Gudrun talks with Polyxeni Spilioti at Aarhus university about spectral geometry. Before working in Aarhus Polyxeni was a postdoctoral researcher in the group of Anton Deitmar at the University of Tübingen. She received her PhD from the University of Bonn, under the supervision of Werner Mueller after earning her Master's at the National and Technical University of Athens (Faculty of Applied Mathematics and Physics). As postdoc she was also guest at the MPI for Mathematics in Bonn, the Institut des Hautes Etudes Scientifiques in Paris and the Oberwolfach Research Institute for Mathematics. In her research she works on questions like: How can one obtain information about the geometry of a manifold, such as the volume, the curvature, or the length of the closed geodesics, provided that we can study the spectrum of certain differential operators? Harmonic analysis on locally symmetric spaces provides a powerful machinery in studying various invariants, such as the analytic torsion, as well as the dynamical zeta functions of Ruelle and Selberg.

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. (...)

In March 2018 Gudrun had a day available in London when travelling back from the FENICS workshop in Oxford. She contacted a few people working in mathematics at the University College London (ULC) and asked for their time in order to talk about their research. In the end she brought back three episodes for the podcast. This is the second of these conversations. Gudrun talks to Marta Betcke. Marta is associate professor at the UCL Department of Computer Science, member of Centre for Inverse Problems and Centre for Medical Image Computing. She has been in London since 2009. Before that she was a postdoc in the Department of Mathematics at the University of Manchester working on novel X-ray CT scanners for airport baggage screening. This was her entrance into Photoacoustic tomography (PAT), the topic Gudrun and Marta talk about at length in the episode. PAT is a way to see inside objects without destroying them. It makes images of body interiors. There the contrast is due to optical absorption, while the information is carried to the surface of the tissue by ultrasound. This is like measuring the sound of thunder after lightning. Measurements together with mathematics provide ideas about the inside. The technique combines the best of light and sound since good contrast from optical part - though with low resolution - while ultrasound has good resolution but poor contrast (since not enough absorption is going on). In PAT, the measurements are recorded at the surface of the tissue by an array of ultrasound sensors. Each of that only detects the field over a small volume of space, and the measurement continues only for a finite time. In order to form a PAT image, it is necessary to solve an inverse initial value problem by inferring an initial acoustic pressure distribution from measured acoustic time series. In many practical imaging scenarios it is not possible to obtain the full data, or the data may be sub-sampled for faster data acquisition. Then numerical models of wave propagation can be used within the variational image reconstruction framework to find a regularized least-squares solution of an optimization problem. Assuming homogeneous acoustic properties and the absence of acoustic absorption the measured time series can be related to the initial pressure distribution via the spherical mean Radon transform. Integral geometry can be used to derive direct, explicit inversion formulae for certain sensor geometries, such as e.g. spherical arrays. At the moment PAT is predominantly used in preclinical setting, to image tomours and vasculature in small animals. Breast imaging, endoscopic fetus imaging as well as monitoring of perfusion and drug metabolism are subject of intensive ongoing research. The forward problem is related to the absorption of the light and modeled by the wave equation assuming instanteneous absorption and the resulting thearmal expansion. In our case, an optical ultrasound sensor records acoustic waves over time, (...)

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 second of three conversation recorded Conference on mathematics of wave phenomena 23-27 July 2018 in Karlsruhe. Gudrun is in conversation with Mariana Haragus about Benard-Rayleigh problems. On the one hand this is a much studied model problem in Partial Differential Equations. There it has connections to different fields of research due to the different ways to derive and read the stability properties and to work with nonlinearity. On the other hand it is a model for various applications where we observe an interplay between boyancy and gravity and for pattern formation in general. An everyday application is the following: If one puts a pan with a layer of oil on the hot oven (in order to heat it up) one observes different flow patterns over time. In the beginning it is easy to see that the oil is at rest and not moving at all. But if one waits long enough the still layer breaks up into small cells which makes it more difficult to see the bottom clearly. This is due to the fact that the oil starts to move in circular patterns in these cells. For the problem this means that the system has more than one solutions and depending on physical parameters one solution is stable (and observed in real life) while the others are unstable. In our example the temperature difference between bottom and top of the oil gets bigger as the pan is heating up. For a while the viscosity and the weight of the oil keep it still. But if the temperature difference is too big it is easier to redistribute the different temperature levels with the help of convection of the oil. The question for engineers as well as mathematicians is to find the point where these convection cells evolve in theory in order to keep processes on either side of this switch. In theory (not for real oil because it would start to burn) for even bigger temperature differences the original cells would break up into even smaller cells to make the exchange of energy faster. In 1903 Benard did experiments similar to the one described in the conversation which fascinated a lot of his colleagues at the time. The equations where derived a bit later and already in 1916 Lord Rayleigh found the 'switch', which nowadays is called the critical Rayleigh number. Its size depends on the thickness of the configuration, the viscositiy of the fluid, the gravity force and the temperature difference. Only in the 1980th it became clear that Benards' experiments and Rayleigh's analysis did not really cover the same problem since in the experiment the upper boundary is a free boundary to the surrounding air while Rayleigh considered fixed boundaries. And this changes the size of the critical Rayleigh number. For each person doing experiments it is also an observation that the shape of the container with small perturbations in the ideal shape changes the convection patterns. Maria does study the dynamics of nonlinear waves and patterns. This means she is interested in understanding processes which (...)

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 to Anna Geyer. Anna is Assistant professer at TU Delft in the Mathematical Physics group at the Delft Institute of Applied Mathematics. She is interested in the behaviour of solutions to equations which model shallow water waves. The day before (04.07.2019) Anna gave a talk at the Kick-off meeting for the second funding period of the CRC Wave phenomena at the mathematics faculty in Karlsruhe, where she discussed instability of peaked periodic waves. Therefore, Gudrun asks her about the different models for waves, the meaning of stability and instability, and the mathematical tools used in her field. For shallow water flows the solitary waves are especially fascinating and interesting. Traveling waves are solutions of the form u(t,x)=f(x-ct) representing waves of permanent shape f that propagate at constant speed c. These waves are called solitary waves if they are localized disturbances, that is, if the wave profile f decays at infinity. If the solitary waves retain their shape and speed after interacting with other waves of the same type, we say that the solitary waves are solitons. One can ask the question if a given model equation (sometimes depending on parameters in the equation or the size of the initial conditions) allows for solitary or periodic traveling waves, and secondly whether these waves are stable or unstable. Peaked periodic waves are an interesting phenomenon because at the wave crest (the peak) they are not smooth, a situation which might lead to wave breaking. For which equations are peaked waves solutions? And how stable are they? Anna answers these questions for the reduced Ostrovsky equation, which serves as model for weakly nonlinear surface and internal waves in a rotating ocean. The reduced Ostrovsky equation is a modification of the Korteweg-de Vries equation, for which the usual linear dispersive term with a third-order derivative is replaced by a linear nonlocal integral term, representing the effect of background rotation. Peaked periodic waves of this equation are known to exist since the late 1970's. Anna presented recent results in which she answers the long standing open question whether these solutions are stable. In particular, she proved linear instability of the peaked periodic waves using semi-group theory and energy estimates. Moreover, she showed that the peaked wave is unique and that the equation does not admit Hölder continuous solutions, which implies that the reduced Ostrovsky equation does not admit cusps. Finally, it turns out that the peaked wave is also spectrally unstable. This is joint work with Dmitry Pelinovsky. For the stability analysis it is really delicate how to choose the right spaces such that their norms measure the behaviour of the solution. The Camassa-Holm equation allows for solutions with peaks which are stable with respect to certain perturbations and unstable with respect to others, and can model breaking waves. (...)

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).

In June 2019 Gudrun talked with Serena Carelli. Serena is member of the Research Training Group (RTG) Simet, which is based in Karlsruhe, Ulm and Offenburg. It started its work in 2017 and Gudrun is associated postdoc therein. The aim of that graduate school is to work on the better understanding of Lithium-ion batteries. For that it covers all scales, namley from micro (particles), meso (electrodes as pairs) to macro (cell) and involves scientists from chemistry, chemical engineering, material sciences, electro engineering, physics and mathematics. The group covers the experimental side as well as modeling and computer simulations. Serena is one of the PhD-students of the program. She is based in Offenburg in the group of Wolfgang Bessler (the deputy speaker of the RTG). Her research focusses on End-of-life prediction of a lithium-ion battery cell by studying the mechanistic ageing models of the graphite electrode among other things. Mathematical modelling and numerical simulation have become standard techniques in Li-ion battery research and development, with the purpose of studying the issues of batteries, including performance and ageing, and consequently increasing the model-based predictability of life expectancy. Serena and others work on an electrochemical model of a graphite-based lithium-ion cell that includes combined ageing mechanisms: 1. Electrochemical formation of the solid electrolyte interphase (SEI) at the anode, 2. breaking of the SEI due to mechanical stress from volume changes of the graphite particles, causing accelerated SEI growth, 3. gas formation and dry-out of the electrodes, 4. percolation theory for describing the loss of contact of graphite particles to the liquid electrolyte, 5. formation of reversible and irreversible Li plating. The electrochemistry is coupled to a multi-scale heat and mass transport model based on a pseudo-3D approach. A time-upscaling methodology is developed that allows to simulate large time spans (thousands of operating hours). The combined modeling and simulation framework is able to predict calendaric and cyclic ageing up to the end of life of the battery cells. The results show a qualitative agreement with ageing behavior known from experimental literature. Serena has a Bachelor in Chemistry and a Master's in Forensic Chemistry from the University of Torino. She worked in Spain, the Politécnico de Torino and in Greece (there she was Marie Curie fellow at the Foundation for Research and Technology - Hellas) before she decided to spend time in Australia and India.

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 (...)