Max Planck Institute for Mathematics

Max Planck Institute for Mathematics

From the foundations of information science to string theory and the theory of black holes, from global positioning systems to secure data encryption, the technology of the modern world depends on sophisticated mathematics. In all these examples and in countless others, the mathematics that is used was developed in the course of purely theoretical investigations, and the applications came only later and unexpectedly. It is this more theoretical and foundational side of the field which is primarily studied at the Max Planck Institute for Mathematics. Here research is done in geometry and topology (its more flexible cousin), in number theory and analysis – fields that are centuries old, but that continue to lead to exciting new discoveries and turn out to have surprising connections among each other and to other sciences.


Vivatsgasse 7
53111 Bonn
Phone: +49 228 402-0
Fax: +49 228 402-277

PhD opportunities

This institute has an International Max Planck Research School (IMPRS):

IMPRS for Moduli Spaces

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

New Director of the Max Planck Institute for Mathematics awarded the highest distinction in Mathematics


For many, mathematics is nothing more than an accumulation of abstract formulas and dry recipes for calculating. Not so for Friedrich Hirzebruch, Founding Director of the Max Planck Institute for Mathematics in Bonn, Germany. He had already succumbed to the beauty of the subject in his youth. As the “doyen of German post-war mathematics,” Hirzebruch made this city on the Rhine an attractor for researchers the world over.

Johann Sebastian Bach, Le Corbusier and Maurits Escher: Mathematics has influenced many a creative genius. But also mathematics itself contains an element of beauty.

Starting with completely abstract structures, mathematicians develop new theories and models that precisely formulate and describe real properties of the real world.

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Surfaces in 4-dimensional spaces

2020 Ray, Arunima


Topology is the study of spaces of arbitrary dimensions and their properties. The simplest spaces, called manifolds, locally resemble Euclidean spaces, such as the infinite line, or the infinite plane. Curiously, manifolds of dimension at least five behave similarly to one another, while lower dimensional spaces are less controlled. Seminal work of Freedman from 1982 shed light on 4-dimensional manifolds. Unfortunately this is not well-understood by the mathematical community. Recent work provides an accessible exposition of these beautiful ideas as well as extensions and generalizations.


Symmetry groupoids in classical field theory

2019 Blohmann, Christian


The correspondence between symmetries and conservedquantities is one of the most important principles of physics. Incontrast to classical mechanics and gauge field theories, the con-served quantities of General Relativity do not span a symmetry algebra in the conventional sense. Instead, a so-called Hamiltonian Lie algebroid is obtained from a naturally constructed symmetry groupoid.


Counting surfaces

2018 Borot, Gaëtan


Imagining physical theories taking place in a space M allow mathematicians to extract fine geometric information on M. In particular, quantum field theories and string theories have led to the definition of new and hard-to-compute geometric invariants. The algebraic structures that govern them have in fact a wider range of applications for the enumerative geometry of surfaces, for reasons that are not completely understood yet. This report describes the principle of one these structures, called topological recursion and based on the strategy of cutting surfaces into pairs of pants.


Borcherds products

2016 Kaiser, Christian


After introducing elliptic modular forms we consider Borcherds products as singular theta lifts on orthogonal groups. Finally we discuss a characterization of Borcherds products by symmetries.


Quantum mechanics on a graph

2015 Mnev, Pavel


We discuss the problem of counting paths going along the edges of a graph as a toy model for Feynman’s path integral in quantum mechanics.

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