Modellansatz - English episodes only

This is one of two conversations which Gudrun Thäter recorded alongside the conference Women in PDEs which took place at our Department in Karlsruhe on 27-28 April 2017.

Maria Lopez-Fernandez from the University La Sapienza in Rome was one of the seven invited speakers. She got her university degree at the University of Valladolid in Spain and worked as an academic researcher in Madrid and at the University of Zürich.

Her field of research is numerical analyis and in particular the robust and efficient approximation of convolutions. The conversation is mainly focussed on its applications to wave scattering problems. The important questions for the numerical tools are: Consistency, stability and convergence analysis. The methods proposed by Maria are Convolution Quadrature type methods for the time discretization coupled with the boundary integral methods for the spatial discretization. Convolution Quadrature methods are based on Laplace transformation and numerical integration. They were initially mostly developed for parabolic problems and are now adapted to serve in the context of (hyperbolic) wave equations. Convolution quadrature methods introduce artificial dissipation in the computation, which stabilzes the numerics. However it would be physically more meaningful to work instead with schemes which conserve mass.

She is mainly interested in

• fast algorithms with
• reduced memory requirements and
• adaptivity in time and space.

The motivational example for her talk was the observation of severe acoustic problems inside a new building at the University of Zürich. Any conversation in the atrium made a lot of noise and if someone was speaking loud it was hard to understand by the others. An improvement was provided by specialised engineers who installed absorbing panels. From the mathematical point of view this is an nice application of the modelling and numerics of wave scattering problems. Of course, it would make a lot of sense to simulate the acoustic situation for such spaces before building them - if stable fast software for the distribution of acoustic pressure or the transport of signals was available.

The mathematical challenges are high computational costs, high storage requirements and and stability problems. Due to the nonlocal nature of the equations it is also really hard to make the calculations in parallel to run faster. In addition time-adaptive methods for these types of problems were missing completely in the mathematical literature. In creating them one has to control the numerical errors with the help of a priori and a posteriori estimates which due to Maria's and others work during the last years is in principle known now but still very complicated. Also one easily runs into stability problems when changing the time step size.

The acoustic pressure distribution for the new building in Zürich has been sucessfully simulated by co-workers in Zürich and Graz by using these results together with knowledge about the sound-source and deriving heuristic measures from that in order to find a sequence of time steps which keeps the problem stable and adapt to the computations effectively.

There is a lot of hope to improve the performance of these tools by representing the required boundary element matrices by approximations with much sparser matrices.

• M. López Fernández, S. Sauter: Generalized Convolution Quadrature with Variable Time Stepping. Part II: Algorithm and Numerical Results. Applied Numerical Mathematics, 94, pp. 88 - 105 (2015)
• M. López Fernández, S. Sauter: Generalized Convolution Quadrature based on Runge-Kutta Methods. Numerische Mathematik, 133 (4), pp. 734 - 779 (2016)
• S. Sauter, M. Schanz: Convolution Quadrature for the Wave Equation with Impedance Boundary Conditions. Journal of Computational Physics, Vol 334, pp. 442 - 459 (2017)

• T. Arens: Lärmschutz, Gespräch mit S. Ritterbusch im Modellansatz Podcast, Folge 16, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2014.
• F. Sayas: Acoustic Scattering, Conversation with G. Thäter in the Modellansatz Podcast, Episode 58, Department of Mathematics, Karlsruhe Institute of Technology (KIT), 2016.

Constanza Rojas-Molina is a postdoc at the Institute of Applied Mathematics of the University of Bonn. Gudrun Thäter met her in Bonn to talk about Constanza's blog The Rage of the Blackboard. The blog’s title makes reference to an angry blackboard, but also to the RAGE Theorem, named after the mathematical physicists D. Ruelle, W. Amrein, V. Georgescu, and V. Enss." Standing at a blackboard can be intimidating and quite a few might remember moments of anxiety when being asked to develop an idea in front of others at the blackboard. But as teachers and scientists we work with the blackboard on a daily basis and find a way to "tame" its "rage".

Gudrun and Constanza share that they are working in fields of mathematics strongly intertwined with physics. While Gudrun is interested in Mathematical Fluid dynamics, Constanza's field is Mathematical physics. Results in both fields very much rely on understanding the spectrum of linear (or linearized) operators. In the finite-dimensional case this means to study the eigenvalues of a matrix. They contain the essence of the action of the operator - represented by different matrices in differing coordinate systems. As women in academia and as female mathematicians Gudrun and Constanza share the experience that finding the essence of their actions in science and defining the goals worth to pursue are tasks as challenging as pushing science itself, since most traditional coordinate systems were made by male colleagues and do not work in the same way for women as for men. This is true even when raising own children does not enter the equation.

For that Constanza started to reach out to women in her field to speak about their mathematical results as well as their experiences. Her idea was to share the main findings in her blog with an article and her drawings. When reaching out to a colleague she sends a document explaining the goal of the project and her questions in advance. Constanza prepares for the personal conversation by reading up about the mathematical results. But at the same moment she is interested in questions like: how do you work, how do you come up with ideas, what do you do on a regular day, etc.

The general theme of all conversations is that a regular day does not exist when working at university. It seems that the only recurring task is daily improvisation on any schedule made in advance. One has to optimize how to live with the peculiar situation being pushed to handle several important tasks at once at almost any moment and needs techniques to find compromise and balance. An important question then is: how to stay productive and satisfied under these conditions, how to manage to stay in academia and what personal meaning does the word success then take. In order to distill the answers into a blog entry Constanza uses only a few quotes and sums up the conversation in a coherent text. Since she seeks out very interesting people, there is a lot of interesting material. Constanza focuses on the aspects that stay with her after a longer thought process. These ideas then mainly drive the blog article. Another part of the blog are two drawings: one portrait of the person and one which pictures the themes that were discussed and might not have made it into the text.

Surprisingly it turned out to be hard to find partners to talk to, and the process to make it a blog entry takes Constanza a year or longer. On the other hand, she feels very lucky that she found women which were very generous with their time and in sharing their experiences. Besides the engagement and love for what they do, all the participants had this in common: they were already promoting the participation of women in science. To learn from them as a younger researcher means, for example, to see the own impact on students and that building a community is very important, and a success in its own. Though Constanza invests a lot of time in the blog project, it is worth the effort since it helps her to work towards a future either in or outside academia.

Gudrun and Constanza found out that though both of their projects explore mathematical themes as well as people working in mathematics, the written parts of blog and podcast differ in that what makes it into the notes in Constanza's blog is, so to say, bonus material available only for the listening audience in Gudruns podcast (since it is never in the shownotes). In that sense, Gudrun's podcast and Constanza's blog are complementary views on the life of researchers.

Constanza did her undergraduate studies in La Serena in Chile. She started out with studying physics but soon switched to mathematics in order to understand the basics of physics. When she had almost finished her Masters program in La Serena she wanted to continue in science abroad. She was admitted to a french (one year) Master program at the University Paris 6 and later did her PhD in the nearby University Cergy-Pontoise. After that she applied for a Marie-Curie fellowship in order to continue her research in Germany. She spent time as postdoc at the Mittag-Leffler-Institut in Stockholm and at CAMTP in Maribor (Slovenia) before moving to the LMU Munich for two years with the fellowship. After that she got the position in Bonn and is now preparing for her next step.

Gudrun and Constanza want to thank Tobias Ried who put them in contact.

• Women in Math statement
• Constanza Rojas-Molina: Etude mathématique des propriétés de transport des opérateurs de Schrödinger aléatoires avec structure quasi-cristalline (The mathematical study of electronic transport in random Schrödinger operators with quasicrystalline structures). PhD-thesis Université de Cergy-Pontoise, 2012.
• A. Pohl: Quantenchaos, Conversation with G. Thäter in the Modellansatz Podcast, Episode 79, Fakultät für Mathematik, Karlsruher Institut für Technologie (KIT), 2016. http://modellansatz.de/quantenchaos
• Constanza's illustration blog

Pascal Kraft is a researcher at the Institute for Applied and Numerical Mathematics of the Karlsruhe Institute of Technology (KIT) and he introduces us to Julia Sets which he investigated for his Bachelors Thesis. It is natural for us to think something like this: If I take two simple things and put them together in some sense, nothing too complex should arise from that. A fascinating result of the work of mathematicians like Gaston Julia and Benoît Mandelbrot dating back to the first half of the 20th century show that this assumption doesn't always hold.

In his bachelor's thesis under supervision of Jan-Philipp Weiß, Pascal Kraft worked on the efficient computation of Julia Sets. In laymans terms you can describe these sets as follows: Some electronic calculators have the functions of repeating the last action if you press "=" or "enter" multiple times. So if you used the root function of your calculator on a number and now you want the root of the result you simply press "=" again. Now imagine you had a function on your calculater that didn't only square the input but also added a certain value - say 0.5. Then you put in a number, apply this function and keep repeating it over and over again. Now you ask yourself if you keep pressing the "="-button if the result keeps on growing and tends to infinity or if it stays below some threshold indefinitely.

Using real numbers this concept is somewhat boring but if we use complex numbers we find, that the results are astonishing.

To use a more precise definition: for a function f(z), the Filled Julia Set is defined as the set of values z, for whom the series (fn(z))n stays bounded. The Julia Set is defined as the boundary of this set. A typical example for a suitable function f(z) in this context is f(z)=z2+c. We now look at the complex plane where the x-axis represents the real part of a complex number and the y-axis its imaginary part. For each point on this plane having a coordinate (x,y) we take the corresponding complex number z=x+iy and plug this value into our function f(z) and the results over and over again up to a certain degree until we see if this sequence diverges. Computing a graphical representation of such a Julia Set is a numerically costly task since we have no other way of determining its interior points other then trying out a large amount of starting points and seeing what happens after hundreds of iterations.

The results, however, turn out to be surprising and worth the effort. The geometric representations - images - of filled Julia Sets turn out to be very aesthetically pleasing since they are no simple compositions of elementary shapes but rather consist of intricate shapes and patterns. The reason for these beautiful shapes lie in the nature of multiplication and addition on the complex plane: A multiplication can be a magnification and down-scaling, mirroring and rotation, whereas the complex addition is represented by a translation on the complex plane. Since the function is applied over and over again, the intrinsic features are repeated in scaled and rotated forms over and over again, and this results in a self-similarity on infinite scales. In his bachelor's thesis, Pascal focussed on the efficient computation of such sets which can mean multiple things: it can either mean that the goal was to quickly write a program which could generate an image of a Julia Set, or that a program was sought which was very fast in computing such a program. Lastly it can also mean that we want to save power and seek a program which uses computational power efficiently to compute such an image, i.e. that consumes little energy. This is a typical problem when considering a numerical approach in any application and it arises very naturally here: While the computation of Julia Sets can greatly benefit from parallelization, the benefits are at loss when many tasks are waiting for one calculation and therefore the speedup and computational efficiency breaks down due to Amdahl's law.

The difference of these optimization criteria becomes especially obvious when we want to do further research ontop of our problem solver that we have used so far. The Mandelbrot Set for example is the set of values c, for whom the Filled Julia Set is not equal to the Julia Set (i.e. the Filled Julia Set has interior points). One detail is important for the computation of either of these sets: If we check one single point we can never really say if it is inside the Filled Julia Set for sure (unless we can prove periodicity but that is not really feasible). What we can show however is, that if the magnitude of a point in the series of computations is above a certain bound, the results will tend to infinity from this point on. The approach is therefore to compute steps until either a maximum of steps is reached or a certain threshold is exceeded. Based on this assumption, we see that computing a point which lies inside the filled Julia Set is the bigger effort. So if computing a Julia Set for a given parameter is a lot of work, its complex parameter c most likely lies inside the Mandelbrot Set (as we find many points for whom the computation doesn't abort prematurely and it is therefore likely that some of these points will be interior). If we want to draw the Mandelbrot Set based on this approach, we have to compute thousands of Julia Sets and if the computation of a single image was to take a minute this would not really be feasible anymore.

Since the computation of a Julia Set can even be done in a webbrowser these days, we include below a little tool which lets you set a complex parameter c and compute four different Julia Sets. Have fun with our Interactive Julia Sets!

• J. Dufner, A. Roser, F. Unseld: Fraktale und Julia-Mengen, Harri Deutsch Verlag, 1998.
• H.-O. Peitgen, P. H. Richter: The beauty of fractals: images of complex dynamical systems, Springer Berlin Heidelberg, 1986.
• B. B. Mandelbrot: Fractal aspects of the iteration of z→λz(1−z) for complex λ and z, Annals of the New York Academy of Sciences 357.1: 249-259, 1980.
• P. Kraft: Paralleles Rechnen auf GPUs - Julia Mengen und das magnetische Pendel Fraktal, Bachelor Thesis. 2012.
• J. Gaston: Mémoire sur l’itération des fonctions rationnelles, Journal de Math´ematiques pures et appliqu ´ees 4 (Rep 1968), pp. 47-245 / 121-319, 1918.
• P. Blanchard: Complex analytical dynamics on the Riemann sphere, Bulletin of the American Mathematical Society 11, pp. 84-141, 1984.

Andrii Khrabustovskyi works at our faculty in the group Nonlinear Partial Differential Equations and is a member of the CRC Wave phenomena: analysis and numerics. He was born in Kharkiv in the Ukraine and finished his studies as well as his PhD at the Kharkiv National University and the Institute for Low Temperature Physics and Engineering of the National Academy of Sciences of Ukraine. He joined our faculty in 2012 as postdoc in the former Research Training Group 1294 Analysis, Simulation and Design of Nanotechnological Processes, which was active until 2014. Gudrun Thäter talked with him about one of his research interests Asymptotic analysis and homogenization of PDEs.

Photonic crystals are periodic dielectric media in which electromagnetic waves from certain frequency ranges cannot propagate. Mathematically speaking this is due to gaps in the spectrum of the related differential operators. For that an interesting question is if there are gaps inbetween bands of the spectrum of operators related to wave propagation, especially on periodic geometries and with periodic coeffecicients in the operator. It is known that the spectrum of periodic selfadjoint operators has bandstructure. This means the spectrum is a locally finite union of compact intervals called bands. In general, the bands may overlap and the existence of gaps is therefore not guaranteed. A simple example for that is the spectrum of the Laplacian in L2(Rn) which is the half axis [0,∞).
The classic approach to such problems in the whole space case is the Floquet–Bloch theory.

Homogenization is a collection of mathematical tools which are applied to media with strongly inhomogeneous parameters or highly oscillating geometry. Roughly spoken the aim is to replace the complicated inhomogeneous by a simpler homogeneous medium with similar properties and characteristics. In our case we deal with PDEs with periodic coefficients in a periodic geometry which is considered to be infinite. In the limit of a characteristic small parameter going to zero it behaves like a corresponding homogeneous medium. To make this a bit more mathematically rigorous one can consider a sequence of operators with a small parameter (e.g. concerning cell size or material properties) and has to prove some properties in the limit as the parameter goes to zero. The optimal result is that it converges to some operator which is the right homogeneous one. If this limit operator has gaps in its spectrum then the gaps are present in the spectra of pre-limit operators (for small enough parameter).

The advantages of the homogenization approach compared to the classical one with Floquet Bloch theory are:

1. The knowledge of the limit operator is helpful and only available through homogenization.
2. For finite domains Floquet Bloch does not work well. Though we always have a discrete spectrum we might want to have the gaps in fixed position independent of the size of our domain. Here the homogenization theory works in principle also for the bounded case (it is just a bit technical).

An interesting geometry in this context is a domain with periodically distributed holes. The question arises: what happens if the sizes of holes and the period simultaneously go to zero? The easiest operator which we can study is the Laplace operator subject to the Dirichlet boundary conditions. There are three possible regimes:

1. For holes of the same order as the period (even slightly smaller) the Dirichelet conditions on the boundary of holes dominate -- the solution for the corresponding Poisson equation tends to zero.
2. For significantly smaller holes the influence on the holes is so small that the problem "forgets" about the influence of the holes as the parameter goes to zero.
3. There is a borderline case which lies between cases 1 and 2. It represents some interesting effects and can explain the occurance of so-called strange terms.

A traditional ansatz in homogenization works with the concept of so-called slow and fast variables. The name comes from the following observation. If we consider an infinite layer in cylindrical coordinates, then the variable r measures the distance from the origin when going "along the layer", ϕ the angle in that plane, and z is the variable which goes into the finite direction perpendicular to that plane. When we have functions y(x)=rkf(z,ϕ) then the derivative with respect to r changes the power to rk−1 while the other derivatives leave that power unchanged. In the interesting case k is negative and the r-derivate makes it decreasing even faster. This leads to the name fast variable. The properties in this simple example translate as follows. For any function u we will think of having a set of slow and fast variables (characteristic to the problem) and a small parameter eps and try to find u as u=u0(x)+εu1(x,y)+ε2u2(x,y)+... where in our applications typically y=x/ε. One can formally sort through the ε-levels using the properties of the differential operator. The really hard part then is to prove that this formal result is indeed true by finding error estimates in the right (complicated) spaces.

There are many more tools available like the technique of Tartar/Murat, who use a weak formulation with special test functions depending on the small parameter. The weak point of that theory is that we first have to know the resulat as the parameter goes to zero before we can to construct the test function. Also the concept of Gamma convergence or the unfolding trick of Cioranescu are helpful.

An interesting and new application to the mathematical results is the construction of wave guides. The corresponding domain in which we place a waveguide is bounded in two directions and unbounded in one (e.g. an unbounded cylinder).
Serguei Nazarov proposed to make holes in order to make gaps into the line of the spectrum for a specified wave guide. Andrii Khrabustovskyi suggests to distribute finitely many traps, which do not influence the essential spectrum but add eigenvalues. One interesting effect is that in this way one can find terms which are nonlocal in time or space and thus stand for memory effects of the material.

• P. Exner and A. Khrabustovskyi: On the spectrum of narrow Neumann waveguide with periodically distributed δ′ traps, Journal of Physics A: Mathematical and Theoretical, 48 (31) (2015), 315301.
• A. Khrabustovskyi: Opening up and control of spectral gaps of the Laplacian in periodic domains, Journal of Mathematical Physics, 55 (12) (2014), 121502.
• A. Khrabustovskyi: Periodic elliptic operators with asymptotically preassigned spectrum, Asymptotic Analysis, 82 (1-2) (2013), 1-37.
• S.A. Nazarov, G. Thäter: Asymptotics at infinity of solutions to the Neumann problem in a sieve-type layer, Comptes Rendus Mecanique 331(1) (2003) 85-90.
• S.A. Nazarov: Asymptotic Theory of Thin Plates and Rods: Vol.1. Dimension Reduction and Integral Estimates. Nauchnaya Kniga: Novosibirsk, 2002.

Liliana de Luca Xavier Augusto is PhD student of chemical engineering at the Federal University of São Carlos in Brasil. She spent one year of her PhD (October 2015-2016) at the KIT in Karlsruhe to work with the group developing the software OpenLB at the Mathematical Department and the Department of Chemical Engineering.

Liliana Augusto investigates filtering devices which work on a micro (10−6m) and nano (10−9m) level, and computes the pressure drop between in- and outlet of the filter as well as the collection efficiency. There is a research group conducting experimental setups for these problems, but her research group focuses specifically on mathematical modeling and computer simulation. Due to the small scale and nature of the experiments, one cannot easily take pictures from the physical filters by electronic microsopy, but it is indeed feasible to deduce some important characteristics and geometry such as the size of the fibres for proper modelling and simulation. Appropriate models for the small scale are mesoscopic like Lattice Boltzmann Model where microscopic models are very expensive- too expensive. She is busy with special boundary conditions necessary no-slip boundary condition on the macro scale has to be translated. There is a certain slip to be taken into account to align the results with experimental findings.

Lattice Boltzman methods are not very prominent in Brasil. She was looking for suitable partners and found the development group around OpenLB who had co-operations with Brazil. She tried to apply the software on the problem, and she found out about the possibility to work in Germany through a program of the Brasilian government. It is not so common to go abroad as a PhD-student in Brazil. She learnt a lot not only in an academical manner but highly recommends going abroad to experience new cultures as well.

She does not speak German- everything, from looking for partners to arriving in Germany, happened so fast that she could not learn the language beforehand. At the university, English was more than sufficient for scientific work, but she had difficulties finding a place to stay. In the end, she found a room in a student dorm with German students and a few other international students.

• L.L.X. Augusto e.a.: CFD Simulation of nanofiber-enhanced air filter media - FILTECH 2015 - G6 - Modelling and simulation, 2015.
• L.L.X. Augusto e.a.: Predicting air flow resistance and capture efficiency of fibrous air filter media - Roomvent, 2014.

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 L1-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.

• Press coverage of Crime prevention collected
• Website of Mathematical and Simulation Modeling of Crime

• 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]

• 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.

• AMS - Mathematical Moments Podcast: MM97 - Forecasting Crime

• 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.

• 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.

Nicolas Monod teaches at the École polytechnique fédérale in Lausanne and leads the Ergodic and Geometric Group Theory group there. In May 2016 he was invited to give the Gauß lecture of the German Mathematical Society (DMV) at the Technical University in Dresden. He presented 100 Jahre Zweisamkeit – The Banach-Tarski Paradox. The morning after his lecture we met to talk about paradoxes and hidden assumptions our mind makes in struggling with geometrical representations and measures.

A very well-known game is Tangram. Here a square is divided into seven pieces (which all are polygons). These pieces can be rearranged by moving them around on the table, e.g.. The task for the player is to form given shapes using the seven pieces – like a cat etc.. Of course the Tangram cat looks more like a flat Origami-cat. But we could take the Tangram idea and use thousands or millions of little pieces to build a much more realistic cat with them – as with pixels on a screen. In three dimensions one can play a similar game with pieces of a cube. This could lead to a LEGO-like three-dimensional cat for example. In this traditional Tangram game, there is no fundamental difference between the versions in dimension two and three.

But in 1914 it was shown that given a three-dimensional ball, there exists a decomposition of this ball into a finite number of subsets, which can then be rearranged to yield two identical copies of the original ball. This sounds like a magical trick – or more scientifically said – like a paradoxical situation. It is now known under the name Banach-Tarski paradox. In his lecture, Nicolas Monod dealt with the question: Why are we so surprised about this result and think of it as paradoxical?

One reason is the fact that we think to know deeply what we understand as volume and expect it to be preserved under rearrangements (like in the Tangram game, e.g.).Then the impact of the Banach-Tarski paradox is similar for our understanding of volume to the shift in understanding the relation between time and space through Einstein's relativity theory (which is from about the same time). In short the answer is: In our every day concept of volume we trust in too many good properties of it.

It was Felix Hausdorff who looked at the axioms which should be valid for any measure (such as volume). It should be independent of the point in space where we measure (or the coordinate system) and if we divide objects, it should add up properly. In our understanding there is a third hidden property: The concept "volume" must make sense for every subset of space we choose to measure. Unfortunately, it is a big problem to assign a volume to any given object and Hausdorff showed that all three properties cannot all be true at the same time in three space dimensions. Couriously, they can be satisfied in two dimensions but not in three.

Of course, we would like to understand why there is such a big difference between two and three space dimensions, that the naive concept of volume breaks down by going over to the third dimension. To see that let us consider motions. Any motion can be decomposed into translations (i.e. gliding) and rotations around an arbitrarily chosen common center. In two dimensions the order in which one performs several rotations around the same center does not matter since one can freely interchange all rotations and obtains the same result. In three dimensions this is not possible – in general the outcomes after interchanging the order of several rotations will be different. This break of the symmetry ruins the good properties of the naive concept of volume.

Serious consequences of the Banach-Tarski paradox are not that obvious. Noone really duplicated a ball in real life. But measure theory is the basis of the whole probability theory and its countless applications. There, we have to understand several counter-intuitive concepts to have the right understanding of probabilities and risk. More anecdotally, an idea of Bruno Augenstein is that in particle physics certain transformations are reminiscent of the Banach-Tarski phenomenon.

Nicolas Monod really enjoys the beauty and the liberty of mathematics. One does not have to believe anything without a proof. In his opinion, mathematics is the language of natural sciences and he considers himself as a linguist of this language. This means in particular to have a closer look at our thought processes in order to investigate both the richness and the limitations of our models of the universe.

References:

• F. Hausdorff: Bemerkung über den Inhalt von Punktmengen. Math. Ann. 75 (3), 428–433, 1914.
• S. Banach and A.Tarski: Sur la décomposition des ensembles de points en parties respectivement congruentes, Fundamenta Mathematicae 6, 244–277, 1924.
• J. von Neumann: Zur allgemeinen Theorie des Maßes Fundamenta Mathematicae 13, 73–116, 1929.
• S. Wagon: The Banach–Tarski Paradox. Cambridge University Press, 1994.
• B.W. Augenstein: Links Between Physics and Set Theory, Chaos, Solitons and Fractals, 7 (11), 1761–1798, 1996.
• N. Monod: Groups of piecewise projective homeomorphisms, PNAS 110 (12), 4524-4527, 2013.
• Vsauce-Video on the Banach-Tarksi Paradox

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 10−4 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.

• 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.

• 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.