CRC 478 - Geometric structures in mathematics

Basic data for this project

Description

In mathematics there are certain structures of a universal nature which appear in completely different settings. Understanding them allows to treat seemingly unrelated problems with uniform methods. The collaborative research centre concentrates on geometric structures, their developement as a method and their applications. This concerns fields like Arithmetic Geometry, Differential Geometry, Topology, Analysis and Noncommutative Geometry. Examples of such methods are cohomology and K-theories. They play an important role in every "Projektbereich" of the collaborative research centre. K-theory came from algebraic geometry and topology. Later this theory influenced analysis on manifolds via index theory and now has become one of the main tools in Noncommutative Geometry. Via regulators, special L-values and the theory of motives K-theory now plays an important role in Arithmetic Geometry as well. The collaborative research centre works on a number of the main open problems in mathematics like the Bloch-Kato conjectures, The Baum-Connes conjectures, conjectures of Atiyah, Hopf, Singer and many others. Work on these problems will create a lot of new and interesting mathematics which in turn will help to solve further problems. The intertwining of different mathematical directions via geometric structures will remain a fruitful line of research for a long time to come.