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# 25 years MPI for Mathematics in the Sciences

On March 1, 1996, our institute officially opened its doors at Inselstraße 22. Today we are very proud to be one of the most renowned mathematical research institutions in the world. We owe this to the numerous scientists who have accompanied us through the last 25 years with groundbreaking research projects, successful doctorates, significant research results and, last but not least, a creative and familiar cooperation. Warmest thanks also go to all employees in administration, library, IT and scientific service, who make our institute a perfectly functioning scientific institution.

In this column we would like to remember - together with you - experiences and achievements that have accompanied us over the years. Please check back here from time to time for news from the "good old days".

If you would also like to share a memory with us, please contact us.

## Episode 6 - Annual highlights (16.06.2021)

The Max Planck Society for the Advancement of Science e.V. (MPG) is dedicated to developing fundamental knowledge. Its research spectrum is broadly diversified: the 86 Max Planck Institutes and facilities conduct basic research in the natural sciences, biological sciences, humanities and social sciences. The scientists working there investigate the interior of elementary particles and the origin of our universe, they research the molecular building blocks of life and specific interactions in ecosystems, changes in societies as a result of global migration and international legal comparisons. (MPG yearbook 2019)

Each year the Max Planck Society presents a report on its activities in the reporting year. We also contribute to this yearbook by reporting on selected research highlights in an annual article. We would like to introduce two of them here:

Research report 2005:**Quantum Gravity: No Experiments, but Mathematics**

Christian Fleischhack | Mathematical Physics*General relativity and quantum theory have not been merged into a consistent theory of quantum gravity yet. Unfortunately, to date, there are no experiments available that may disclose parts of the unified theory. Nevertheless, mathematics is already in a position to provide us with rigorous statements on how quantum gravity may look like.*

Research report 2020:**Deep Learning Theory **

Guido Montúfar | ERC Research Group Mathematical Machine Learning*This project develops mathematical theory for deep learning, critical in making these enormously successful machine learning methods more broadly applicable, efficient, interpretable, safe, and reliable. Concretely, we seek to clarify the interplay between the representational power of artificial neural networks as parametric sets of hypotheses, the properties and consequences of the parameter optimization procedures that are employed in order to select a hypothesis based on data, and the performance of trained neural networks at test time on new data.*

You will find all our contributions since 2003 on our institute presence on the MPG website under the column *YEARBOOK* (unfortunately they are only available in German).

## Episode 5 - Facts & Figures (19.05.2021)

What would a mathematical institute be without facts and figures? For today's review, we have rummaged through our databases and elicited some statistics from them.

In the past 25 years, at our institute we had/have:

- 132 staff members working in our administration, library, IT and scientific service, including our trainees.
- More than 300 PhD students.
- More than 550 Postdocs.
- 14 Junior research group leaders.
- 230 scientific conferences and workshops.
- More than 5.000 seminars and lectures.
- More than 3.000 short term guests visited our institute within research collaborations, conferences, seminars and other projects.
- Our institute published more than 3.700 journal articles.
- Our scientists came from 68 different countries. Curious which countries these are? To illustrate this broad spectrum of countries worldwide we prepared a chart for you.

Our directors and their research groups:

Jürgen Jost (since 1996)

Riemannian, Kählerian and algebraic geometry, geometric analysis, nonlinear partial differential equations in mathematical physics, calculus of variations, mathematical and theoretical biology, neural networks, theory of cognition, mathematical methods for complex systems, information theory, dynamical and complex systems in economics and social sciences, theory and history of science.

Felix Otto (since 2010)

Analysis of continuum models in materials science (ferromagnets, shape memory alloys) and fluids by methods from partial differential equations and the calculus of variations. Treatment of random effects (stochastic homogenization, thermal noise), pattern formation, energy landscapes, scaling laws.

Bernd Sturmfels (since 2017)

Algebra, geometry and combinatorics with strong connection to applications, drawn from statistics, computer science and the life sciences. Areas like algebraic geometry, commutative algebra, convex and discrete geometry, multi-linear and tensor algebra, algebraic combinatorics, dynamical systems, and symbolic computation are covered.

Wolfgang Hackbusch (1999 - 2014, retired director)

Numerical solution of partial differential equations and integral equations, Multi-grid methods, Boundary element methods

Stefan Müller (1996 - 2008, former director)

Mathematical foundations of materials science, microstructures, micromagnetism, continuum mechanics, singular perturbations, non-linear partial differential equations and the calculus of variations

Ensuing position at: Hausdorff Center for Mathematics Bonn (Germany)

Eberhard Zeidler (1996 - 2007, †)

Non-linear partial differential equations, non-linear functional analysis, classical and modern mathematical physics, selected questions of mathematical chemistry and mathematical biology

In addition, 8 research groups are currently working at our institute, led by outstanding young scientists. These are:

Daniele Agostini: Algebraic geometry and interactions with probability, statistics and mathematical physics, Riemann's theta function.

Benjamin Gess: Stochastic partial differential equations, regularization and well-posedness by noise, random dynamical systems.

Michael Joswig: Polyhedral combinatorics, algorithmic geometry, mathematical software.

Guido Montúfar: Deep learning theory, geometric analysis of capacity, optimization, and generalization in deep neural networks.

Matteo Smerlak: Theoretical evolutionary dynamics, fitness landscapes.

André Uschmajew: Low-rank tensor approximation, numerical tensor calculus, optimization.

Paul Breiding: Emmy Noether Research Group on Numerical and Probabilistic Nonlinear Algebra

Noémie Combe: Minerva group on Frobenius structures, Operads, Deformation Theory & Applications

We are incredibly proud that all our former group leaders now hold prestigious positions at renowned universities and scientific institutions worldwide.

## Episode 4 - A cool paperclip (30.04.2021)

A very cool and clever paper clip is the protagonist of this video, which we published in 2008 on the occasion of the Science Year of Mathematics. This paper clip is extraordinary because it has a memory. Even if you bend it, it can remember its original shape. All it needs is hot water, and it's back to its old self. The reason for this is its unique material, a so-called shape memory material.

The research group around our former director Prof. Dr. Stefan Müller, developed mathematical models to explore materials with these novel properties. The key is the tiny microstructures of the metal.

More than 500 partners from science, culture and politics participated in the 2008 Science Year of Mathematics. Under the motto "Mathematics. Everything that counts," a total of more than 760 events, exhibitions, competitions, and festivals invited people to experience mathematics. They all showed: Mathematics is fun!

P.S. The paper clip is still alive. and it enjoys great popularity in our public relations campaigns and projects with students. :)

### Cast

**Prof. Dr. Stefan Müller** is one of the founding directors of our institute. From 1996 to 2008 he led the research group "Mathematical foundations of materials science, microstructures, micromagnatism, continuum mechanics, singular perturbations, non-linear partial differential equations and the calculus of variations". Since 2008 he holds a professorship at the University of Bonn. He is also Hausdorff Chair and Associate Director of the Hausdorff Research Institute for Mathematics at the Bonn Cluster of Excellence Hausdorff Center for Mathematics.

**Prof. Dr. Anja Schlömerkemper** was a Ph.D. student in the group of Prof. Stefan Müller and received her Ph.D. in 2002 on "Magnetic forces in discrete and continuous systems". Her research stays led her to the University of Oxford, to the University of Stuttgart and from 2005 to 2009 again back to our institute. After holding positions at the Universities of Bonn and Erlangen-Nuremberg, she was appointed Professor and at the University of Würzburg, where she now holds the Chair of Mathematics in the Sciences and is also Vice President of the University.

**Prof. Dr. Patrick Dondl** was a postdoc in the research group of Prof. Stefan Müller. Now he holds a professorship for Applied Mathematics at the Albert-Ludwigs-Universität Freiburg. His scientific interests still focus on the mathematical modeling and analysis of problems arising in the physical sciences. In particular, his group conducts research on interfaces in random media, phase-field models for thin elastic structures, and microstructures in plasticity.

Please read also the following scientific article published on March 4, 2004 in the journal Nature (Volume 428).

Kaushik Bhattacharya, Sergio Conti, Giovanni Zanzotto, and Johannes Zimmer:*Crystal symmetry and the reversibility of martensitic transformations.*

## Episode 3 - "This institute is an essential piece of my life"

Interview with Prof. Dr. Jürgen Jost (08.04.2021)

Prof. Dr. Jürgen Jost founded our institute in 1996 together with Prof. Dr. Eberhard Zeidler and Prof. Dr. Stefan Müller and accompanied it as director and working group leader through the entire 25 years. We are very pleased that he shares some memories, thoughts and inspirations with us in this interview.

Prof. Jost, what fascinates you about mathematics and what made you decide to become a mathematician?

Mathematics is a very precise tool of thinking and allows me to develop concepts and understand structures. I know this sounds very abstract at first. When asked how I got into it, I could of course say mathematics came easy to me or I was always good at mathematics in school, as is the case with most mathematicians. But actually, it's a little different. I wanted to become a scientist. The fact that it actually became mathematics was a bit coincidental. Ever since my school days, it has always been my dream to understand reality and what surrounds us, what constitutes matter, life, consciousness and so much more. I initially studied various subjects to varying depths and breadths: Mathematics, Physics, Economics and Philosophy. And it turned out that I was able to do a PhD in mathematics very quickly (and at the age of 23) in a very highly topical and rapidly developing field of research. I seized this opportunity, became a mathematician, and focused on special mathematical research, solving concrete problems and thus building my reputation as a scientist.

In 1996, you moved from the Ruhr University Bochum to a Max Planck Institute in eastern Germany and shifted your center of life to Leipzig. What did this mean for you and your family?

Founding a Max Planck Institute and becoming Max Planck Director is, of course, a great opportunity. Such a prominent and well-equipped scientific position can be found practically nowhere else in the world. I was very happy to seize this opportunity, somewhat to the chagrin of my parents, who hoped that I would stay close to them and accept one of the university professorships offered to me. But I feel at home here. I didn't know Leipzig before. Eastern Germany was completely unknown to me until then, and I only got to know the city of Leipzig during the institute negotiations. Myself and also my family have always felt very comfortable in Leipzig. It is a very pleasant and charming city and it has also been fascinating to see how much Leipzig has changed and developed in these 25 years that I have been here now.

Which experience or event within these 25 years of the institute's history do you particularly remember?

There have been many highlights in our institute's history. However, I particularly remember a wonderful international conference that took place shortly after our institute was founded, the interdisciplinary Walter Andrejewski Symposium "Mathematics in the Sciences". Some of the world's greatest scientists gave lectures - among them the winner of the Nobel Prize in Chemistry Manfred Eigen, Sir Roger Penrose, who received the Nobel Prize in Physics for his contributions to the general theory of relativity last year, or Misha Gromov, one of the world's leading geometers, who was awarded the Abel Prize in 2009. This was an outstanding scientific event, which has firmly anchored and positioned our institute in the international scientific landscape and of which we can be very proud.

Which research topics did you consider particularly promising in the start-up phase and how have they developed? And vice versa: Are there any research topics that were not initially in your focus, but have become significantly more important over time?

For this answer I have to elaborate a bit. Researchers establish their reputation by solving special problems in a special research field. This is what I did in mathematics. I was a mathematician in my field who was obviously considered capable of co-founding such an institute. But I already had some things in mind that I wanted to develop - for example, the relationships with physics and neurobiology. I had already started this research, but over time it developed a considerable momentum of its own as I came to understand better and better how all the sciences are interrelated. I was able to expand my research profile due to the opportunities afforded by a Max Planck Institute so that I could touch and cover a wide range of sciences and infuse them with new mathematical ideas. Over time, we no longer worked only in mathematics, but also concentrated on theoretical physics - especially in the early years and essentially influenced by the interests of Prof. Zeidler. In addition, we extended our research to the fields of mathematical neurobiology and mathematical biology, which led us to appoint the bioinformatician Peter Stadler as an external scientific member of our institute.

Over the years, I also developed numerous contacts with psychologists, social scientists, and economists. I also got involved in the history of science and I pursued the philosophy of science, and much more. Doing science in such a universal and versatile way was the great opportunity that crystallized over the years and then developed in an extraordinarily dynamic manner. As a result, I think we have succeeded in establishing a working group here that is quite unique in the world. We can interconnect the different sciences on the basis of very solid and profound mathematics. We can cover the broad spectrum of science. We can also go into conceptual depths. And we can work in an interdisciplinary fashion. This is something very special that I would not have been able to realize anywhere else - to be given both the freedom to do what one thinks is right and to be provided with the appropriate resources to do so. Of course, we must be evaluated by the success of our projects, but we have succeeded well in this so far. In addition to our basic funding, we have also been able - particularly in Europe - to attract significant and innovative third-party funding projects.

You have mentored numerous talented young people and guided them to their scientific careers. What advice do you give your former PhD students and postdocs?

First of all, I would like to say that it is always a great pleasure for me to work with so many talented, motivated, competent, hard-working, and eager young people and to introduce them to new scientific disciplines so that they can later find their own way. I cannot give a general advice, because this depends on the individual personality. There is no universal concept to being successful. But I think I've learned quite well over the years to get a good grasp of each person's strengths and to steer them in a direction where they can develop those strengths and where, at the same time, a path can open up for a successful career.

Not all topics that we consider important and innovative, and which may already be well established, are useful for building up a scientific career. So, my advice to young people is: be open-minded, be curious, create something new, but also make sure that this resonates enough with established institutions to open up suitable scientific positions. And, of course, there are many talented young people who, after completing their doctorates, pursue positions in industry, business, and other fields, and they are also very successful. I think it is extremely important that we, as a distinguished research and educational institution, train young people to do something very useful and important outside of science as well. When you have a PhD in mathematics, you have a lot of options. Mathematics teaches you to think precisely and to grasp structures abstractly. These skills are highly appreciated in many fields, even if the concrete mathematical methods that one has learned may no longer be applied.

As you already described, your research group explores a wide range of mathematical topics. Where do you see our institute in the coming years?

I have covered such a broad spectrum of scientific topics and touched on so many sciences that I am sure my successor will not be able to. In a certain way, I am a scientific exotic in mathematics, perhaps also within the Max Planck Society, so that a narrower focus will probably have to be applied again in the future. This is quite good, because mathematics cannot live without very successful and deeply penetrating specialized research being carried out. The research spectrum will therefore certainly narrow again somewhat, which is definitely positive for the institute. Of course, I hope that the interdisciplinary breadth that I have tried to build up can somehow - at least as a spirit - live on in the background. And that maybe I could inspire other people to try something similar.

How do you assess the role of mathematics for the future in general?

It is becoming increasingly evident that mathematics is one of the great sciences of the future. 25 years ago, mathematics was important in many fields of application and became more and more important, which was also the reason for the foundation of our institute. But at that time, it was not yet clear, for example, how far data science and the associated amounts of data we have to deal with today would develop. It is a great challenge for mathematics to systematically penetrate these heterogeneous, complex, very large and very different data. In the meantime, the application scientists are overburdened with this which will also result in a shift of emphasis within mathematics. Thus, mathematics will merge with statistics and machine learning, it will be about the geometry of high-dimensional spaces, which has not been so much in focus so far, and much more. Therefore, I believe, and this is shared by many of my colleagues, that the role of mathematics will increase enormously in the future. Mathematics will be of immense scientific as well as social relevance in order to be able to deal wisely with these amounts of data we are confronted with, to discover structures in them, and to creatively explore how to shape our future based on them. It will take some time to see what is really possible. I myself will have to retire in a few years. To what extent this will be realized at our institute or within the Max Planck Society or at other research institutions, I do not yet know, of course.

Your wife is also a mathematician, and some of your children have also chosen for scientific careers. What role does mathematics play in your personal environment? Is there a day for you without science?

Days without science are actually very rare for me, because I do so many things and find them so exciting that I want to spend as much of my energy and time as possible on these exciting questions. Science and daily life merge in a certain way. I read a lot - things that are partly related to my research or things that just stick and maybe inspire me to do something new 10 years later. I deal with topics that are simply delightful, exciting and interesting. That is not so easy to separate.

This year, you are not only celebrating the institute's anniversary, but also your 65th birthday. What do you wish for your personal and the Institute’s future?

For the future of our institute, I wish that the institute will remain innovative, that it will be able to conduct science at the highest level, that it will continue to attract excellent young researchers and open up them successful scientific careers, and much more. And I am sure that it will happen. We will certainly get very good successors for the currently vacant director positions, and the Max Planck Society has well-established mechanisms to ensure that the institutes operate very successfully. I'm not really worried about that.

For myself, of course, I still have many projects. I will retire in three years, and I will certainly remain active after that. I have written about 20 books over the years, but I still have many ideas in my mind. I have started, and even partially finished, manuscripts for at least 10 more books. I can still do a lot in the next few years, as long as I can maintain my health and creativity.

Dear Professor Jost, many thanks! We wish you all the best.

### A small selection of his books

**Jost, Jürgen: Leibniz und die moderne Naturwissenschaft**

What does a 17th century scholar still have to say for today's natural sciences? Quite a lot, as this book shows.

**Jost, Jürgen: Riemannian Geometry and Geometric Analysis**

His most successful book, already published in 7 editions. This established reference work continues to provide its readers with a gateway to some of the most interesting developments in contemporary geometry.

## Episode 2 - In memory of our founding director Professor Eberhard Zeidler (22.03.2021)

We dedicate today's article to an outstanding scientist, warm-hearted human being, leadership mentor, and extraordinary communicator of mathematics - our founding director,

Prof. Dr. Dr. h.c. Eberhard Zeidler.

Eberhard Zeidler was born in Leipzig October 6th, 1940. In 1959, he began studying mathematics at the University of Leipzig, but was expelled from the University for political reasons in 1961. He was allowed to resume his studies only in 1964. After his PhD in 1967 with Herbert Beckert, the head of the Leipzig school of mathematical analysis, he became one of the leading scientists in the field, as for instance testified by his monumental 5 volume treatise on »Nonlinear Functional Analysis and its Applications« that quickly became a classic text in the field. In 1974, he became a full professor. After the collapse of the GDR in 1989, he played a leading role in the reformation and restructuring of the University.

It was due to his high international scientific standing, his proven leadership and his grand vision that the Max Planck Institute for Mathematics in the Sciences was founded in 1996 in Leipzig. He lead the institute during its first years, quickly turning it into an internationally recognized and highly innovative research institute. His own scientific vision was concerned with the deep unity of mathematics and theoretical physics, and this gave him the incredible

energy and enthusiasm for his multi-volume treatise on »Quantum Field Theory« whose subtitle »A Bridge between Mathematicians and Physicists« so nicely expresses his aim. He wanted to build bridges and explore for himself and show to others the deep conceptual unity between pure mathematics and theoretical physics. Also, the several German and English versions of his Handbook of Mathematics have reached a wide and enthusiastic readership. Thus, he was one of the very few scientists of our time who could still oversee and explore mathematics as a whole, and with all its deep connections with and applications in physics, instead of just small and isolated subfields, as is nowadays so common.

In 1994, he was elected a member of the German National Academy Leopoldina. In 2006, he was awarded the »Alfried-Krupp-Wissenschaftspreis«, and in 2014, the first Wissenschaftspreis der Teubner-Stiftung zur Förderung der Mathematischen Wissenschaften«. In 2004, he obtained the Honorary Doctorate of the Vietnam

Academy of Science and Technology in Hanoi.

He was a great scientist, but also a great human being, always concerned for the well being of others, always helping other people, warmhearted and generous, and with a profound sense of humor. Our institute owes him its existence, its vision, its structure, and its success. We shall always remain grateful to him for what he has taught us as a scientist and as a human being. (Prof. Jürgen Jost)

Read more about the life and work of Eberhard Zeidler in an obituary by Prof. Jürgen Jost:*Jost, Jürgen*: Eberhard Zeidler 1940-2016

Jahresbericht der Deutschen Mathematiker-Vereinigung, 120 (2018) 3, p. 221-228

DOI: 10.1365/s13291-017-0175-4

Watch a short slideshow in memory of our founding director Professor Eberhard Zeidler (more resolutions and download).

We have displayed this exhibition in our library in honor of Prof. Zeidler.

### A small selection of his books

**Quantum Field Theory - A Bridge between Mathematicians and Physicists**

Quantum Field Theory builds a bridge between mathematicians and physicists, based on challenging questions about the fundamental forces in the universe (macrocosm), and in the world of elementary particles (microcosm).

- Quantum Field Theory I: Basics in Mathematics and Physics (2006)
- Quantum Field Theory II: Quantum Electrodynamics (2009)
- Quantum Field Theory III: Gauge Theory (2011)

**Die Faszination der Wechselwirkungen zwischen Mathematik und Naturwissenschaften**

Jahresbericht der Deutschen Mathematiker-Vereinigung, 109 (2007) Sonderheft, p. 19-40

## Episode 1 - That’s how we started (01.03.2021)

The official decision to found a "Max Planck Institute for Mathematics in the Sciences" was made as early as 1995 on the suggestion of Bert-Wolfgang Schulze, head of the MPG working group "Partial Differential Equations and Complex Analysis" at the University of Potsdam. It was prepared by a founding commission headed by the great mathematician Friedrich Hirzebruch, the long-time director of the Max Planck Institute for Mathematics in Bonn.

On November 17, 1995 Jürgen Jost, Stefan Müller and Eberhard Zeidler were appointed Scientific Members of the Max Planck Society and directors of the new mathematical MPI. Jürgen Moser from ETH Zurich contributed to the establishment of our institute as an External Scientific Member.

On March 1, 1996, the first research group "Methods of Mathematical Physics" was launched, led by Professor Eberhard Zeidler, who also served as the first executive director of our institute. On August 1, Professor Jürgen Jost followed him with his group on "Geometric Methods, Complex Systems", and on October 1 the group "Analytical Methods, Mathematical Aspects of Material Sciences" headed by Professor Stefan Müller was launched.

Experience a few official as well as personal impressions from our first days in the following video (more resolutions and download).

Read more about the founding phase of our institute in the following article written by our founding directors in 1996 for the "Notices of the American Mathematical Society".