Modellansatz - English episodes only

Gudrun talks with Debajyoti Choudhuri. He is staying at KIT as a short term guest. He is Associate Professor in the School of Basic Sciences at IIT Bhubaneswar, India. He did his M.Sc. and Ph.D. in Mathematics at the University of Hyderabad. His research interest lies in the analysis of elliptic PDEs using Functional Analytic and topological methods. In this he touches and has a slight overlap with the research of Gudrun. The conversation starts with the discussion about a small paper which Debajyoti put on the archiv. It is about understanding how to work with the Fractional Laplacian. This means extending the classical Laplace operator Δ to non-integer powers. This operator is the main part in PDEs which model, e.g, anomalous diffusion, probability theory, image processing, finance, and nonlocal mechanics. (-Δ)s, where 0 < s < 1. What makes It different to the ordinary Laplacian? While the traditional Laplace operator is local, i.e. it depends only on values of u and its derivatives near x, the fractional Laplacian is nonlocal, it depends on values of u everywhere in space. Thus, for the analytical and numerical treatment one needs very different methods. There are several possible definitions. Some of them can be found in the Wikipedia article which is cited below. On ℝn, the cleanest definition is the Fourier definition which follows the idea: Take the Fourier transform. Multiply by |ξ|2s. Transform back. In the short paper which is discussed the singular integral definition is used: For 0 < s < 1: (-Δ)^s u(x) = C(n,s) PV ∫ [u(x) - u(y)] / |x - y|^(n + 2s) dy This makes the nonlocality explicit: every point y contributes to the value at x. The method central in studying Laplace problems is variational. It considers an (infinite) family of generalised problems and works on the existence of so-called weak solutions. These problems are formulated with the help of Sobolev spaces. The weak solution for the Laplace problem is an element of the space H1=W1,2. This means the solution and its (generalised) gradient are bounded in L2 in the domain in which the problem is solved. This has physical meaning and due to known properties (embedding) of Sobolev spaces the pointwise (strong) solutions often can be constructed when enough regularitiy of the weak solutions is proved. Fractional Laplacians naturally live in fractional Sobolev spaces. These are not that easy to connect to physical properties and a few of the equivalent definitions in the context of classical Sobolev spaces are not equivalent any more everywhere. Common approaches for numerics for PDEs including the fractional Laplacian are: Fourier spectral methods (periodic domains) Finite element methods for fractional PDEs Matrix-function methods (As) Caffarelli–Silvestre extension methods Quadrature approximations of singular integrals The Extension trick introduced by Caffarelli and Silvestre in 2007 (their original paper is cited below) is also discussed as part of the short note. p-laplacian augurs well in the sense because the unicity of the definitions of the s-laplacian is still lacking. The conversation then turns to how Debajyoti found his way into mathematics and the topic of PDEs and how life and work feel like in his university. More information: Webpage of Debajyoti Choudhuri Debajyoti Choudhuri: A quick sneak-peek at the s-fractional Laplacian operator (2022) Wikipedia on the Fractional Laplace operator Mateusz Kwaśnicki: Ten equivalent definitions of the fractional Laplace operator (2015) E. Di Nezza, G. Palatucci, E. Valdinoci, Hitchhiker’s guide to the fractional Sobolev spaces, Bull. Sci. Math., 136(5), 521–573 (2012) L. Caffarelli, L. Silvestre, An extension problem related to the fractional Laplacian, Communications in Partial Differential Equations, 32, 1245–1260 (2007)

In this episode Gudrun speaks with Nadja Klein and Moussa Kassem Sbeyti who work at the Scientific Computing Center (SCC) at KIT in Karlsruhe. Since August 2024, Nadja has been professor at KIT leading the research group Methods for Big Data (MBD) there. She is an Emmy Noether Research Group Leader, and a member of AcademiaNet, and Die Junge Akademie, among others. In 2025, Nadja was awarded the Committee of Presidents of Statistical Societies (COPSS) Emerging Leader Award (ELA). The COPSS ELA recognizes early career statistical scientists who show evidence of and potential for leadership and who will help shape and strengthen the field. She finished her doctoral studies in Mathematics at the Universität Göttingen before conducting a postdoc at the University of Melbourne as a Feodor-Lynen fellow by the Alexander von Humboldt Foundation. Afterwards she was a Professor for Statistics and Data Science at the Humboldt-Universität zu Berlin before joining KIT. Moussa joined Nadja's lab as an associated member in 2023 and later as a postdoctoral researcher in 2024. He pursued a PhD at the TU Berlin while working as an AI Research Scientist at the Continental AI Lab in Berlin. His research primarily focuses on deep learning, developing uncertainty-based automated labeling methods for 2D object detection in autonomous driving. Prior to this, Moussa earned his M.Sc. in Mechatronics Engineering from the TU Darmstadt in 2021. The research of Nadja and Moussa is at the intersection of statistics and machine learning. In Nadja's MBD Lab the research spans theoretical analysis, method development and real-world applications. One of their key focuses is Bayesian methods, which allow to incorporate prior knowledge, quantify uncertainties, and bring insights to the “black boxes” of machine learning. By fusing the precision and reliability of Bayesian statistics with the adaptability of machine and deep learning, these methods aim to leverage the best of both worlds. The KIT offers a strong research environment, making it an ideal place to continue their work. They bring new expertise that can be leveraged in various applications and on the other hand Helmholtz offers a great platform in that respect to explore new application areas. For example Moussa decided to join the group at KIT as part of the Helmholtz Pilot Program Core-Informatics at KIT (KiKIT), which is an initiative focused on advancing fundamental research in informatics within the Helmholtz Association. Vision models typically depend on large volumes of labeled data, but collecting and labeling this data is both expensive and prone to errors. During his PhD, his research centered on data-efficient learning using uncertainty-based automated labeling techniques. That means estimating and using the uncertainty of models to select the helpful data samples to train the models to label the rest themselves. Now, within KiKIT, his work has evolved to include knowledge-based approaches in multi-task models, eg. detection and depth estimation — with the broader goal of enabling the development and deployment of reliable, accurate vision systems in real-world applications. (...)

One of the reasons we started this podcast in 2013 was to provide a more realistic picture of mathematics and of the way mathematicians work. On Nov. 19 2021 Gudrun talked to Stephanie Anne Salomone who is Professor and Chair in Mathematics at the University of Portland. She is also Director of the STEM Education and Outreach Center and Faculty Athletic Representative at UP. She is an Associate Director of Project NExT, a program of the Mathematical Association of America that provides networking and professional development opportunities to mathematics faculty who are new to our profession. She is a wife and mother of three boys, Milo (13), Jude (10), and Theodore (8). This conversation started on Twitter in the summer of 2021. There Stephanie (under the twitter handle @SitDownPee) and @stanyoshinobu Dr. Stan Yoshinobu invited their fellow mathematicians to the following workshop: Come help us build gender equity in mathematics! Picture a Mathematician workshop led by @stanyoshinobu Dr. Stan Yoshinobu and me, designed for men in math, but all genders welcome. Gudrun was curious to learn more and followed the provided link: Gender equity in the mathematical sciences and in the academy broadly is not yet a reality. Women (and people of color, and other historically excluded groups) are confronted with systemic biases, daily experiences, feelings of not being welcome or included, that in the aggregate push them out of the mathematical sciences. This workshop is designed primarily for men in math (although all genders are welcome to participate) to inform and inspire them to better see some of the key issues with empathy, and then to take action in creating a level-playing field in the academy. Workshop activities include viewing “Picture a Scientist” before the workshop, a 2-hour synchronous workshop via zoom, and follow-up discussions via email and Discord server. *All genders welcome AND this workshop is designed for men to be allies. This idea resonated strongly with Gudrun's experiences: Of course women and other groups which are minorities in research have to speak out to fight for their place but things move forward only if people with power join the cause. At the moment people with power in mathematical research mostly means white men. That is true for the US where Stephanie is working as well as in Germany. Allyship is a concept which was introduced by people of colour to name white people fighting for racial justice at their side. Of course, it is a concept which helps in all situations where a group is less powerful than another. Men working for the advancement of non-male mathematicians is strictly necessary in order for equality of chances and a diversity of people in mathematics to be achieved in the next generation. And to be clear: this has nothing to do with counting heads but it is about not ruining the future of mathematics as a discipline by creating obstacles for mathematicians with minoritized identities. (...)

This is the third of three conversation recorded during the Conference on mathematics of wave phenomena 23-27 July 2018 in Karlsruhe. Gudrun is in conversation with Anne-Sophie Bonnet-BenDhia from ENSTA in Paris about transmission properties in perturbed waveguides. The spectral theory is essential to study wave phenomena. For instance, everybody has experimented with resonating frequencies in a bathtube filled with water. These resonant eigenfrequencies are eigenvalues of some operator which models the flow behaviour of the water. Eigenvalue problems are better known for matrices. For wave problems, we have to study eigenvalue problems in infinite dimension. Like the eigenvalues for a finite dimensional matrix the Spectral theory gives access to intrinisic properties of the operator and the corresponding wave phenomena. Anne-Sophie is interested in waveguides. For example, optical fibres can guide optical waves while wind instruments are guides for acoustic waves. Electromagnetic waveguides also have important applications. A practical objective is to optimize the transmission in a waveguide, even if there are some perturbations inside. It is known that for certain frequencies, there is no reflection by the perturbations but it is not apriori clear how to find these frequencies. Anne-Sophie uses complex analysis for that. The idea is to complexify the (originally real) coordinates by analytic extension. It is a classic idea for resonances that she adapts to the problem of transmission. This mathematical method of complex scaling is linked to the method of perfectly matched layers in numerics. It is used to solve problems set in unbounded domains on a computer by finite elements. Thanks to the complex scaling, she can solve a problem in a bounded domain, which reproduces the same behaviour as in the infinite domain. Finally, Anne-Sophie is able to get numerically a complex spectrum of frequencies, related to the quality of the transmission in a perturbed waveguide. The imaginary part of the complex quantity gives an indication of the quality of the transmission in the waveguide. The closer to the real axis the better the transmission.

This is the first of three conversation recorded Conference on mathematics of wave phenomena 23-27 July 2018 in Karlsruhe. Gudrun talked to Fioralba Cakoni about the Linear Sampling Method and Scattering. The linear sampling method is a method to reconstruct the shape of an obstacle without a priori knowledge of either the physical properties or the number of disconnected components of the scatterer. The principal problem is to detect objects inside an object without seeing it with our eyes. So we send waves of a certain frequency range into an object and then measure the response on the surface of the body. The waves can be absorbed, reflected and scattered inside the body. From this answer we would like to detect if there is something like a tumor inside the body and if yes where. Or to be more precise what is the shape of the tumor. Since the problem is non-linear and ill posed this is a difficult question and needs severyl mathematical steps on the analytical as well as the numerical side. In 1996 Colton and Kirsch (reference below) proposed a new method for the obstacle reconstruction problem in inverse scattering which is today known as the linear sampling method. It is a method to solve the above stated problem, which scientists call an inverse scattering problem. The method of linear sampling combines the answers to lots of frequencies but stays linear. So the problem in itself is not approximated but the interpretation of the response is. The central idea is to invert a bounded operator which is constructed with the help of the integral over the boundary of the body. Fioralba got her Diploma (honor’s program) and her Master's in Mathematics at the University of Tirana. For her Ph.D. she worked with George Dassios from the University of Patras but stayed at the University of Tirana. After that she worked with Wolfgang Wendland at the University of Stuttgart as Alexander von Humboldt Research Fellow. During her second year in Stuttgart she got a position at the University of Delaware in Newark. Since 2015 she has been Professor at Rutgers University. She works at the Campus in Piscataway near New Brunswick (New Jersey).

Gudrun talks with Changjing Zhuge. He is a guest in the group of Lennart Hilbert and works at the College of applied sciences and the Beijing Institute for Scientific and Engineering Computing (BISEC) at the Beijing University of Technology. He is a mathematician who is interested in system biology. In some cases he studies delay differential equations or systems of ordinary differential equations to characterize processes and interactions in the context of cancer research. The inbuilt delays originate e.g. from the modeling of hematopoietic stem cell populations. Hematopoietic stem cells give rise to other blood cells. Chemotherapy is frequently accompanied by unwished for side effects to the blood cell production due to the character of the drugs used. Often the production of white blood cells is hindered, which is called neutropenia. In an effort to circumvent that, together with chemotherapy, one treats the patient with granulocyte colony stimulating factor (G-CSF). To examine the effects of the typical periodic chemotherapy in generating neutropenia, and the corresponding response of this system to given to G-CSF Changjing and his colleagues studied relatively simple but physiologically realistic mathematical models for the hematopoietic stem cells. And these models are potential for modeling of other stem-like biosystems such as cancers. The delay in the system is related to the platelet maturation time and the differentiation rate from hematopoietic stem cells into the platelet cell. Changjing did his Bachelor in Mathematics at the Beijing University of Technology (2008) and continued with a PhD-program in Mathematics at the Zhou-Peiyuan Center for Applied Mathematics, Tsinghua University, China. He finished his PhD in 2014. During his time as PhD student he also worked for one year in Michael C Mackey's Lab at the Centre for Applied Mathematics in Bioscience and Medicine of the McGill University in Montreal (Canada).

Gudrun met Magdalena Gonciarz in Dresden. They sat down in a very quiet Coffeeshop in Dreikönigskirche and talked about their experiences as scientists giving science an image. Magda started Portrait of science in 2016 with two objectives: to show that science is a process with many contributors at all carreer levels and to have a get-away from a demanding PhD-project, to express her creativity and have tangible results. The person who pointed Gudrun in Magda's direction is Lennart Hilbert, a former co-worker of Magda in Dresden who is now working at KIT on Computational Architectures in the Cell Nucleus (he will be a podcast guest very soon). On the Portrait of Science page one can find photographs of people from Dresden's Life Science campus. Apart from the photographs, one can also find their stories. How and why did they become scientists? What do they do, what are they passionate about? Magda invites us: "Forget the tubes and Erlenmeyer flasks. Science is only as good as the people who do it. So sit back, scroll down and get to know them looking through the lens of Magdalena Gonciarz. Have you ever wondered what kind of people scientists are? Would you like to know what are they working on? What drives and motivates them - spending days in the basement without the sun? Portrait of Science project aims at uncovering more about people who contribute to science at all levels - Research Group Leaders, Postdocs, PhD Students, Staff Scientists and Technicians. All of them are vital for progress of scientific research and all of them are passionate people with their own motivations." When she started the Portrait of Science project, Magda challenged herself to take more pictures. She wanted to show the real people behind science and their personality. This was a creative task, quite different from her work as scientist - done with comparably little time. On top of taking the pictures, interviewees were asked to fill out a questionaire to accompany the story told by the photographs. Surprisingly, the stories told by her co-workers turned out to be quite inspiring. The stories told have shown the passion and the diverse motivations. People mentioned their failures as well. There were stories about accidents and their crucial role in carreers, about coincidence of finding a fascinating book or the right mentor - even as far back as in early childhood sometimes. Sharing ups and downs and the experience that there is a light at the end of the tunnel was a story she needed and which was worth to be shared. Knowing how hard scientific work can be, and how multiple friends and colleagues struggled more than she herself, Magda still strongly feels that it is useful to show that this is not a private and unique experience, but probably a part of the life of every scientist. This struggle can be overcome with time, effort, and help. Magda comes from Poland. During her Master's studies, she had an opportunity to do a research placement (...)

In this episode Gudrun talks with her new colleague Xian Liao. In November 2018 Xian has been appointed as Junior Professor (with tenure track) at the KIT-Faculty of Mathematics. She belongs to the Institute of Analysis and works in the group Nonlinear Partial Differential Equations. She is very much interested in Dispersive Partial Differential Equations. These equations model, e.g., the behaviour of waves. For that it is a topic very much in the center of the CRC 1173 - Wave phenomena at our faculty. Her mathematical interest was always to better understand the solutions of partial differential equations. But she arrived at dispersive equations through several steps in her carreer. Originally she studied inhomogeneous incompressible fluids. This can for example mean that the fluid is a mixture of materials with different viscosities. If we have a look at the Navier-Stokes equations for materials like water or oil, one main assumption therein is, that the viscosity is a material constant. Nevertheless, the equations modelling their flows are already nonlinear and there are a few serious open questions. Studying flows of inhomogneous materials brings in further difficulties since there occur more and more complex nonlinearities in the equations. It is necessary to develop a frame in which one can characterise the central properties of the solutions and the flow. It turned out that for example finding and working with quantities which remain conserved in the dynamics of the process is a good guiding line - even if the physical meaning of the conserved quantitiy is not always clear. Coming from classical theory we know that it makes a lot of sense to have a look at the conservation of mass, energy and momentum, which translate to conserved quantities as combinations of velocity, its derivatives, pressure and density. Pressure and density are not independent in these simplified models but are independent in the models Xiao studies. In the complex world of inhomogeneous equations we lose the direct concept to translate between physics and mathematics but carry over the knowledge that scale invarance and conservation are central properties of the model. It is interesting to characterize how the complex system develops with a change of properties. To have a simple idea - if it is more developing in the direction of fast flowing air or slow flowing almost solid material. One number which helps to see what types of waves one has to expect is the Mach number. It helps to seperate sound waves from fluid waves. A mathematical/physical question then is to understand the process of letting the Mach number go to zero in the model. It is not that complicated to make this work in the formulae. But the hard work is done in proving that the solutions to the family of systems of PDEs with lower and lower Mach number really tend to the solutions of the derived limit system. For example in order to measure if solutions are similar to each other (...)

Gudrun talks with the Scotish engineer Claire Harvey. After already having finished a Master's degree in Product design engineering at the University of Glasgow for the last two years Claire has been a student of the Energy Technologies (ENTECH) Master program. This is an international and interdisciplinary program under the label of the European Institute of Innovation and Technology (EIT) inbetween a number of European technical universities. She spent her first year in Lisbon at Instituto Superior Técnico (IST) and the second master year at the Karlsruhe Institute of Technology (KIT). Gudrun had the role of her supervisor at KIT while she worked on her Master's thesis at the EUREF Campus in Berlin for the Startup inno2grid. Her study courses prepared her for very diverse work in the sector of renewable energy. Her decision to work with inno2grid in Berlin was based on the fact, that it would help to pave the way towards better solutions for planning micro grids and sustainable districts. Also, she wanted to see an actual micro grid at work. The office building of Schneider Electric, where the Startup inno2grid has its rooms is an experiment delivering data of energy production and consumption while being a usual office building. We will hear more about that in the episode with Carlos Mauricio Rojas La Rotta soon. Micro grids are small scale electrical grid systems where self-sufficient supply is achieved. Therefore, the integration of micro grid design within district planning processes should be developed efficiently. In the planning process of districts with decentralised energy systems, unique and customised design of micro grids is usually required to meet local technical, economical and environmental needs. From a technical standpoint, a detailed understanding of factors such as load use, generation potential and site constraints are needed to correctly and most efficiently design and implement the network. The presence of many different actors and stakeholders contribute to the complexity of the planning process, where varying levels of technical experience and disparate methods of working across teams is commonplace. Large quantities of digital information are required across the whole life-cycle of a planning project, not just to do with energetic planning but also for asset management and monitoring after a micro grid has been implemented. In the design of micro grids, large amounts of data must be gathered, there are initial optimization objectives to be met, and simulating control strategies of a district which are adapted to customer requirements is a critical step. Linking these processes - being able to assemble data as well as communicate the results and interactions of different "layers" of a project to stakeholders are challenges that arise as more cross-sector projects are carried out, with the growing interest in smart grid implementation. (...)

Gudrun talks to Jousef Murad about the computing platform SimScale. Jousef is currently studying mechanical engineering at the Karlsruhe Institute of Technology (KIT) and focuses on turbulence modelling and computational mechanics in his Master's studies. He first learned about the existence of SimScale early in the year 2015 and started as a FEA (finite element analysis) simulation assistant in November 2016. Meanwhile he switched to Community Management and now is Community and Academic Program Manager at the company being responsible for user requests and Formula student teams all over the world. Formula student is a name for design competitions for teams of students constructing racing cars. SimScale is a cloud-based platform that gives instant access to computational fluid dynamics (CFD) and finite element analysis (FEA) simulation technology, helping engineers and designers to easily test performance, optimize durability or improve efficiency of their design. SimScale is accessible from a standard web browser and from any computer, eliminating the hurdles that accompany traditional simulation tools: high installation costs, licensing fees, deployment of high-performance computing hardware, and required updates and maintenance. Via the platform, several state-of-the-art open solvers are made available like,e.g., OpenFOAM and Meshing with SnappyHexMesh. More information about the packages being used can be found at https://www.simscale.com/open-source/ . On top of having easier access to open source software, the connected user forum is very active and helps everybody to enter the field even as a person without experience. Founded in 2012 in Munich (Germany), nowadays SimScale is an integral part of the design validation process for many companies worldwide and individual users. It is mainly used by product designers and engineers working in Architecture, Engineering and Construction or Heating, Ventilation and Air-Conditioning. Also in the Electronics, Consumer Goods and Packaging and Containers industries SimScale is useful for testing and optimizing designs in the early development stages. SimScale offers pricing plans that can be customized, from independent professionals to SMEs and multinational companies. The Community plan makes it possible to use SimScale for free, with 3000 core hours/year using up to 16 cloud computing cores.

Gudrun met the USA-based mathematician Roza Aceska from Macedonia in Turin at the Conference MicroLocal and Time-Frequency Analysis 2018. The topic of the recorded conversation is dynamical sampling. The situation which Roza and other mathematician study is: There is a process which develops over time which in principle is well understood. In mathematical terms this means we know the equation which governs our model of the process or in other words we know the family of evolution operators. Often this is a partial differential equation which accounts for changes in time and in 1, 2 or 3 spatial variables. This means, if we know the initial situation (i.e. the initial conditions in mathematical terms), we can numerically calculate good approximations for the instances the process will have at all places and at all times in the future. But in general when observing a process life is not that well sorted. Instead we might know the principal equation but only through (maybe only a few) measurements we can find information about the initial condition or material constants for the process. This leads to two questions: How many measurements are necessary in order to obtain the full information (i.e. to have exact knowledge)? Are there possibilities to choose the time and the spatial situation of a measurement so clever as to gain as much as possible new information from any measurement? These are mathematical questions which are answered through studying the equations. The science of sampling started in the 1940s with Claude Shannon who found fundamental limits of signal processing. He developed a precise framework - the so-called information theory. Sampling and reconstruction theory is important because it serves as a bridge between the modern digital world and the analog world of continuous functions. It is surprising to see how many applications rely on taking samples in order to understand processes. A few examples in our everyday life are: Audio signal processing (electrical signals representing sound of speech or music), image processing, and wireless communication. But also seismology or genomics can only develop models by taking very intelligent sample measurements, or, in other words, by making the most scientific sense out of available measurements. The new development in dynamical sampling is, that in following a process over time it might by possible to find good options to gain valuable information about the process at different time instances, as well as different spatial locations. In practice, increasing the number of spatially used sensors is more expensive (or even impossible) than increasing the temporal sampling density. These issues are overcome by a spatio-temporal sampling framework in evolution processes. The idea is to use a reduced number of sensors with each being activated more frequently. Roza refers to a paper by Enrique Zuazua (...)

Gudrun talks with Asher Zarth. He finished his Master thesis in the Lattice Boltzmann Research group at the Karlsruhe Institute for Technology (KIT) in April 2018. Lattice Boltzmann methods (LBM) are an established method of computational fluid dynamics. Also, the solution of temperature-dependent problems - modeled by the Boussinesq approximation - with LBM has been done for some time. Moreover, LBM have been used to solve optimization problems, including parameter identification, shape optimization and topology optimization. Usual optimization approaches for partial differential equations are strongly based on using the corresponding adjoint problem. Especially since this method provides the sensitivities of quantities in the optimization process as well. This is very helpful. But it is also very hard to find the adjoint problem for each new problem. This needs a lot of experience and deep mathematical understanding. For that, Asher uses automatic differentiation (AD) instead, which is very flexible and user friendly. His algorithm combines an extension of LBM to porous media models as part of the shape optimization framework. The main idea of that framework is to use the permeability as a geometric design parameter instead of a rigid object which changes its shape in the iterative process. The optimization itself is carried out with line search methods, whereby the sensitivities are calculated by AD instead of using the adjoint problem. The method benefits from a straighforward and extensible implementation as the use of AD provides a way to obtain accurate derivatives with little knowledge of the mathematical formulation of the problem. Furthermore, the simplicity of the AD system allows optimization to be easily integrated into existing simulations - for example in the software package OpenLB which Asher used in his thesis. One example to test the algorithm is the shape of an object under Stokes flow such that the drag becomes minimal. It is known that it looks like an american football ball. The new algorithm converges fast to that shape.

In March 2018 Gudrun visited University College London and recorded three conversations with mathematicians working there. Her first partner was Karen Page. She works in Mathematical Biology and is interested in mathematical models for pattern formation. An example would be the question why (and how) a human embryo develops five fingers on each hand. The basic information for that is coded into the DNA but how the pattern develops over time is a very complicated process which we understand only partly. Another example is the patterning of neurons within the vertebrate nervous system. The neurons are specified by levels of proteins. Binding of other proteins at the enhancer region of DNA decides whether a gene produces protein or not. This type of work needs a strong collaboration with biologists who observe certain behaviours and do experiments. Ideally they are interested in the mathematical tools as well. One focus of Karen's work is the development of the nervous system in its embryonic form as the neural tube. She models it with the help of dynamical systems. At the moment they contain three ordinary differential equations for the temporal changes in levels of three proteins. Since they influence each other the system is coupled. Moreover a fourth protein enters the system as an external parameter. It is called sonic hedgehog (Shh). It plays a key role in regulating the growth of digits on limbs and organization of the brain. It has different effects on the cells of the developing embryo depending on its concentration. Concerning the mathematical theory the Poincaré Bendixson theorem completely characterizes the long-time behaviour of two-dimensional dynamical systems. Working with three equations there is room for more interesting long-term scenarios. For example it is possible to observe chaotic behaviour. Karen was introduced to questions of Mathematical Biology when starting to work on her DPhil. Her topic was Turing patterns. These are possible solutions to systems of Partial differential equations that are thermodynamically non-equilibrium. They develop from random perturbations about a homogeneous state, with the help of an input of energy. Prof. Page studied mathematics and physics in Cambridge and did her DPhil in Oxford in 1999. After that she spent two years at the Institute for Advanced Study in Princeton and has been working at UCL since 2001.