Max Planck Institute for Mathematics in the Sciences

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

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

Scientists have developed new tools to understand how networks fsynchronize

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

Researchers study historical developments of the periodic system of chemical elements

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.

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.

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.

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.

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.

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.