Prof. Dr. Clara Löh privat, UR / Löh
At its meeting on May 12 and 13, 2026, the German Research Foundation (DFG) approved four years of funding for Collaborative Research Center (CRC) 1785, “Generalized Motivic Methods in Geometry.” The CRC extends the established, abstract concept of motivic methods to subfields of mathematics for which it has previously been of little significance. Under the term “motivic thinking,” guiding principles are to be established to formulate and solve a wide variety of geometric problems. In addition to researchers from the University of Regensburg, the geometry department at the University of Augsburg, as well as a researcher from TUM and a researcher from JGU Mainz, are involved in the application.
“Motivic thinking in mathematics deals with universal structures that provide a common, conceptual explanation for individual phenomena that appear similar. In the new Collaborative Research Center, we view motivic thinking as a fundamental principle and extend it from its origins in algebraic geometry to a variety of geometric contexts. Of particular interest is the challenge of finding the right balance between overarching abstract methods and very specific geometric features,” explains Prof. Dr. Clara Löh of the Department of Mathematics at the University of Regensburg, the designated spokesperson for the Collaborative Research Center. “The funding for the CRC enables us to create an attractive environment for this research program, one that both requires and fosters intensive interaction between different areas of geometry. In particular, we look forward to the collaboration and exchange within the framework of the projects as well as the visiting scholar and workshop programs,” Löh continues.
For University President Professor Dr. Udo Hebel, the success of Regensburg’s scholars confirms the outstanding research work carried out by the researchers involved in the SFB: “The German Research Foundation’s approval of Collaborative Research Center 1785 in Mathematics impressively underscores the excellent and internationally recognized basic research conducted at the University of Regensburg. I would like to extend my special thanks to Prof. Dr. Clara Löh and all the participating scientists for their extraordinary dedication, their scientific creativity, and their successful collaboration across the various subfields of mathematics. The CRC will provide important impetus for the further development of geometry and, at the same time, sustainably strengthen our university’s profile in the field of cutting-edge mathematical research.”
About the CRC 1785
Recent developments in higher category theory suggest a new unifying viewpoint in geometry. Geometry is a multi-faceted area of mathematics, traditionally driven by insights from analysis, algebra, topology, and homotopy theory. In contrast, recent breakthroughs are based on the interplay between general abstract concepts and concrete calculations. A fundamental example are motivic methods in algebraic geometry, which resulted in spectacular applications to number
theory. The idea of motivic methods is to pass from individual invariants to universal solutions, to perform calculations in the associated universal category, and then to interpret these calculations in the context of the original geometric problem.
In this CRC, we consider generalised motivic methods as guiding principles, based on the fundamental concepts of universal structures, linearisation, and parametrisation. The impact of this motivic point of view is twofold: It gives fresh perspectives on important problems in geometry and it opens up new, sometimes surprising, directions of research.
We will apply motivic thinking to challenging open problems in algebraic geometry, topology, and Riemannian geometry. On the one hand, we expect substantial progress in contexts where motivic methods have already reached a high level of sophistication, such as for the construction of six-functor formalisms without 𝔸¹-invariance, the independence of 𝓁 in 𝓁-adic cohomology theories, problems in relative diophantine geometry, and for problems on special L-values. On the
other hand, applying the same philosophy in original ways in other geometric fields will generate new unexpected strategies for, among others, the calculation of logarithmic torsion homology growth, the calculation of E-theory and KK-theory of operator algebras, for the understanding of relative moduli spaces of topological manifolds, and for invariance properties of moduli spaces of Riemannian manifolds with spectral constraints.
To realise this programme, we will extend motivic ideas and frameworks to geometric contexts in which they have not yet been considered, we will perform motivic computations at the universal level, and we will explore the reach of generalised motivic methods in specific examples in geometry. In particular, this will generate novel interactions between different fields: We will transfer ideas from established motivic setups such as motivic homotopy theory to geometric topology and Riemannian geometry. Conversely, concrete geometric problems and calculations will inspire the creation and computation of universal structures.
The long-term vision of this CRC is to establish generalised motivic methods and motivic thinking as a powerful unifying framework in geometry.
Contact for scientific information:
Prof. Dr. Clara Löh
Universität Regensburg
Fakultät für Mathematik
Phone: +49 (0)941 943-2572
Mail: Clara.Loeh@mathematik.uni-regensburg.de
