Max Planck Institute for Mathematics in the Sciences

Max Planck Institute for Mathematics in the Sciences

Without mathematics, everyday life as we know it would be inconceivable. Telephone networks, timetables and stock inventories are all optimised using modern methods of discrete mathematics. The rapid transmission of images by means of data compression uses concepts from mathematical analysis. The highly-efficient encoding of data, for example, in bank transactions carried out over the Internet, is an application of number theory. High-resolution computer tomography was also made possible by the development of new mathematical processes for image reconstruction. The list of examples is endless. Mathematical models and methods also play an increasingly important role in the optimisation of entire production processes. Moreover, the connection between mathematics and its applications is not a one-way street: basic questions posed by the sciences, engineering and economics have always inspired mathematicians to search for new mathematical methods and structures. The interaction between mathematics and the sciences forms the core of the work carried out at this Institute.

Contact

Inselstraße 22
04103 Leipzig
Phone: +49 341 9959-50
Fax: +49 341 9959-658

PhD opportunities

This institute has an International Max Planck Research School (IMPRS):
IMPRS Mathematik in den Naturwissenschaften

In addition, there is the possibility of individual doctoral research. Please contact the directors or research group leaders at the Institute.

Department Geometric Methods, Complex Structures in Biology and Cognition more
Department Pattern Formation, Energy Landscapes, and Scaling Laws more
For a chimpanzee, one good turn deserves another
Apes only provide food to conspecifics that have previously assisted them more
Sound of words is no coincidence

Sound of words is no coincidence

Forschungsmeldung September 13, 2016
Particular sounds are preferred or avoided in non-related languages far more often than previously assumed more
Robots: the curiosity of the body

Robots: the curiosity of the body

Forschungsmeldung November 25, 2015
A new learning rule could help robots to acquire new movements and explain how people develop sensorimotor intelligence more

Languages less arbitrary than long assumed

Forschungsmeldung October 01, 2015
Tonal languages require humidity

Tonal languages require humidity

Forschungsmeldung January 22, 2015
Languages with a wide range of tone pitches have primarily developed in regions with high levels of humidity more
The pathways of epidemics

The pathways of epidemics

Forschungsmeldung October 30, 2012
A new computer model rapidly and accurately estimates who spreads an infection particularly extensively, thereby facilitating countermeasures more
Robots start learning

Robots start learning

Forschungsmeldung March 23, 2009

Ingeniously designed machines learn to move without receiving any instructions from control programs. Similarly, robots are learning about their bodies and their environment.

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Evolving slower gets you the bigger piece of the pie

Evolving slower gets you the bigger piece of the pie

Forschungsmeldung January 17, 2003
Scientists from the Max Planck Institute for Mathematics in the Sciences and from the University of Washington reveal in a recent paper in PNAS that evolving faster, adapting better, and learning more quickly is not always beneficial for a species. more
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Grain growth is an intricate process during which the grain structure of a polycrystal coarsens. Efficient numerical schemes shed light on the statistical behavior of the overall structure. The underlying differential equation for the interfaces is mean curvature flow. The mathematical structure of the equation as a steepest descent in an energy landscape gives new insights and allows to develop and to analyze numerical algorithms.

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How to choose a suitable detail level describing a complex system.

2016 Pfante, Oliver; Bertschinger, Nils; Olbrich, Eckehard; Ay, Nihat; Jost, Jürgen
Mathematics
Analyzing complex systems, the question occurs what one needs to know at the detail level in order to understand the dynamics at the system level. Ideally, one could use a higher level of description at which the system dynamics unfolds autonomously. That means once the initial state is known, one no longer needs to check the details at the lower level in order to predict the system. Formal methods have been developed to analyze the issues involved. In particular, the question of the flow of information between levels has been linked to the question of memory effects at the respective levels. more

Hysteresis

2015 Tikhomirov, Sergey
Mathematics
Hysteresis often appears in nature as a mechanism for self-organisation. A mathematical description of this phenomenon is given and discussed in some example cases motivated by real life applications. These examples are decomposed into simple (transverse) and difficult (non transverse) cases. The solution for the transverse case is given, the non transverse case is partially solved and open problems and questions are discussed. more

Effective description of heterogeneous media

2014 Marahrens, Daniel; Otto, Felix
Mathematics

In applications, it is often desirable to obtain macroscopic properties of heterogeneous materials with microscopic structures. It is possible to obtain these by simulating a representative volume element. In order to do so efficiently, one requires accurate error estimates which are derived by combining ideas from analysis and probability theory.

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Knowledge and awareness of economic actors

2013 Kabalak, Alihan
Mathematics
That a subject knows something is conceptualized as it ruling out certain facts while considering other facts possible. Thus the distinction between ‘ruled out’ and ‘possible’ draws upon a given set of conceivable facts. If in a model the knowledge of several subjects is based on the same conceivable facts, everyone receives the same information about what anyone can possibly know (identical ‘awareness’). Overcoming this restriction allows more differentiated analyses of economic decision making. more

Numerical computation of wave propagation

2012 Banjai, Lehel; Gruhne, Volker
Mathematics
The understanding and prediction of the behaviour of, for example, acoustic, electromagnetic, and elastic waves in different media is of great importance in a number of applications. The propagation of waves is often investigated in unbounded spatial domains. To obtain a numerical approximation in such situations the method of boundary integral equations is particularly attractive. The analysis and development of numerical methods for boundary integral equations for wave equations, in contrast to the time-harmonic Helmholtz equation, has until recently been in its infancy. more

Domain and wall patterns in ferromagnetic films

2011 Döring, Lukas; Otto, Felix; Steiner, Jutta
Mathematics
The magnetization of a ferromagnetic material forms a variety of patterns, but the scale separation between large domains of almost constant magnetization and narrow transition layers, so-called domain walls, hinders the numerical simulation of these structures. However, rigorous asymptotic analysis makes it possible to derive reduced models that explain the observed phenomena and make them accessible to numerical simulation, as we will illustrate using the example of the deformed Landau state, asymmetric domain walls and the concertina pattern. more

Autonomy: an information theoretic perspective

2010 Bertschinger, Nils
Complex Systems Mathematics
Many living systems can survive thanks to sophisticated and flexible information processing capabilities enabling them to act autonomously. Even though this ability is increasingly demanded from technical systems, it is still largely unknown which properties are required for a system to obtain autonomy. To fill this gap, an information theoretic measure of autonomy is proposed here. For illustration the autonomy of simple systems is measured and a close relationship between autonomy and memory is established. more

Fast Solvers and the Curse of Dimension

2009 Grasedyck, Lars
Mathematics
In many applications one has to solve large-scale (linear) systems. The task to solve these as fast as possible is the goal of “fast solvers”. At the beginning only sparse systems were considered, but due to the recently at the MPI-MIS developed “hierarchical matrices” the scope has become much broader. However, even fast solvers are limited by the dimensionality of the problem under consideration – a challenge for modern mathematical methods which the group “Scientific Computing” at the MPI-MIS wants to meet. more

H²-Matrices

2007 Börm, Steffen; Hackbusch, Wolfgang
Mathematics
The numerical approximation of physical or biological models leads to large systems of equations that have to be solved as rapidly as possible by a computer. If these systems are approximated by H2-matrices, they can be handled far more efficiently than by standard methods. Under certain conditions it is even possible to reach the optimal order of complexity, i.e., to ensure that the number of operations is proportional to the size of the solution vector. more

Stochastic effects in models for materials with multiple scales

2007 Dirr, Nicolas
Material Sciences Mathematics
Models for materials which are valid on the length scales of day-to-day experience describe quantities which are derived by averaging over many degrees of freedom of a finer length scale. The deviations from this average due to thermal effects or impurities are often negligible. In some situations, however, they can trigger effects which are observable on the rough length scale. This is exemplified by the mathematical investigation of a prototypical model. more

Quantum Gravity: No Experiments, but Mathematics

2006 Fleischhack, Christian
Mathematics Particle 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. more

Principles of robustness

2006 Ay, Nihat
Complex Systems Mathematics
At the MPI for Mathematics in the Sciences general principles that underlie the robustness of evolved systems are studied and formalized. The aim is to formulate a mathematical theory of robustness that can be used for developing artificial adaptive systems. more

Coupled dynamical systems and structure formation

2005 Atay, Fatihcan; Jost, Jürgen
Complex Systems Mathematics
We investigate the formation of structure in coupled dynamical systems. Although the individual elements may possess complex dynamics of their own, the system can globally synchronize. This is even possible when the activities between elements are transmitted with temporal delays, that is, when the system does no longer have an intrinsic notion of simultaneity. For suitable parameter constellations, it can also be observed that the restriction of the individual degrees of freedom of the participating elements caused by the coupling leads to the emergence of global dynamical patterns on a longer time scale. This suggests a new mathematical approach to the formation of structure in coupled systems. more

Wavelets in Quantum Chemistry

2004 Flad, Heinz-Jürgen; Hackbusch, Wolfgang
Chemistry Complex Systems Mathematics
The vision to perform chemical experiments routinely on a computer has been realized nowadays, at least to some extent, by modern techniques in quantum chemistry. For future applications, however, it will be a major challenge to better incorporate the multiscale character of quantum chemical problems. Wavelets provide local and systematic decompositions of the quantities involved into their characteristic length- and energy-scales. Furthermore these decompositions lead to numerical algorithms with almost optimal complexity. We have studied within our research project various aspects of this approach for concrete model problems in many-particle theory. Thereby we are aiming to bridge the gap between quantum chemistry and the relevant disciplines in applied and numerical mathematics. more