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.

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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Topological periodic homology

2018 Thomas; Nikolaus
It is a classical algebraic problem to study solutions of polynomial equations. Since this can be very hard, people have invented certain algebraic invariants, called cohomology theories, to get a qualitative picture. Crystalline cohomology is one of these cohomology theories which is especially important in counting solutions over finite fields. It is an algebraic invariant, but it has recently been possible to express crystalline cohomology in terms of a topological invariant - called topological periodic homology. This leads to natural generalizations, extensions and computations. more

Borcherds products

2017 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. more

Quantum mechanics on a graph

2016 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. more

Diophantine equations

2015 von Känel, Rafael; Matschke, Benjamin
We discuss certain classical Diophantine equations and we explain how geometry can help to understand their solutions. In particular we consider some highlights in the field of Diophantine equations, including Faltings' resolution of the Mordell conjecture and Wiles' proof of Fermat's last theorem. We also discuss a project at the MPIM Bonn which combines the methods of Faltings and Taylor-Wiles in order to make some progress concerning the classical problem of finding all squares and cubes of given difference. more

Deformations of foliations

2014 Vogel, Thomas
We discuss an application of contact topology to the space of foliations by surfaces on 3-manifolds. Without further geometric restrictions on the foliation, the connected components of the space of foliations are well understood. In contrast to this, if one requires that all foliations under consideration are taut, then the space of these foliations is more complicated. So far, its connected components are classified only in few special situations. more

Higgs bundles and representation varieties

2014 Swoboda, Jan
We discuss the moduli space of solutions to Hitchin's selfduality equations and its connection with representation varieties. Afterwards some recent results concerning the degeneration profile of solutions for large Higgs fields will be presented. more

Counting algebraic curves - old, new and refined

2013 Göttsche, Lothar
The counting of algebraic curves on surfaces is a classical topic of algebraic geometry, which has gained new interest because of the advent of string theory from physics. After a short introduction into enumerative invariants of curves a refinement is considered. Instead of a number of curves one gets a polynomial, which can be related to invariants of real and tropical algebraic geometry. more
We give a glimpse into the world of reflection groups, Coxeter groups, Lie groups and Kazhdan-Lusztig polynomials. more

Deformation quantization and the formula of Kontsevich

2012 Rossi, Carlo Antonio
After a brief comparison between the commutative algebra of classical observables in Hamiltonian mechanics and non-commutative quantum observables in quantum mechanics, an algebraic way of realizing the quantization process (deformation quantization) is introduced and in particular the universal formula of Kontsevich is explained in some details. Finally, some (open) problems related to deformation quantization are considered, some of which have been positively addressed at the Max Planck Institute for Mathematics in Bonn. more

Constants in arithmetic: periods and their relations

2012 Raum, Martin; Raum, Sven
Periods are a special kind of numbers, which play an outstanding role in number theory. An important aspect to consider are relations between different representations of periods. These can be studied in great detail for L-values, which are more accessible than most other examples are. more

Duality of C*-algebras

2011 Schneider, Ansgar
C*-algebras are of central importance in Physics, Mathematics and in the border area between. Mathematicians at the MPIM in Bonn try to find new insights between C*-algebras and other mathematical structures. more

Geometry in the tropical limit

2011 Ilia Itenberg, Grigory Mikhalkin
Complex algebraic varieties become easy piecewise-linear objects after passing to the so-called tropical limit. Geometry of these limiting objects is known as tropical geometry. In this survey we look at a few simple examples of correspondence principle between classical and tropical geometries. more

L-functions and Deligne-Beilinson conjectures

2010 Mellit, Anton
We explain the important idea of L-function in mathematics and Deligne-Beilinson conjectures about values of L-functions at integral points. As main examples we consider here the Riemann zeta function and L-functions of elliptic curves. In the end we explain relation between values of L-functions of elliptic curves at 2 and logarithmic Mahler measures. more

Open books and applications

2010 Vogel, Thomas
Open books are particular decompositions of manifolds which are useful for the construction of foliations. There are also important applications to the theory of contact structures. more

Multiplicity one theorems in representation theory

2009 Aizenbud, Avraham; Gourevitch, Dmitry
An important question of Representation Theory is "what happens to an irreducible representation of a group when one restricts it to a subgroup?". Usually it stops being irreducible and it is interesting to know whether it decomposes to distinct irreducible representations. This report contains an exposition on this question and a short description of a recent proof that in the case of the pair (GL(n+1),GL(n)) over local fields the answer is positive. more
Sequences defined by recurrence relations are known since the work of Fibonacci. They occur in number theory as well as for the solutions of differential equations. This report explains why special billiard tables give rise to recurrence relations such that the corresponding sequences consists unexpectedly of integers only. more

q-Series and Modular Forms

2008 Zagier, Don
Three sources of q-hypergeometric series will be discussed: Mock theta functions, "characters" of rational conformal field theories and the Witten-Reshitikhin-Turaev invariant of the icosahedral Poincaré sphere. more

Arithmetic and Arakelov Geometry

2008 Durov, Nikolai
The aim of this note is to give a very informal introduction into the world of arithmetic geometry, and especially Arakelov geometry. more

Generalized triangle inequalities

2007 Kapovich, Mikhail
The purpose of this article is to give an overview of generalized triangle inequalities in symmetric spaces and their application in representation theory. more
From a polygon in the plane with additional symmetries one can glue together closed surfaces. On the space of all such surfaces two flows act. Statistical properties of the thus obtained are closely related to statistical properties of the action of the transformation class group. more
Geometry studies geodesics in various settings, in particular on hyperbolic surfaces. The distribution of geodesics on arithmetic hyperbolic surfaces gives information on the arithmetic of quadratic forms, an important branch of number theory. more

Renormalisation as Galois-Symmetry

2006 Marcolli, Matilde
In the sixties and seventies of the last century physicists developed procedures to renormalize infinities appearing in quantum field theory. We explain recent work of Connes-Kreimer and Connes-Marcolli, giving a conceptual understanding of these procedures and indicating unexpected links to number theory. more

The multiplicative order

2005 Moree, Pieter
The notion of multiplicative order is very old and has interested mathematicians since the days of Fermat. It plays a role in different branches of mathematics. We will give a short overview of the history of research in this area up to recent work concerning the distribution of the order over residue classes. more
The theory of dynamical systems provides powerful methods for studying diverse problems from planetary orbits to number theory. Recent work by Einsiedler, Katok and Lindenstrauss used methods from dynamical systems to make significant progress on a classical conjecture in number theory. more

Noncommutative geometry and number theory

2004 Marcolli, Prof. Matilde
We describe how noncommutative geometry, a mathematical formulation of geometry adapted to quantum phenomena, interacts with number theory through quantum statistical mechanical systems with phase transitions and spontaneous symmetry breaking. This provides a unified setting for many important arithmetic results including the spectral realization of zeros of the Riemann zeta function and the Galois theory of modular functions. more

Maps between spheres

2004 Baues, Hans-Joachim; Jibladze, Mamuka
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