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

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Department Pattern Formation, Energy Landscapes, and Scaling Laws

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For a chimpanzee, one good turn deserves another

Apes only provide food to conspecifics that have previously assisted them

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Sound of words is no coincidence

Particular sounds are preferred or avoided in non-related languages far more often than previously assumed

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Robots: the curiosity of the body

A new learning rule could help robots to acquire new movements and explain how people develop sensorimotor intelligence

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Ingeniously designed machines learn to move without receiving any instructions from control programs. Similarly, robots, whose brains are developed by Ralf Der and Nihat Ay at the Max Planck Institute for Mathematics in the Sciences, are learning about their bodies and their environment.

Some metals have a memory: They can be bent and, if subject to the right treatment, can reassume their original shape. Stefan Müller, Director at the Max Planck Institute for Mathematics in the Sciences, and his colleague Anja Schlömerkemper are exploring the mathematical laws on which this memory is based, and through which it may be improved.

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Tensor decomposition

2018 Michałek, M.; Sturmfels, B.

Mathematics

Tensors are higher-dimensional matrices. They provide an efficient way to store big data sets and to analyze them statistically. Using the decomposition into tensors of rank 1, hidden structures and correlations can be discovered. The geometry of tensors plays an important role in this research area.

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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.

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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.

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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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