Podcasts about british applied mathematics colloquium

This is the last of four conversation Gudrun had during the British Applied Mathematics Colloquium which took place 5th – 8th April 2016 in Oxford. Andrea Bertozzi from the University of California in Los Angeles (UCLA) held a public lecture on The Mathematics of Crime. She has been Professor of Mathematics at UCLA since 2003 and Betsy Wood Knapp Chair for Innovation and Creativity (since 2012). From 1995-2004 she worked mostly at Duke University first as Associate Professor of Mathematics and then as Professor of Mathematics and Physics. As an undergraduate at Princeton University she studied physics and astronomy alongside her major in mathematics and went through a Princeton PhD-program. For her thesis she worked in applied analysis and studied fluid flow. As postdoc she worked with Peter Constantin at the University of Chicago (1991-1995) on global regularity for vortex patches. But even more importantly, this was the moment when she found research problems that needed knowledge about PDEs and flow but in addition both numerical analysis and scientific computing. She found out that she really likes to collaborate with very different specialists. Today hardwork can largely be carried out on a desktop but occasionally clusters or supercomputers are necessary. The initial request to work on Mathematics in crime came from a colleague, the social scientist Jeffrey Brantingham. He works in Anthropology at UCLA and had well established contacts with the police in LA. He was looking for mathematical input on some of his problems and raised that issue with Andrea Bertozzi. Her postdoc George Mohler came up with the idea to adapt an earthquake model after a discussion with Frederic Paik Schoenberg, a world expert in that field working at UCLA. The idea is to model crimes of opportunity as being triggered by crimes that already happend. So the likelihood of new crimes can be predicted as an excitation in space and time like the shock of an earthquake. Of course, here statistical models are necessary which say how the excitement is distributed and decays in space and time. Mathematically this is a self-exciting point process. The traditional Poisson process model has a single parameter and thus, no memory - i.e. no connections to other events can be modelled. The Hawkes process builds on the Poisson process as background noise but adds new events which then are triggering events according to an excitation rate and the exponential decay of excitation over time. This is a memory effect based on actual events (not only on a likelihood) and a three parameter model. It is not too difficult to process field data, fit data to that model and make an extrapolation in time. Meanwhile the results of that idea work really well in the field. Results of field trials both in the UK and US have just been published and there is a commercial product available providing services to the police. In addition to coming up with useful ideas and having an interdisciplinary group of people committed to make them work it was necessery to find funding in order to support students to work on that topic. The first grant came from the National Science Foundation and from this time on the group included George Tita (UC Irvine) a criminology expert in LA-Gangs and Lincoln Chayes as another mathematician in the team. The practical implementation of this crime prevention method for the police is as follows: Before the policemen go out on a shift they ususally meet to divide their teams over the area they are serving. The teams take the crime prediction for that shift which is calculated by the computer model on the basis of whatever data is available up to shift. According to expected spots of crimes they especially assign teams to monitor those areas more closely. After introducing this method in the police work in Santa Cruz (California) police observed a significant reduction of 27% in crime. Of course this is a wonderful success story. Another success story involves the career development of the students and postdocs who now have permanent positions. Since this was the first group in the US to bring mathematics to police work this opened a lot of doors for young people involved. Another interesting topic in the context of Mathematics and crime are gang crime data. As for the the crime prediction model the attack of one gang on a rival gang usually triggers another event soon afterwards. A well chosen group of undergraduates already is mathematically educated enough to study the temporary distribution of gang related crime in LA with 30 street gangs and a complex net of enemies. We are speaking about hundreds of crimes in one year related to the activity of gangs. The mathematical tool which proved to be useful was a maximum liklihood penalization model again for the Hawkes process applied on the expected retaliatory behaviour. A more complex problem, which was treated in a PhD-thesis, is to single out gangs which would be probably responsable for certain crimes. This means to solve the inverse problem: We know the time and the crime and want to find out who did it. The result was published in Inverse Problems 2011. The tool was a variational model with an energy which is related to the data. The missing information is guessed and then put into the energy . In finding the best guess related to the chosen energy model a probable candidate for the crime is found. For a small number of unsolved crimes one can just go through all possible combinations. For hundreds or even several hundreds of unsolved crimes - all combinations cannot be handled. We make it easier by increasing the number of choices and formulate a continuous instead of the discrete problem, for which the optimization works with a standard gradient descent algorithm. A third topic and a third tool is Compressed sensing. It looks at sparsitiy in data like the probability distribution for crime in different parts of the city. Usually the crime rate is high in certain areas of a city and very low in others. For these sharp changes one needs different methods since we have to allow for jumps. Here the total variation enters the model as the -norm of the gradient. It promotes sparsity of edges in the solution. Before coming up with this concept it was necessary to cross-validate quite a number of times, which is computational very expensive. So instead of in hours the result is obtained in a couple minutes now. When Andrea Bertozzi was a young child she spent a lot of Sundays in the Science museum in Boston and wanted to become a scientist when grown up. The only problem was, that she could not decide which science would be the best choice since she liked everything in the museum. Today she says having chosen applied mathematics indeed she can do all science since mathematics works as a connector between sciences and opens a lot of doors. References Press coverage of Crime prevention collected Website of Mathematical and Simulation Modeling of Crime Examples for work of undergraduates M. Allenby, e.a.: A Point Process Model for Simulating Gang-on-Gang Violence, Project Report, 2010. K. Louie: Statistical Modeling of Gang Violence in Los Angeles, talk at AMS Joint meetings San Francisco, AMS Session on Mathematics in the Social Sciences, 2010] Publications of A. Bertozzi and co-workers on Crime prevention G.O. Mohler e.a.: Randomized controlled field trials of predictive policing, J. Am. Stat. Assoc., 111(512), 1399-1411, 2015. J. T. Woodworth e.a.: Nonlocal Crime Density Estimation Incorporating Housing Information, Phil. Trans. Roy. Soc. A, 372(2028), 20130403, 2014. J. Zipkin, M. B. Short & A. L. Bertozzi: Cops on the dots in a mathematical model of urban crime and police response, Discrete and Continuous Dynamical Systems B, 19(5), pp. 1479-1506, 2014. H. Hu e.a.: A Method Based on Total Variation for Network Modularity Optimization using the MBO Scheme, SIAM J. Appl. Math., 73(6), pp. 2224-2246, 2013. L.M. Smith e.a.: Adaptation of an Ecological Territorial Model to Street Gang Spatial Patterns in Los Angeles Discrete and Continuous Dynamical Systems A, 32(9), pp. 3223 - 3244, 2012. G. Mohler e.a.. (2011): Self- exciting point process modeling of crime, Journal of the American Statistical Association, 106(493):100–108, 2011. A. Stomakhin, M. Short, and A. Bertozzi: Reconstruction of missing data in social networks based on temporal patterns of interactions. Inverse Problems, 27, 2011. N. Rodriguez & A. Bertozzi: Local Existence and Uniqueness of Solutions to a PDE model for Criminal Behavior , M3AS, special issue on Mathematics and Complexity in Human and Life Sciences, Vol. 20, Issue supp01, pp. 1425-1457, 2010. Related Podcasts AMS - Mathematical Moments Podcast: MM97 - Forecasting Crime British Applied Mathematics Colloquium 2016 Special J.Dodd: Crop Growth, Conversation with G. Thäter in the Modellansatz Podcast episode 89, Department of Mathematics, Karlsruhe Institute of Technology (KIT), 2016. http://modellansatz.de/crop-growth H. Wilson: Viscoelastic Fluids, Conversation with G. Thäter in the Modellansatz Podcast episode 92, Department of Mathematics, Karlsruhe Institute of Technology (KIT), 2016. http://modellansatz.de/viscoelastic-fluids A. Hosoi: Robots, Conversation with G. Thäter in the Modellansatz Podcast, episode 108, Department for Mathematics, Karlsruhe Institute of Technologie (KIT), 2016. http://modellansatz.de/robot A. Bertozzi: Crime Prevention, Conversation with G. Thäter in the Modellansatz Podcast, episode 109, Department for Mathematics, Karlsruhe Institute of Technologie (KIT), 2016. http://modellansatz.de/crime-prevention

This is the last of four conversation Gudrun had during the British Applied Mathematics Colloquium which took place 5th – 8th April 2016 in Oxford. Andrea Bertozzi from the University of California in Los Angeles (UCLA) held a public lecture on The Mathematics of Crime. She has been Professor of Mathematics at UCLA since 2003 and Betsy Wood Knapp Chair for Innovation and Creativity (since 2012). From 1995-2004 she worked mostly at Duke University first as Associate Professor of Mathematics and then as Professor of Mathematics and Physics. As an undergraduate at Princeton University she studied physics and astronomy alongside her major in mathematics and went through a Princeton PhD-program. For her thesis she worked in applied analysis and studied fluid flow. As postdoc she worked with Peter Constantin at the University of Chicago (1991-1995) on global regularity for vortex patches. But even more importantly, this was the moment when she found research problems that needed knowledge about PDEs and flow but in addition both numerical analysis and scientific computing. She found out that she really likes to collaborate with very different specialists. Today hardwork can largely be carried out on a desktop but occasionally clusters or supercomputers are necessary. The initial request to work on Mathematics in crime came from a colleague, the social scientist Jeffrey Brantingham. He works in Anthropology at UCLA and had well established contacts with the police in LA. He was looking for mathematical input on some of his problems and raised that issue with Andrea Bertozzi. Her postdoc George Mohler came up with the idea to adapt an earthquake model after a discussion with Frederic Paik Schoenberg, a world expert in that field working at UCLA. The idea is to model crimes of opportunity as being triggered by crimes that already happend. So the likelihood of new crimes can be predicted as an excitation in space and time like the shock of an earthquake. Of course, here statistical models are necessary which say how the excitement is distributed and decays in space and time. Mathematically this is a self-exciting point process. The traditional Poisson process model has a single parameter and thus, no memory - i.e. no connections to other events can be modelled. The Hawkes process builds on the Poisson process as background noise but adds new events which then are triggering events according to an excitation rate and the exponential decay of excitation over time. This is a memory effect based on actual events (not only on a likelihood) and a three parameter model. It is not too difficult to process field data, fit data to that model and make an extrapolation in time. Meanwhile the results of that idea work really well in the field. Results of field trials both in the UK and US have just been published and there is a commercial product available providing services to the police. In addition to coming up with useful ideas and having an interdisciplinary group of people committed to make them work it was necessery to find funding in order to support students to work on that topic. The first grant came from the National Science Foundation and from this time on the group included George Tita (UC Irvine) a criminology expert in LA-Gangs and Lincoln Chayes as another mathematician in the team. The practical implementation of this crime prevention method for the police is as follows: Before the policemen go out on a shift they ususally meet to divide their teams over the area they are serving. The teams take the crime prediction for that shift which is calculated by the computer model on the basis of whatever data is available up to shift. According to expected spots of crimes they especially assign teams to monitor those areas more closely. After introducing this method in the police work in Santa Cruz (California) police observed a significant reduction of 27% in crime. Of course this is a wonderful success story. Another success story involves the career development of the students and postdocs who now have permanent positions. Since this was the first group in the US to bring mathematics to police work this opened a lot of doors for young people involved. Another interesting topic in the context of Mathematics and crime are gang crime data. As for the the crime prediction model the attack of one gang on a rival gang usually triggers another event soon afterwards. A well chosen group of undergraduates already is mathematically educated enough to study the temporary distribution of gang related crime in LA with 30 street gangs and a complex net of enemies. We are speaking about hundreds of crimes in one year related to the activity of gangs. The mathematical tool which proved to be useful was a maximum liklihood penalization model again for the Hawkes process applied on the expected retaliatory behaviour. A more complex problem, which was treated in a PhD-thesis, is to single out gangs which would be probably responsable for certain crimes. This means to solve the inverse problem: We know the time and the crime and want to find out who did it. The result was published in Inverse Problems 2011. The tool was a variational model with an energy which is related to the data. The missing information is guessed and then put into the energy . In finding the best guess related to the chosen energy model a probable candidate for the crime is found. For a small number of unsolved crimes one can just go through all possible combinations. For hundreds or even several hundreds of unsolved crimes - all combinations cannot be handled. We make it easier by increasing the number of choices and formulate a continuous instead of the discrete problem, for which the optimization works with a standard gradient descent algorithm. A third topic and a third tool is Compressed sensing. It looks at sparsitiy in data like the probability distribution for crime in different parts of the city. Usually the crime rate is high in certain areas of a city and very low in others. For these sharp changes one needs different methods since we have to allow for jumps. Here the total variation enters the model as the -norm of the gradient. It promotes sparsity of edges in the solution. Before coming up with this concept it was necessary to cross-validate quite a number of times, which is computational very expensive. So instead of in hours the result is obtained in a couple minutes now. When Andrea Bertozzi was a young child she spent a lot of Sundays in the Science museum in Boston and wanted to become a scientist when grown up. The only problem was, that she could not decide which science would be the best choice since she liked everything in the museum. Today she says having chosen applied mathematics indeed she can do all science since mathematics works as a connector between sciences and opens a lot of doors. References Press coverage of Crime prevention collected Website of Mathematical and Simulation Modeling of Crime Examples for work of undergraduates M. Allenby, e.a.: A Point Process Model for Simulating Gang-on-Gang Violence, Project Report, 2010. K. Louie: Statistical Modeling of Gang Violence in Los Angeles, talk at AMS Joint meetings San Francisco, AMS Session on Mathematics in the Social Sciences, 2010] Publications of A. Bertozzi and co-workers on Crime prevention G.O. Mohler e.a.: Randomized controlled field trials of predictive policing, J. Am. Stat. Assoc., 111(512), 1399-1411, 2015. J. T. Woodworth e.a.: Nonlocal Crime Density Estimation Incorporating Housing Information, Phil. Trans. Roy. Soc. A, 372(2028), 20130403, 2014. J. Zipkin, M. B. Short & A. L. Bertozzi: Cops on the dots in a mathematical model of urban crime and police response, Discrete and Continuous Dynamical Systems B, 19(5), pp. 1479-1506, 2014. H. Hu e.a.: A Method Based on Total Variation for Network Modularity Optimization using the MBO Scheme, SIAM J. Appl. Math., 73(6), pp. 2224-2246, 2013. L.M. Smith e.a.: Adaptation of an Ecological Territorial Model to Street Gang Spatial Patterns in Los Angeles Discrete and Continuous Dynamical Systems A, 32(9), pp. 3223 - 3244, 2012. G. Mohler e.a.. (2011): Self- exciting point process modeling of crime, Journal of the American Statistical Association, 106(493):100–108, 2011. A. Stomakhin, M. Short, and A. Bertozzi: Reconstruction of missing data in social networks based on temporal patterns of interactions. Inverse Problems, 27, 2011. N. Rodriguez & A. Bertozzi: Local Existence and Uniqueness of Solutions to a PDE model for Criminal Behavior , M3AS, special issue on Mathematics and Complexity in Human and Life Sciences, Vol. 20, Issue supp01, pp. 1425-1457, 2010. Related Podcasts AMS - Mathematical Moments Podcast: MM97 - Forecasting Crime British Applied Mathematics Colloquium 2016 Special J.Dodd: Crop Growth, Conversation with G. Thäter in the Modellansatz Podcast episode 89, Department of Mathematics, Karlsruhe Institute of Technology (KIT), 2016. http://modellansatz.de/crop-growth H. Wilson: Viscoelastic Fluids, Conversation with G. Thäter in the Modellansatz Podcast episode 92, Department of Mathematics, Karlsruhe Institute of Technology (KIT), 2016. http://modellansatz.de/viscoelastic-fluids A. Hosoi: Robots, Conversation with G. Thäter in the Modellansatz Podcast, episode 108, Department for Mathematics, Karlsruhe Institute of Technologie (KIT), 2016. http://modellansatz.de/robot A. Bertozzi: Crime Prevention, Conversation with G. Thäter in the Modellansatz Podcast, episode 109, Department for Mathematics, Karlsruhe Institute of Technologie (KIT), 2016. http://modellansatz.de/crime-prevention

This is another conversation Gudrun had during the British Applied Mathematics Colloquium which took place 5th – 8th April 2016 in Oxford. Since 2002 Anette Hosoi has been Professor of Mechanical Engineering at MIT (in Cambridge, Massachusetts). She is also a member of the Mathematical Faculty at MIT. After undergraduate education in Princeton she changed to Chicago for a Master's and her PhD in physics. Anette Hosoi wanted to do fluid dynamics even before she had any course on that topic. Then she started to work as Assistant Professor at MIT where everyone wanted to build robots. So she had to find an intersection between fluid and roboters. Her first project were Robo-snailes with her student Brian Chan. Snails move using a thin film of fluid under their foot (and muscles). Since then she has been working on the fascinating boundary of flow and biomechanics. At the BAM Colloquium she was invited for a plenary lecture on "Marine Mammals and Fluid Rectifiers: The Hydrodynamics of Hairy Surfaces". It started with a video of Boston dynamics which showed the terrific abilities some human-like robots have today. Nevertheless, these robots are rigid systems with a finite number of degrees of freedom. Anette Hosoi is working in control and fluid mechanics and got interested in soft systems in the context of robots of a new type. Soft systems are a completely new way to construct robots and for that one has to rethink everything from the bottom up.You are a dreamer she was told for that more than once. For example Octopuses (and snails) move completely different to us and most animals the classcallly designed robots with two, four or more legs copy. At the moment the investigation of those motions is partially triggered by the plausible visualization in computer games and in animated movie sequences. A prominent example for that is the contribution of two mathematicians at UCLA to represent all interactions with snow in the animated movie Frozen. The short verison of their task was to get the physics right when snow falls off trees or people fall into snow - otherwise it just doesn't look right. To operate robots which are not built with mechanical devices but use properties of fluids to move one needs valves and pumps to control flow. They should be cheap and efficient and without any moving parts (since moving parts cause problems). A first famous example for such component is a fluid rectifier which was patented by Nicola Tesla in the 1920ies. His device relied on inertia. But in the small devices as necessary for the new robots there are no inertia. For that Anette Hosoi and her group need to implement new mechnisms. A promising effect is elasticity - especially in channels. Or putting hair on the boundary of channels. Hair can cause asymmetric behaviour in the system. In one direction it bends easily with the flow while in the opposite direction it might hinder flow. While trying to come up with clever ideas for the new type of robots the group found a topic which is present (almost) everywhere in biology - which means a gold mine for research and open questions. Of course hair is interacting with the flow and not just a rigid boundary and one has to admit that in real life applications the related flow area usually is not small (i.e. not negligible in modelling and computations). Mathematically spoken, the model needs a change in the results for the boundary layer. This is clear from the observations and the sought after applications. But it is clear from the mathematical model as well. At the moment they are able to treat the case of low Reynolds number and the linear Stokes equation which of course, is a simplification. But for that case the new boundary conditions are not too complicated and can be treated similar as for porous media (i.e. one has to find an effective permeability). Fortunately even analytic solutions could be calculated. As next steps it would be very interesting to model plunging hairy surfaces into fluids or withdrawing hairy surfaces from fluids (which is even more difficult). This would have a lot of interesting applications and a first question could be to find optimal hair arrangements. This would mean to copy tricks of bat tongues like people at Brown University are doing. References I. E. Block Community Lecture: Razor Clams to Robots: The Mathematics Behind Biologically Inspired Design , A. Hosoi at SIAM Annual meeting, 2013. B. Chan, N.J. Balmforth and A.E. Hosoi: Building a better snail: Lubrication and adhesive locomotion, Phys. Fluids, 17, 113101, 2005.

This is another conversation Gudrun had during the British Applied Mathematics Colloquium which took place 5th – 8th April 2016 in Oxford. Since 2002 Anette Hosoi has been Professor of Mechanical Engineering at MIT (in Cambridge, Massachusetts). She is also a member of the Mathematical Faculty at MIT. After undergraduate education in Princeton she changed to Chicago for a Master's and her PhD in physics. Anette Hosoi wanted to do fluid dynamics even before she had any course on that topic. Then she started to work as Assistant Professor at MIT where everyone wanted to build robots. So she had to find an intersection between fluid and roboters. Her first project were Robo-snailes with her student Brian Chan. Snails move using a thin film of fluid under their foot (and muscles). Since then she has been working on the fascinating boundary of flow and biomechanics. At the BAM Colloquium she was invited for a plenary lecture on "Marine Mammals and Fluid Rectifiers: The Hydrodynamics of Hairy Surfaces". It started with a video of Boston dynamics which showed the terrific abilities some human-like robots have today. Nevertheless, these robots are rigid systems with a finite number of degrees of freedom. Anette Hosoi is working in control and fluid mechanics and got interested in soft systems in the context of robots of a new type. Soft systems are a completely new way to construct robots and for that one has to rethink everything from the bottom up.You are a dreamer she was told for that more than once. For example Octopuses (and snails) move completely different to us and most animals the classcallly designed robots with two, four or more legs copy. At the moment the investigation of those motions is partially triggered by the plausible visualization in computer games and in animated movie sequences. A prominent example for that is the contribution of two mathematicians at UCLA to represent all interactions with snow in the animated movie Frozen. The short verison of their task was to get the physics right when snow falls off trees or people fall into snow - otherwise it just doesn't look right. To operate robots which are not built with mechanical devices but use properties of fluids to move one needs valves and pumps to control flow. They should be cheap and efficient and without any moving parts (since moving parts cause problems). A first famous example for such component is a fluid rectifier which was patented by Nicola Tesla in the 1920ies. His device relied on inertia. But in the small devices as necessary for the new robots there are no inertia. For that Anette Hosoi and her group need to implement new mechnisms. A promising effect is elasticity - especially in channels. Or putting hair on the boundary of channels. Hair can cause asymmetric behaviour in the system. In one direction it bends easily with the flow while in the opposite direction it might hinder flow. While trying to come up with clever ideas for the new type of robots the group found a topic which is present (almost) everywhere in biology - which means a gold mine for research and open questions. Of course hair is interacting with the flow and not just a rigid boundary and one has to admit that in real life applications the related flow area usually is not small (i.e. not negligible in modelling and computations). Mathematically spoken, the model needs a change in the results for the boundary layer. This is clear from the observations and the sought after applications. But it is clear from the mathematical model as well. At the moment they are able to treat the case of low Reynolds number and the linear Stokes equation which of course, is a simplification. But for that case the new boundary conditions are not too complicated and can be treated similar as for porous media (i.e. one has to find an effective permeability). Fortunately even analytic solutions could be calculated. As next steps it would be very interesting to model plunging hairy surfaces into fluids or withdrawing hairy surfaces from fluids (which is even more difficult). This would have a lot of interesting applications and a first question could be to find optimal hair arrangements. This would mean to copy tricks of bat tongues like people at Brown University are doing. References I. E. Block Community Lecture: Razor Clams to Robots: The Mathematics Behind Biologically Inspired Design , A. Hosoi at SIAM Annual meeting, 2013. B. Chan, N.J. Balmforth and A.E. Hosoi: Building a better snail: Lubrication and adhesive locomotion, Phys. Fluids, 17, 113101, 2005.

This is the second of four conversations Gudrun had during the British Applied Mathematics Colloquium which took place 5th – 8th of April 2016 in Oxford. Helen Wilson always wanted to do maths and had imagined herself becoming a mathematician from a very young age. But after graduation she did not have any road map ready in her mind. So she applied for jobs which - due to a recession - did not exist. Today she considers herself lucky for that since she took a Master's course instead (at Cambridge University), which hooked her to mathematical research in the field of viscoelastic fluids. She stayed for a PhD and after that for postdoctoral work in the States and then did lecturing at Leeds University. Today she is a Reader in the Department of Mathematics at University College London. So what are viscoelastic fluids? If we consider everyday fluids like water or honey, it is a safe assumption that their viscosity does not change much - it is a material constant. Those fluids are called Newtonian fluids. All other fluids, i.e. fluids with non-constant viscosity or even more complex behaviours, are called non-Newtonian and viscoelastic fluids are a large group among them. Already the name suggests, that viscoelastic fluids combine viscous and elastic behaviour. Elastic effects in fluids often stem from clusters of particles or long polymers in the fluid, which align with the flow. It takes them a while to come back when the flow pattern changes. We can consider that as keeping a memory of what happened before. This behaviour can be observed, e.g., when stirring tinned tomato soup and then waiting for it to go to rest again. Shortly before it finally enters the rest state one sees it springing back a bit before coming to a halt. This is a motion necessary to complete the relaxation of the soup. Another surprising behaviour is the so-called Weissenberg effect, where in a rotation of elastic fluid the stretched out polymer chains drag the fluid into the center of the rotation. This leads to a peak in the center, instead of a funnel which we expect from experiences stirring tea or coffee. The big challenge with all non-Newtonian fluids is that we do not have equations which we know are the right model. It is mostly guess work and we definitely have to be content with approximations. And so it is a compromise of fitting what we can model and measure to the easiest predictions possible. Of course, slow flow often can be considered to be Newtonian whatever the material is. The simplest models then take the so-called retarded fluid assumption, i.e. the elastic properties are considered to be only weak. Then, one can expand around the Newtonian model as a base state. The first non-linear model which is constructed in that way is that of second-order fluids. They have two more parameters than the Newtonian model, which are called normal stress coefficients. The next step leads to third-order fluids etc. In practice no higher than third-order fluids are investigated. Of course there are a plethora of interesting questions connected to complex fluids. The main question in the work of Helen Wilson is the stability of the flow of those fluids in channels, i.e. how does it react to small perturbations? Do they vanish in time or could they build up to completely new flow patterns? In 1999, she published results of her PhD thesis and predicted a new type of instability for a shear-thinning material model. It was to her great joy when in 2013 experimentalists found flow behaviour which could be explained by her predicted instability. More precisely, in the 2013 experiments a dilute polymer solution was sent through a microchannel. The material model for the fluid is shear thinning as in Helen Wilson's thesis. They observed oscillations from side to side of the channel and surprising noise in the maximum flow rate. This could only be explained by an instability which they did not know about at that moment. In a microchannel inertia is negligible and the very low Reynolds number of suggested that the instability must be caused by the non-Newtonian material properties since for Newtonian fluids instabilities can only be observed if the flow configuration exeeds a critical Reynolds number. Fortunately, the answer was found in the 1999 paper. Of course, even for the easiest non-linear models one arrives at highly non-linear equations. In order to analyse stability of solutions to them one firstly needs to know the corresponding steady flow. Fortunately, if starting with the easiest non-linear models in a channel one can still find the steady flow as an analytic solution with paper and pencil since one arrives at a 1D ODE, which is independent of time and one of the two space variables. The next question then is: How does it respond to small perturbation? The classical procedure is to linearize around the steady flow which leads to a linear problem to solve in order to know the stability properties. The basic (steady) flow allows for Fourier transformation which leads to a problem with two scalar parameters - one real and one complex. The general structure is an eigenvalue problem which can only be solved numerically. After we know the eigenvalues we know about the (so-called linear) stability of the solution. An even more interesting research area is so-called non-linear stability. But it is still an open field of research since it has to keep the non-linear terms. The difference between the two strategies (i.e. linear and non-linear stability) is that the linear theory predicts instability to the smallest perturbations but the non-linear theory describes what happens after finite-amplitude instability has begun, and can find larger instability regions. Sometimes (but unfortunately quite rarely) both theories find the same point and we get a complete picture of when a stable region changes into an unstable one. One other really interesting field of research for Helen Wilson is to find better constitutive relations. Especially since the often used power law has inbuilt unphysical behaviour (which means it is probably too simple). For example, taking a power law with negative exponent says that In the middle of the flow there is a singularity (we would divide by zero) and perturbations are not able to cross the center line of a channel. Also, it is unphysical that according to the usual models the shear-thinning fluid should be instantly back to a state of high viscosity after switching off the force. For example most ketchup gets liquid enough to serve it only when we shake it. But it is not instantly thick after the shaking stops - it takes a moment to solidify. This behaviour is called thixotropy. Literature and additional material H. Wilson: UCL Lunch Hour Lectures, Feb. 2016. H.J. Wilson and J.M. Rallison: Instability of channel flow of a shear-thinning White–Metzner fluid, Journal of Non-Newtonian Fluid Mechanics 87 (1999) 75–96. Hugues Bodiguel, Julien Beaumont, Anaïs Machado, Laetitia Martinie, Hamid Kellay, and Annie Colin: Flow Enhancement due to Elastic Turbulence in Channel Flows of Shear Thinning Fluids, Physical Review Letters 114 (2015) 028302. Non-Newtonian Fluids Explained, Science Learning.

This is the second of four conversations Gudrun had during the British Applied Mathematics Colloquium which took place 5th – 8th of April 2016 in Oxford. Helen Wilson always wanted to do maths and had imagined herself becoming a mathematician from a very young age. But after graduation she did not have any road map ready in her mind. So she applied for jobs which - due to a recession - did not exist. Today she considers herself lucky for that since she took a Master's course instead (at Cambridge University), which hooked her to mathematical research in the field of viscoelastic fluids. She stayed for a PhD and after that for postdoctoral work in the States and then did lecturing at Leeds University. Today she is a Reader in the Department of Mathematics at University College London. So what are viscoelastic fluids? If we consider everyday fluids like water or honey, it is a safe assumption that their viscosity does not change much - it is a material constant. Those fluids are called Newtonian fluids. All other fluids, i.e. fluids with non-constant viscosity or even more complex behaviours, are called non-Newtonian and viscoelastic fluids are a large group among them. Already the name suggests, that viscoelastic fluids combine viscous and elastic behaviour. Elastic effects in fluids often stem from clusters of particles or long polymers in the fluid, which align with the flow. It takes them a while to come back when the flow pattern changes. We can consider that as keeping a memory of what happened before. This behaviour can be observed, e.g., when stirring tinned tomato soup and then waiting for it to go to rest again. Shortly before it finally enters the rest state one sees it springing back a bit before coming to a halt. This is a motion necessary to complete the relaxation of the soup. Another surprising behaviour is the so-called Weissenberg effect, where in a rotation of elastic fluid the stretched out polymer chains drag the fluid into the center of the rotation. This leads to a peak in the center, instead of a funnel which we expect from experiences stirring tea or coffee. The big challenge with all non-Newtonian fluids is that we do not have equations which we know are the right model. It is mostly guess work and we definitely have to be content with approximations. And so it is a compromise of fitting what we can model and measure to the easiest predictions possible. Of course, slow flow often can be considered to be Newtonian whatever the material is. The simplest models then take the so-called retarded fluid assumption, i.e. the elastic properties are considered to be only weak. Then, one can expand around the Newtonian model as a base state. The first non-linear model which is constructed in that way is that of second-order fluids. They have two more parameters than the Newtonian model, which are called normal stress coefficients. The next step leads to third-order fluids etc. In practice no higher than third-order fluids are investigated. Of course there are a plethora of interesting questions connected to complex fluids. The main question in the work of Helen Wilson is the stability of the flow of those fluids in channels, i.e. how does it react to small perturbations? Do they vanish in time or could they build up to completely new flow patterns? In 1999, she published results of her PhD thesis and predicted a new type of instability for a shear-thinning material model. It was to her great joy when in 2013 experimentalists found flow behaviour which could be explained by her predicted instability. More precisely, in the 2013 experiments a dilute polymer solution was sent through a microchannel. The material model for the fluid is shear thinning as in Helen Wilson's thesis. They observed oscillations from side to side of the channel and surprising noise in the maximum flow rate. This could only be explained by an instability which they did not know about at that moment. In a microchannel inertia is negligible and the very low Reynolds number of suggested that the instability must be caused by the non-Newtonian material properties since for Newtonian fluids instabilities can only be observed if the flow configuration exeeds a critical Reynolds number. Fortunately, the answer was found in the 1999 paper. Of course, even for the easiest non-linear models one arrives at highly non-linear equations. In order to analyse stability of solutions to them one firstly needs to know the corresponding steady flow. Fortunately, if starting with the easiest non-linear models in a channel one can still find the steady flow as an analytic solution with paper and pencil since one arrives at a 1D ODE, which is independent of time and one of the two space variables. The next question then is: How does it respond to small perturbation? The classical procedure is to linearize around the steady flow which leads to a linear problem to solve in order to know the stability properties. The basic (steady) flow allows for Fourier transformation which leads to a problem with two scalar parameters - one real and one complex. The general structure is an eigenvalue problem which can only be solved numerically. After we know the eigenvalues we know about the (so-called linear) stability of the solution. An even more interesting research area is so-called non-linear stability. But it is still an open field of research since it has to keep the non-linear terms. The difference between the two strategies (i.e. linear and non-linear stability) is that the linear theory predicts instability to the smallest perturbations but the non-linear theory describes what happens after finite-amplitude instability has begun, and can find larger instability regions. Sometimes (but unfortunately quite rarely) both theories find the same point and we get a complete picture of when a stable region changes into an unstable one. One other really interesting field of research for Helen Wilson is to find better constitutive relations. Especially since the often used power law has inbuilt unphysical behaviour (which means it is probably too simple). For example, taking a power law with negative exponent says that In the middle of the flow there is a singularity (we would divide by zero) and perturbations are not able to cross the center line of a channel. Also, it is unphysical that according to the usual models the shear-thinning fluid should be instantly back to a state of high viscosity after switching off the force. For example most ketchup gets liquid enough to serve it only when we shake it. But it is not instantly thick after the shaking stops - it takes a moment to solidify. This behaviour is called thixotropy. Literature and additional material H. Wilson: UCL Lunch Hour Lectures, Feb. 2016. H.J. Wilson and J.M. Rallison: Instability of channel flow of a shear-thinning White–Metzner fluid, Journal of Non-Newtonian Fluid Mechanics 87 (1999) 75–96. Hugues Bodiguel, Julien Beaumont, Anaïs Machado, Laetitia Martinie, Hamid Kellay, and Annie Colin: Flow Enhancement due to Elastic Turbulence in Channel Flows of Shear Thinning Fluids, Physical Review Letters 114 (2015) 028302. Non-Newtonian Fluids Explained, Science Learning.

This is the first of four conversation Gudrun had during the British Applied Mathematics Colloquium which took place 5th – 8th of April 2016 in Oxford. Josie Dodd finished her Master's in Mathematical and Numerical Modelling of the Atmosphere and Oceans at the University of Reading. In her PhD project she is working in the Mathematical Biology Group inside the Department of Mathematics and Statistics in Reading. In this group she develops models that describe plant and canopy growth of the Bambara Groundnut - especially the plant interaction when grown as part of a crop. The project is interdisciplinary and interaction with biologists is encouraged by the funding entity. Why is this project so interesting? In general, the experimental effort to understand crop growth is very costly and takes a lot of time. So it is a great benefit to have cheaper and faster computer experiments. The project studies the Bambara Groundnut since it is a candidate for adding to our food supply in the future. It is an remarkably robust crop, draught tolerant and nitrogent inriching, which means the production of yield does not depend on fertilizer. The typical plant grows 150 days per year. The study will find results for which verfication and paramater estimations from actual green house data is available. On the other hand, all experience on the modelling side will be transferable to other plants up to a certain degree. The construction of the mathematical model includes finding equations which are simple enough but cover the main processes as well as numerical schemes which solve them effectively. At the moment, temperature and solar radiation are the main input to the model. In the future, it should include rain as well. Another important parameter is the placement of the plants - especially in asking for arrangements which maximize the yield. Analyzing the available data from the experimental partners leads to three nonlinear ODEs for each plant. Also, the leave production has a Gaussian distribution relationship with time and temperature. The results then enter the biomass equation. The growth process of the plant is characterized by a change of the rate of change over time. This is a property of the plant that leads to nonlinearity in the equations. Nevertheless, the model has to stay as simple as possible, while firstly, bridging the gap to complicated and more precise models, and secondly, staying interpretable to make people able to use it and understand its behaviour as non-mathematicians. This is the main group for which the models should be a useful tool. So far, the model for interaction with neighbouring plants is the computational more costly part, where - of course - geometric consideration of overlapping have to enter the model. Though it does not yet consider many plants (since green house sized experimental data are available) the model scales well to a big number of plants due to its inherent symmetries. Since at the moment the optimizaition of the arrangements of plants has a priority - a lot of standardization and simplifying assumptions are applied. So for the future more parameters such as the input of water should be included, and it would be nice to have more scales. Such additional scales would be to include the roots system or other biological processes inside the plant. Of course, the green house is well controlled and available field data are less precise due to the difficulty of measurements in the field. During her work on the project and as a tutor Josie Dodd found out that she really likes to do computer programming. Since it is so applicable to many things theses skills open a lot of doors. Therefore, she would encourage everybody to give it a try. Literature and additional material Crops for the Future website Asha Sajeewani Karunaratne: Modelling the response of Bambara groundnut (Vigna subterranea (L.) Verdc) for abiotic stress, PhD thesis, University of Nottingham (2009). A.S. Karunaratne e.a.: Modelling the canopy development of bambara groundnut, Agricultural and Forest Meteorology 150, (7–8) 2010, 1007–1015.

This is the first of four conversation Gudrun had during the British Applied Mathematics Colloquium which took place 5th – 8th of April 2016 in Oxford. Josie Dodd finished her Master's in Mathematical and Numerical Modelling of the Atmosphere and Oceans at the University of Reading. In her PhD project she is working in the Mathematical Biology Group inside the Department of Mathematics and Statistics in Reading. In this group she develops models that describe plant and canopy growth of the Bambara Groundnut - especially the plant interaction when grown as part of a crop. The project is interdisciplinary and interaction with biologists is encouraged by the funding entity. Why is this project so interesting? In general, the experimental effort to understand crop growth is very costly and takes a lot of time. So it is a great benefit to have cheaper and faster computer experiments. The project studies the Bambara Groundnut since it is a candidate for adding to our food supply in the future. It is an remarkably robust crop, draught tolerant and nitrogent inriching, which means the production of yield does not depend on fertilizer. The typical plant grows 150 days per year. The study will find results for which verfication and paramater estimations from actual green house data is available. On the other hand, all experience on the modelling side will be transferable to other plants up to a certain degree. The construction of the mathematical model includes finding equations which are simple enough but cover the main processes as well as numerical schemes which solve them effectively. At the moment, temperature and solar radiation are the main input to the model. In the future, it should include rain as well. Another important parameter is the placement of the plants - especially in asking for arrangements which maximize the yield. Analyzing the available data from the experimental partners leads to three nonlinear ODEs for each plant. Also, the leave production has a Gaussian distribution relationship with time and temperature. The results then enter the biomass equation. The growth process of the plant is characterized by a change of the rate of change over time. This is a property of the plant that leads to nonlinearity in the equations. Nevertheless, the model has to stay as simple as possible, while firstly, bridging the gap to complicated and more precise models, and secondly, staying interpretable to make people able to use it and understand its behaviour as non-mathematicians. This is the main group for which the models should be a useful tool. So far, the model for interaction with neighbouring plants is the computational more costly part, where - of course - geometric consideration of overlapping have to enter the model. Though it does not yet consider many plants (since green house sized experimental data are available) the model scales well to a big number of plants due to its inherent symmetries. Since at the moment the optimizaition of the arrangements of plants has a priority - a lot of standardization and simplifying assumptions are applied. So for the future more parameters such as the input of water should be included, and it would be nice to have more scales. Such additional scales would be to include the roots system or other biological processes inside the plant. Of course, the green house is well controlled and available field data are less precise due to the difficulty of measurements in the field. During her work on the project and as a tutor Josie Dodd found out that she really likes to do computer programming. Since it is so applicable to many things theses skills open a lot of doors. Therefore, she would encourage everybody to give it a try. Literature and additional material Crops for the Future website Asha Sajeewani Karunaratne: Modelling the response of Bambara groundnut (Vigna subterranea (L.) Verdc) for abiotic stress, PhD thesis, University of Nottingham (2009). A.S. Karunaratne e.a.: Modelling the canopy development of bambara groundnut, Agricultural and Forest Meteorology 150, (7–8) 2010, 1007–1015.