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

### Publications

Zhang W
(2016)

*Matrix Depot: an extensible test matrix collection for Julia*in PeerJ Computer Science*Matrix Depot: an extensible test matrix collection for Julia*in PeerJ Computer Science

Woodhouse M
(2012)

*Segregation-induced fingering instabilities in granular free-surface flows*in Journal of Fluid Mechanics
Willoughby N
(2012)

*Homogenization methods to approximate the effective response of random fibre-reinforced Composites*in International Journal of Solids and Structures
Wiederseiner S
(2011)

*Experimental investigation into segregating granular flows down chutes*in Physics of Fluids
Van Der Vaart K
(2015)

*Underlying Asymmetry within Particle Size Segregation.*in Physical review letters
Thompson A
(2014)

*Sequential deposition of overlapping droplets to form a liquid line*in Journal of Fluid Mechanics
Sidorov N
(2011)

*Optimizing properties of balanced words*
Shearer T
(2013)

*Torsional wave propagation in a pre-stressed hyperelastic annular circular cylinder*in The Quarterly Journal of Mechanics and Applied Mathematics
Shearer T
(2015)

*Antiplane wave scattering from a cylindrical cavity in pre-stressed nonlinear elastic media.*in Proceedings. Mathematical, physical, and engineering sciences
Shearer T
(2020)

*A Recruitment Model of Tendon Viscoelasticity That Incorporates Fibril Creep and Explains Strain-Dependent Relaxation.*in Journal of biomechanical engineering
Razis D
(2014)

*Arrested coarsening of granular roll waves*in Physics of Fluids
Prieto K
(2017)

*Sparsity and level set regularization for diffuse optical tomography using a transport model in 2D*in Inverse Problems
Pihler-Puzovic D
(2013)

*Modelling the suppression of viscous fingering in elastic-walled Hele-Shaw cells*in Journal of Fluid Mechanics
Perrey-Debain E
(2015)

*TE Mode Mixing Dynamics in Curved Multimode Optical Waveguides*in Communications in Computational Physics
Pearce P
(2016)

*Image-Based Modeling of Blood Flow and Oxygen Transfer in Feto-Placental Capillaries.*in PloS one
Parnell W
(2015)

*Antiplane Wave Scattering from a Cylindrical Void in a Pre-Stressed Incompressible Neo-Hookean Material*in Communications in Computational Physics
Parnell W
(2016)

*Transient Thermal Mixed Boundary Value Problems in the Half-Space*in SIAM Journal on Applied Mathematics
Montaldi J
(2014)

*Transformations of measures via their generalized densities*in Russian Journal of Mathematical Physics
Melchor J
(2019)

*Damage prediction via nonlinear ultrasound: A micro-mechanical approach.*in Ultrasonics
Martin P
(2015)

*One-dimensional reflection by a semi-infinite periodic row of scatterers*in Wave Motion
Mansuro lu N
(2013)

*FREE CENTRE-BY-METABELIAN LIE RINGS*in The Quarterly Journal of Mathematics
Lionheart WRB
(2017)

*Non-linearity in monochromatic transmission tomography*in Non-linearity in monochromatic transmission tomography
Lin L
(2014)

*Covariance structure regularization via entropy loss function*in Computational Statistics & Data Analysis
Levasseur T.
(2015)

*Higher symmetries of powers of the Laplacian and rings of differential operators*in arXiv e-prints
Levasseur T
(2017)

*Higher symmetries of powers of the Laplacian and rings of differential operators*in Compositio Mathematica
Kovács L
(2014)

*Free centre-by-metabelian Lie algebras in characteristic 2*in Bulletin of the London Mathematical Society
Kokelaar B
(2014)

*Fine-grained linings of leveed channels facilitate runout of granular flows*in Earth and Planetary Science Letters
Kitapbayev Y
(2015)

*Stochastic control and real options valuation of thermal storage-enabled demand response from flexible district energy systems*in Applied Energy
Johnson CG
(2017)

*On the buckling of an elastic holey column*in Proceedings of the Royal Society A
JOHNSON C
(2011)

*Granular jets and hydraulic jumps on an inclined plane*in Journal of Fluid Mechanics
Johnson C
(2012)

*Grain-size segregation and levee formation in geophysical mass flows LEVEE FORMATION IN GEOPHYSICAL FLOWS*in Journal of Geophysical Research: Earth Surface
Higham N
(2017)

*Computing the Action of Trigonometric and Hyperbolic Matrix Functions*in SIAM Journal on Scientific Computing
Higham N
(2016)

*Estimating the Largest Elements of a Matrix*in SIAM Journal on Scientific Computing
Higham N
(2016)

*Bounds for the Distance to the Nearest Correlation Matrix*in SIAM Journal on Matrix Analysis and Applications
Higham N
(2015)

*Anderson acceleration of the alternating projections method for computing the nearest correlation matrix*in Numerical Algorithms
Higham N
(2016)

*Restoring Definiteness via Shrinking, with an Application to Correlation Matrices with a Fixed Block*in SIAM Review
Heil M
(2017)

*Journal of Fluid Mechanics*in Topological fluid mechanics of the formation of the Karman-vortex street
Heil M
(2012)

*Quasi-resonances in sound-insulating coatings*in Journal of Sound and Vibration
Heil M
(2016)

*A poroelastic fluid-structure interaction model of syringomyelia*in Journal of Fluid Mechanics
Hazel AL
(2017)

*On the buckling of elastic rings by external confinement.*in Philosophical transactions. Series A, Mathematical, physical, and engineering sciences
Gray R
(2017)

*Amenability and geometry of semigroups*in Transactions of the American Mathematical Society
Gray J
(2014)

*A depth-averaged -rheology for shallow granular free-surface flows*in Journal of Fluid Mechanics
Gray J
(2015)

*Particle-size and -density segregation in granular free-surface flows*in Journal of Fluid Mechanics
Gray J
(2015)

*Particle-size segregation in dense granular avalanches*in Comptes Rendus Physique
GRAY J
(2011)

*Multi-component particle-size segregation in shallow granular avalanches*in Journal of Fluid Mechanics
Glendinning P
(2016)

*Classification of boundary equilibrium bifurcations in planar Filippov systems.*in Chaos (Woodbury, N.Y.)
Glendinning P
(2015)

*Invariant Measures for the n-Dimensional Border Collision Normal Form*in International Journal of Bifurcation and ChaosDescription | 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 | There have been many ways in which this Platform grant has had impact outside academia. We mention a few representative examples below: 1. A two-day workshop (November 2015) on compressive sensing included an interdisciplinary audeince including a number of non academic attendees. As a follow up, Lonheart attended a meeting of the newly formed Alan Turing Institute in Edinburgh. 2. A recent paper by Evatt et al. on buried meteorites in Antarctica, published in Nature Communications (2016), was the result of two interdisciplinary workshops supported by MAPLE. There has been large exposure in the media to the results of the study, including the BBC, most UK newspapers, Smithsonian etc (see e.g. http://www.nature.com/ncomms/2016/160216/ncomms10679/metrics). New research actviity has since been funded by Leverhulme Trust and the British Antarctic Survey to support a prospecting mission to Antarctica in 2019/20. 3. One collaborative project fiunded by MAPLE resulted in software designed to optimise extraction of ore from a mine. This has now been licensed by a complany. 4. MAPLE funding allowed a team from the School of Mathematics to attend the Big Bang Fair 2015 to present a selection of research from the School complete with hands-on activities. It is estimated that they interacted with approximately 5,000 schoolchildren, teachers and members of the public. The stand was so successful that they have been invited back in 2016. 6. An early cereer researcher undertoook a summer research project on drying of thin films. This work ultimately lead to Cambridge Display Technologies (CDT Ltd) funding a CASE studentship that is currently in progress. 7. A number of short term research projects (mainly organised as internships for overseas early career researchers) led to significant developments of oomph-lib, an Object Oriented Multi-PHysics finite element LIBrary; the software is open source and has many users around the world and overseas from academe as well as industry. For example, Thales UK are using this to run their in-house predictive software tools. 8. One colleague, on an overseas research trip in 2015, was approached after a talk by a NASA contractor to help them with an open problem. Using optimal stopping techniques (developed as a part of a MAPLE grant and sponsored industrial collaboration with Thales) NASA had detected the presence of a mysterious drift in the clock error on the New Horizons spacecraft which passed Pluto last July (after launch in 2006). The result was announced at a space conference in the USA in February 2016 and is due to appear in print shortly. It is also indicative that all other methods employed (including the Galileo project community in the EU) have failed to detect any drift. The clock accuracy is vital for communication with the spacecraft which continues its mission to outer parts of the solar system (the next object to be met and examined in about two years). The researcher has been invited in May to the US to give another talk to a wider audience, to continue working on this problem, and to discuss related problems of central interest to NASA and US government. The MAPLE grant was critical in this work establishing the link between the School of Mathematics and NASA. 9. Maple contributed to support for the European Study Group with Industry that was held in Manchester in March 2015. This was a large meeting whereby mathematicians worked on problems from industry, commerce, security/defence sectors etc. 10. MAPLE supported the first (two-day) Festival of Mathematics in June 2014 (held in Manchester in July 2014). The School of Mathematics hosted over 2000 school students, industrialists and academics, over the two-days. |

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 |