Max Planck Institute for Mathematics

Geordie Williamson receives the Max Planck-Humboldt Research Award 2024, and Max Planck-Humboldt Medals go to Laura Waller and Torsten Hoefler

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.

Higher algebra is concerned with extending the integers to new domains of counting, specifically the so-called stable homotopy groups of spheres, which encode rich information about algebraic, geometric, and topological objects. Determining these groups proves to be a fundamental and immensely difficult problem in mathematics. In this paper, we describe the chromatic approach to this problem and its connection to arithmetic geometry, which allows for the confirmation of a 50-year-old prediction.

Embedding theory is the part of topology that studies the question of the embeddability of a space in another space. Much is known about embeddings of graphs in the plane, as well as embeddings of codimension greater than two. Recent work has focused on the question of embeddability in the case of codimension two, linking it to the solvability of certain equations in groups.

The Langlands correspondence predicts a profound connection between algebra and analysis. This connection is deep and still largely conjectural. However, in recent years there has been exciting new progress toward its resolution, thanks to the introduction of striking new geometric techniques. In this report, I give an impressionistic introduction to the Langlands correspondence, and indicate the ideas behind these recent developments.

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.

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.