation:
University of Manchester

Department Name: Mathematics

### Abstract

The challenges presented to the scientific community by the complexities of modern society are inherently interdisciplinary. The success of mathematics in underpinning scientific investigation is evident by the fact that mathematics is the 'language of the physical sciences'; however many areas remain an open challenge to the international mathematics community. These include better engagement with the pressing global concerns of climate change and energy (e.g., developing new mathematical tools for dealing with complexity) and broad and challenging scientific goals such as the mathematization of the life sciences (e.g., mathematical models of cognition and perception, data analysis for DNA). The long-term scientific, technological and economic health of the UK requires mathematicians who (i) communicate fluently between the mathematical sub-disciplines, (ii) establish dialogues across discipline boundaries, and (iii) engage with professions and communities outside academe (in industry, commerce, financial sector, government etc). The award of a Platform Grant to the School of Mathematics at the University of Manchester will enable the initiation of an internal funding mechanism to initiate, nurture and support new and adventurous research initiatives that address these goals.The creation of a platform account offers an integrative approach to the way mathematics is carried out in the School, and in its application outside the School. This should be seen not just as a fundingvehicle but a process whereby innovative and interdisciplinary research will be continuously stimulated and prosecuted. The platform account would also provide an opportunity to add value to the core research activity in pure mathematics, applied mathematics and probability and statistics within the School.

### Planned Impact

The impact of MAPLE takes a number of forms: the direct impact of the research done within the lifetime of MAPLE on neighbouring areas of mathematics, on cognate academic disciplines and those more distant from mathematics, and on research groups outwith academe, e.g. in industry and commerce. Finally, the long term impact will be to the broader society through the spin-offs of innovative and adventurous interdisciplinary research. So, who in particular are likely to benefit? A very conservative estimate of this can be provided by listing the present organisations who currently support, or are partners with, research groups within the School. These span a broad range of companies and industrial and commercial sectors including, as exemplars, the nuclear industry (AMEC, EDF), the environment (Proudman Oceanographic Lab), the pharmaceutical industry (AstraZenica), telecommunications (BT, Photon Design), Defence (DSTL, QinetiQ), underwater systems (Thales), aerospace (Airbus, BAESystems, UMARI), numerical software (NAG and The MathWorks), the health service (NHS), social behaviour and marketing (Unilever), metrology (NPL), scanning and imaging for security systems and in the health service (Rapiscan, Wolfson Molecular Imaging Centre). Of course, MAPLE aims to substantially increase the number of interactions of members of School with all areas of industry and commerce. It will do this through direct mechanisms for establishing links, and via the leverage potential of MAPLE to gain new and diverse funds for significant research collaborations. How will they benefit? Clearly the above organisations, and many others over the lifetome of MAPLE, will benefit by direct involvement in the study groups, conferences and workshops, fellowship schemes etc., and the consequent research output from these, and other substantial interactions. However, the long-term benefits should be far greater. MAPLE will instill a culture of, and offer incentives for, novelty and adventure in research. Thus, we expect much greater effort to be directed towards the EPSRC strategic aims, societal problems, and grand challenges. What will be done to ensure that they benefit? MAPLE will incorporate a process which will create a conduit for dialogue between potential beneficiaries and mathematicians. This should help to provide innovative methods and novel approaches to the problems of new technologies and industries. It will also help create an environment in the School where mathematics from across the whole range of the discipline can be brought to bear on problems. We will use the Technology Translator to facilitate interaction, and to exploit all the opportunities available, including, for example, the Industrial Mathematics KTN.

### Organisations

- University of Manchester (Collaboration, Lead Research Organisation)
- University of Wuppertal (Collaboration)
- Schlumberger Limited (Collaboration)
- St George's University (Collaboration)
- University of Rennes 1 (Collaboration)
- University of California (Collaboration)
- École Normale Supérieure de Cachan (Collaboration)
- University of Limerick (Collaboration)
- École Normale Supérieure, Paris (Collaboration)
- Bavarian Academy of Sciences and Humanities (Collaboration)
- University of Cambridge (Collaboration)
- University of Sheffield (Collaboration)
- École Polytechnique (Collaboration)
- Claude Bernard University Lyon 1 (UCBL) (Collaboration)
- University of Oxford (Collaboration)
- National Polytechnic Engineering Institude of Electrotechnology, Electronics,Computer Sciences, Hydraulics & Telecommunications (Collaboration)
- British Antarctic Survey (Collaboration)

### Publications

Abrahams I
(2011)

*Scattering of flexural waves by a semi-infinite crack in an elastic plate carrying an electric current*in Mathematics and Mechanics of Solids
Amelunxen D
(2014)

*Gordon's inequality and condition numbers in conic optimization*in arXiv
Amelunxen Dennis
(2014)

*Measures on polyhedral cones: characterizations and kinematic formulas*in arXiv e-prints
Aprahamian M
(2016)

*Matrix Inverse Trigonometric and Inverse Hyperbolic Functions: Theory and Algorithms*in SIAM Journal on Matrix Analysis and Applications
Aprahamian M
(2016)

*Matching exponential-based and resolvent-based centrality measures*in Journal of Complex Networks
Baker J
(2015)

*A two-dimensional depth-averaged -rheology for dense granular avalanches*in Journal of Fluid Mechanics
Ballantyne J
(2015)

*The maximal subgroups of*in LMS Journal of Computation and Mathematics
Barker T
(2015)

*Well-posed and ill-posed behaviour of the -rheology for granular flow*in Journal of Fluid Mechanics
Barnwell E
(2016)

*Antiplane elastic wave propagation in pre-stressed periodic structures; tuning, band gap switching and invariance*in Wave Motion
Berkman A
(2012)

*Groups of finite Morley rank with a pseudoreflection action*in Journal of Algebra
Berljafa M
(2015)

*The RKFIT algorithm for nonlinear rational approximation*in SIAM Journal on Scientific Computing
Berljafa M
(2017)

*The RKFIT Algorithm for Nonlinear Rational Approximation*in SIAM Journal on Scientific Computing
Biggs A
(2014)

*The Existence of a Canonical Lifting of Even Poisson Structures to the Algebra of Densities*in Letters in Mathematical Physics
BOROVIK A
(2016)

*RANK 3 BINGO*in The Journal of Symbolic Logic
Botella-Soler V
(2013)

*Lyapunov exponent and topological entropy plateaus in piecewise linear maps*in Journal of Physics A: Mathematical and Theoretical
Botella-Soler V
(2014)

*Hierarchy and polysynchrony in an adaptive network.*in Physical review. E, Statistical, nonlinear, and soft matter physics
Carson E
(2017)

*A New Analysis of Iterative Refinement and Its Application to Accurate Solution of Ill-Conditioned Sparse Linear Systems*in SIAM Journal on Scientific Computing
Cimpeanu R
(2015)

*A parameter-free perfectly matched layer formulation for the finite-element-based solution of the Helmholtz equation*in Journal of Computational Physics
Cotterill P
(2015)

*The time-harmonic antiplane elastic response of a constrained layer*in Journal of Sound and Vibration
Cui X
(2013)

*Gravity-driven granular free-surface flow around a circular cylinder*in Journal of Fluid Mechanics
Dawson G
(2013)

*The trapping and release of bubbles from a linear pore*in Journal of Fluid Mechanics
De Pascalis R
(2014)

*On nonlinear viscoelastic deformations: a reappraisal of Fung's quasi-linear viscoelastic model.*in Proceedings. Mathematical, physical, and engineering sciences
De Pascalis R
(2013)

*Predicting the pressure-volume curve of an elastic microsphere composite*in Journal of the Mechanics and Physics of Solids
De Pascalis R
(2015)

*Simple shear of a compressible quasilinear viscoelastic material*in International Journal of Engineering Science
Desai N
(2016)

*An explicit reconstruction algorithm for the transverse ray transform of a second rank tensor field from three axis data*in Inverse ProblemsDescription | The nature of the grant was to stimulate new intra and interdisciplinary research within the School of Mathematics at Manchester. This ambitious 5 year programme has stimulated a very wide variety of activities (and outputs), ranging from Creativity@home workshops patent support, research visits both to and from Manchester, internships, workshops and conferences, and outreach initiatives. |

Exploitation Route | The main spin-off academically to this programme of activity has been numerous new research partnerships, often in very diverse and unexpected areas. To demonstrate this, we offer three examples which have led to long-term collaborations and research poublications. 1. There is now a close connection between the Schools Mathematics and Earth and Environemental Sciences at Manchester. Investigators have been looking at the effect of debris cover on glacial erosion rates, models of meteorite submergence within ice in the Antarctic, and recently scattering of radiation by ice crystals in clouds. 2. There is now one of the most active groups in the world working at Manchester on the intradisciplinary area of tropical geometry/max-plus mathematics and its applications. The research spans pure mathematics, applied mathematics, and application areas in the natural sciences and engineering disciplines. 3. MAPLE provided support for a Member of the School to deliver a plenary lecture at a conference in Armenia in 2013. The conference, on Compact Stars and Relativistic Gravity, was some way from his expertise (he is a pure mathematician). However, whilst there, he was reminded that about thirty years ago he had had long and fruitful discussions with several of the attendees. He then revisitied these old ideas, which offered a relation between projective geometry and the physics of stars, and the results have been so successful that it is now one of the main foci of his present researches. |

Sectors | Aerospace, Defence and Marine,Agriculture, Food and Drink,Construction,Digital/Communication/Information Technologies (including Software),Education,Energy,Environment,Financial Services, and Management Consultancy,Healthcare,Leisure Activities, including Sports, Recreation and Tourism,Manufacturing, including Industrial Biotechology,Pharmaceuticals and Medical Biotechnology,Retail,Security and Diplomacy,Transport,Other |

Description | The long-term scientific, technological and economic health of the UK requires mathematicians who (i) communicate fluently between the mathematical sub-disciplines, (ii) establish dialogues across discipline boundaries, and (iii) engage with professions and communities outside academe (in industry, commerce, financial sector, government etc). The award of a Platform Grant to the School of Mathematics at the University of Manchester has enabled the initiation of an internal funding mechanism to initiate, nurture and support new and adventurous research initiatives that address these goals. |

First Year Of Impact | 2011 |

Sector | Aerospace, Defence and Marine,Digital/Communication/Information Technologies (including Software),Education,Energy,Environment,Financial Services, and Management Consultancy,Healthcare,Manufacturing, including Industrial Biotechology,Pharmaceuticals and Medical Biotechnology,Transport,Other |

Impact Types | Cultural,Societal,Economic |

Description | Active cloaking for flexural waves in thin plates |

Organisation | University of Rennes 1 |

Country | France |

Sector | Academic/University |

PI Contribution | We had already done some work on active cloaking and this problem was a natural and important extension. |

Collaborator Contribution | Gregory was interested in working with us due to our expertize in metamaterials. |

Impact | A journal publication has been submitted: • Futhazar, G., Parnell, W.J. and Norris, A.N. Active cloaking for flexural waves in thin plates. Submitted to J. Sound Vibration. Aug 2014. |

Start Year | 2012 |

Description | Bubble oscillations in microchannels due to geometrical constraint s |

Organisation | University of Manchester |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | Supervised student. |

Collaborator Contribution | Performed experiments on bubble propagation in partially occluded tubes. |

Impact | Report. |

Start Year | 2011 |

Description | Bubble trapping and sorting in microchannels of complex geometry |

Organisation | University of California |

Country | United States |

Sector | Academic/University |

PI Contribution | Developed a model of bubble trapping in sudden tube expansions. |

Collaborator Contribution | Sungyon worked with my PhD student Geoffrey Dawson, who was performing experiments on the topic. |

Impact | D. Pihler-Puzovic, P. Illien, M. Heil & A. Juel (2012) Suppression of Complex Fingerlike Patterns at the Interface between Air and a Viscous Fluid by Elastic Membranes. Phys. Rev. Lett. 108, 074502, doi:10.1103/PhysRevLett.108.07450 |

Start Year | 2011 |

Description | Effect of debris on the melting of glaciers |

Organisation | Bavarian Academy of Sciences and Humanities |

Country | Germany |

Sector | Academic/University |

PI Contribution | In this research a quantitative analysis is made of the dynamic process by which ice underneath a dry porous debris layer melts. We show that the incorporation of debris layer air flow into a theoretical model of glacial melting can capture the empirically observed features of the so-called Ostrem curve (a plot of the melt-rate as a function of debris depth). Our team inputted the mathematical modelling elements to the project. |

Collaborator Contribution | Partners added expertise on the theoretical and experimental sides of glaciology. |

Impact | Several papers submitted or in progress. Further research work undertaken, especially in regard to meteorite detection in the antarctic. |

Start Year | 2012 |

Description | Effect of debris on the melting of glaciers |

Organisation | British Antarctic Survey |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | In this research a quantitative analysis is made of the dynamic process by which ice underneath a dry porous debris layer melts. We show that the incorporation of debris layer air flow into a theoretical model of glacial melting can capture the empirically observed features of the so-called Ostrem curve (a plot of the melt-rate as a function of debris depth). Our team inputted the mathematical modelling elements to the project. |

Collaborator Contribution | Partners added expertise on the theoretical and experimental sides of glaciology. |

Impact | Several papers submitted or in progress. Further research work undertaken, especially in regard to meteorite detection in the antarctic. |

Start Year | 2012 |

Description | Effect of debris on the melting of glaciers |

Organisation | University of Limerick |

Country | Ireland |

Sector | Academic/University |

PI Contribution | In this research a quantitative analysis is made of the dynamic process by which ice underneath a dry porous debris layer melts. We show that the incorporation of debris layer air flow into a theoretical model of glacial melting can capture the empirically observed features of the so-called Ostrem curve (a plot of the melt-rate as a function of debris depth). Our team inputted the mathematical modelling elements to the project. |

Collaborator Contribution | Partners added expertise on the theoretical and experimental sides of glaciology. |

Impact | Several papers submitted or in progress. Further research work undertaken, especially in regard to meteorite detection in the antarctic. |

Start Year | 2012 |

Description | Effect of debris on the melting of glaciers |

Organisation | University of Oxford |

Department | Mathematical Institute Oxford |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | |

Collaborator Contribution | Partners added expertise on the theoretical and experimental sides of glaciology. |

Impact | |

Start Year | 2012 |

Description | Effect of debris on the melting of glaciers |

Organisation | University of Sheffield |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | |

Collaborator Contribution | Partners added expertise on the theoretical and experimental sides of glaciology. |

Impact | |

Start Year | 2012 |

Description | Fingering under elastic membranes |

Organisation | University of Cambridge |

Country | United Kingdom |

Sector | Academic/University |

PI Contribution | Characterised suppression of viscous fingering under elastic membranes. Led to a joint EPSRC Project Grant submission (Juel, Heil), which was awarded (EP/J007927/1). |

Collaborator Contribution | Characterised suppression of viscous fingering under elastic membranes. Led to a joint EPSRC Project Grant submission (Juel, Heil), which was awarded (EP/J007927/1). |

Impact | D. Pihler-Puzovic, P. Illien, M. Heil & A. Juel (2012) Suppression of Complex Fingerlike Patterns at the Interface between Air and a Viscous Fluid by Elastic Membranes. Phys. Rev. Lett. 108, 074502, doi:10.1103/PhysRevLett.108.07450 |

Start Year | 2011 |

Description | Mathematical models for the nonlinear viscoelastic response of tendon and ligament |

Organisation | Ecole Polytechnique |

Country | France |

Sector | Academic/University |

PI Contribution | Our techniques allow macroscopic nonlinear behaviour to be incorporated via homogenization and this results in the models of tendon and ligament. |

Collaborator Contribution | The Ecole Polytechnique team has excellence in experimentation on soft tissue but require models based on microstructural behaviour, i.e using homogenization. |

Impact | One journal publication which is almost at the point of submission: • Lynch, B., Parnell, W.J., Shearer, T., Abrahams, I.D. and Allain, J.-M. "Micromechanical models for the nonlinear viscoelastic response of tendon and ligament". In prep for submission to J. Biomechanics. |

Start Year | 2013 |

Description | Nonlinear homogenization of damaged media |

Organisation | St George's University |

Country | Grenada |

Sector | Academic/University |

PI Contribution | Our team in Manchester has strength in nonlinear elasticity and homogenization. Hence the combined team has great strength in this area |

Collaborator Contribution | The Grenada team work on characterizing materials via acoustic nonlinearity. |

Impact | Two journal publications are almost at the point of submission: • Melchor, J., Parnell, W.J. and Rus, G. Damage prediction via nonlinear quantitative ultrasound: a micromechanical approach, in preparation for submission to J. Mech. Phys. Solids • Melchor, J., Parnell, W.J. and Rus, G. Nonlinear elastic constitutive behaviour of cracked and damaged materials. In preparation for submission to J. Elasticity |

Start Year | 2011 |

Description | Perfectly matched layers for indefinite Helmholtz problems |

Organisation | Schlumberger Limited |

Country | United States |

Sector | Private |

PI Contribution | The 1 week visit of Dr Vladimir Druskin enabled us to complete a paper which is now under revision with SIAM Review and likely to be accepted soon. Dr. Druskin is a scientist at Schlumberger and this is truly an interdisciplinary collaboration with Geophysicists. My contribution was the implementation of a rational Krylov method for the computation of complex grid steps in a perfectly matched layer. |

Collaborator Contribution | Dr Druskin provided the required knowledge about optimal H_2 model order reduction and Zolotarev problems, and gave insights into the Geophysical problems. |

Impact | Schlumberger uses the developed layers. |

Start Year | 2014 |

Description | Restarted Krylov methods for matrix functions. |

Organisation | University of Wuppertal |

Country | Germany |

Sector | Academic/University |

PI Contribution | MAPLE funded a 1 week visit of Marcel Schweitzer from University of Wuppertal. Our collaboration was related to the convergence analysis of restarted Krylov methods. My main contribution was the explanation of superlinear convergence effects and possible extensions to inexact shift-and-invert Krylov methods. |

Collaborator Contribution | Marcel Schweitzer is an expert in the convergence analysis of polynomial Krylov methods. He is also an author of the funm_quad software that has been developed in this collaboration. |

Impact | None yet. |

Start Year | 2013 |

Description | The curling of ribbons |

Organisation | École Normale Supérieure, Paris |

Country | France |

Sector | Academic/University |

PI Contribution | Supervised student. |

Collaborator Contribution | Performed experiments on curling of ribbon and developed model with O.E. Jensen. |

Impact | Report. Paper in preparation. |

Start Year | 2013 |

Description | The effect of vibration on free-surface flows of yield-stress fluids |

Organisation | National Polytechnic Engineering Institude of Electrotechnology, Electronics,Computer Sciences, Hydraulics & Telecommunications |

Sector | Academic/University |

PI Contribution | Supervised student, |

Collaborator Contribution | Experiments on the fluidisation of yield stress fluids under vibration. |

Impact | Report. |

Start Year | 2014 |

Description | Wrinkling In Sheets |

Organisation | Claude Bernard University Lyon 1 (UCBL) |

Country | France |

Sector | Academic/University |

PI Contribution | Supervised student incollaboration with D. Pihler-Puzovic. |

Collaborator Contribution | Experiments on viscous fingering under elastic membranes. |

Impact | Report. |

Start Year | 2013 |

Description | Wrinkling and fingering under elastic membrane . |

Organisation | École Normale Supérieure de Cachan |

Country | France |

Sector | Academic/University |

PI Contribution | Supervised student. |

Collaborator Contribution | Performed experiments on front propagation in elastic Hele-Shaw cells. |

Impact | Report. Paper in preparation. |

Start Year | 2012 |