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 Mathematics in the Sciences

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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Department Geometry, Groups, and Dynamics

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New mathematical model of genetic interaction identifies master regulators in biological networks

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This year's ERC Synergy Grantees of the Max Planck Society

The scientists and their research teams receive around 40 million euros in funding for their work

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Scientists have developed new tools to understand how networks fsynchronize

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Volunteers at the main railway station in Berlin offering food and beverages to refugees from the Ukraine. From the viewer's perspective, they are standing in front of a long table with light brown cups and drinks. The focus is on a helper with a green waistcoat and FFP-2 mask, who can be seen from diagonally behind. On the other side are people helping themselves to the beverages on offer.

Coupling two approaches of game theory can shed light on how moral norms evolve

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Researchers study historical developments of the periodic system of chemical elements

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Studies show that the louder political minorities shout on social networks, the quieter the democratic majority becomes. Hate, hate speech, and propaganda thrive in echo chambers and distort perceptions in political discourse. Researchers investigate this phenomenon from the perspective of social science, law, and mathematics.

Self-learning algorithms are turning our society upside down. But all too frequently, even their developers do not fully understand how they work. Researchers at the Max Planck Institute for Mathematics in the Sciences want to remedy the situation with fundamental theories of machine learning.

Nowadays, political debates often turn into verbal brawls – especially on social media. In order to counteract this, Eckehard Olbrich and Sven Banisch of the Max Planck Institute for Mathematics in the Sciences in Leipzig and Philipp Lorenz-Spreen of the Max Planck Institute for Human Development are investigating how polarization occurs and how opinion formation in groups works.

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From chemical reaction networks to hypergraphs 

2023 Jost, Jürgen

Mathematics

Structures where more than two elements may have relations, like substances in chemical reaction or joint publications by scientists, are mathematically described as hypergraphs. Scientists at the Max Planck Institute for Mathematics in the Sciences are developing a rich mathematical theory that can identify the specific properties of all kinds of such networks and describe their dynamical behavior. With these methods, also new types of collective patterns in networks of coupled oscillators can be detected and analysed.

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Polynomials, polyhedra and algorithms

2022 Marta Panizzut

Mathematics

Algebraic geometry describes shapes using polynomials. Discrete geometry instead uses matrices and linear equations to describe polyhedra. This different approach is reflected in the type of algorithms used for the computer-aided study of geometric objects. Tropical geometry is a recent mathematical theory that leads to innovative computational methods and exciting connections between algebraic and discrete geometry. Our research group is working to advance this further.

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Solving polynomial equations

2021 Breiding, Paul

Mathematics

Many problems from the sciences can be modelled as the problem of computing the solutions to a system of polynomial equations. Starting from an example application, we will discuss basic strategies for solving such systems of equations and we will explore what solving means in this context. We will underline our philosophy that applied and theoretic scientific questions are not mutually exclusive, but that they complement each other.

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Deep Learning Theory

2020 Montúfar, Guido

Mathematics

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.

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Inverse problems are ubiquitous in nature, in medical, scientific and engineering applications and in our daily lives. In all of these problems it is the objective to recover properties of the underlying (physical) system by means of indirect measurements. As these problems are typically in a mathematically precise sense "ill-posed", this turns out to be a difficult task in general. In this article some of the main challenges are introduced by discussing prototypical problems which are investigated at the MPI MiS.

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