Geometric invariants play an important role in mathematics. They associate simpler structures to complicated geometric objects with the goal to describe and classify these objects. Many deep results in mathematics rely on the successful application of this principle.In recent years, new ideas changed our understanding of classical geometric invariants and showed how these can be refined with sophisticated techniques to higher invariants. This development is driven mainly by Arithmetic Geometry and Global Analysis. Despite different goals in both fields, there is an increasing mutual influence of techniques and concepts. Many of the sometimes surprising relations between the higher invariants in both fields remain mysterious and are not fully understood. A more systematic transfer of concepts and results from one field to the other, as initiated by this CRC, leads often to conceptual explanations and allows a unifying perspective.The research in the first and second funding period of this CRC revealed that currently prospering theories, namely higher category theory, motivic homotopy theory, and derived algebraic geometry, provide powerful frameworks and techniques to study and construct higher invariants in Arithmetic Geometry and Global Analysis. This is reflected in the third period with new projects and PIs with a focus on these theories.The expectation that the joint perspective yields new insights and results was successfully realized in the second funding period and led to the solution of difficult problems and to new perspectives on fundamental questions. The use of higher invariants, higher structures, and the interaction between researchers in Arithmetic Geometry and in Global Analysis was decisive for this success.The main goal of this CRC shall be realized in two interdependent and complementary lines of research: the study of questions on specific higher invariants and the discovery of the principles of the construction of higher invariants. The joint study of both aspects should lead to a unification and a general theory of higher invariants in Arithmetic Geometry and Global Analysis.
