TUM involvement in Higher Invariants project

Collaborative Research Center of the German Research Foundation

3 March 2022
A decomposition of a torus as a higher cobordism

Collaborative Research Center Higher Invariants: Decompoosition of a donut as a higher cobordism.

As of 1st January 2022 the Department of Mathematics of the Technical University of Munich (TUM) is involved in the Collaborative Research Center (CRC) 1085 through the participation of Prof. Claudia Scheimbauer. The CRC "Higher Invariants: Interactions between Arithmetic Geometry and Global Analysis", coordinated by the University of Regensburg, began its third funding period at the beginning of the year and is funded by the German Research Foundation (DFG) with the sum of 8 Million Euros over 4 years.

Prof. Claudia Scheimbauer is Principal Investigator in two of the CRCs projects, "Higher Categories of Correspondences" and "Higher Structures in Functorial Field Theory". For these two projects, funding has been secured for a postdoctoral position and a PhD position within her group at the TUM.

About the Collaborative Research Center 1085

In mathematics, geometric invariants play an important role. They assign to complicated geometric objects simpler structures, enabling the description and classification of these objects. Many deep mathematical findings have been possible through the successful application of this principle. New ideas have changed the understanding behind classical geometric invariants and shown how these can be refined to higher invariants through systematic, technically demanding methods.

This development is particularly driven by applications to arithmetic geometry and global analysis. Despite the differing directions of these two mathematical fields, the techniques and concepts used in both fields are increasingly influential. However, many of the – sometimes surprising – relations between higher invariants in both mathematical fields remain unexplained. A systematic transfer of ideas and results between the two, as initiated in the CRC Higher Invariants, can in this case often lead to conceptional explanations and simplifications.

Standardization of higher invariants

"In the first two funding periods of this CRC, it has been shown, that the currently dominant theories on higher category theory, motivic homotopy theory, and derived algebraic geometry are powerful tools for the study and construction of higher invariants", explains project leader Professor Guido Kings. This development will be furthered in the third funding period and the CRC has been strengthened in these areas, in particular through the participation of the TUM with Professor Claudia Scheimbauer.

The scientists of the CRC Higher Invariants aim to achieve their main goals by following two research directions, which are dependent upon and complement each other: the study of specific higher invariants and the discovery of the principles of construction of higher invariants. This should then lead to a standardization and general theory of higher invariants within arithmetic geometry and global analysis.