TRR 358:  Integral Structures in Geometry and Representation Theory

• Subject Area: Mathematics
• Term: since 2023
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 491392403

Integral structures arise in many places throughout mathematics: as lattices in Euclidean space, as integral models of reductive groups and algebraic schemes, or as integral representations of groups and associative algebras. Even questions about the most basic example of an integral structure, the ring of integers Z, very soon lead into the fields of analysis, algebra, or geometry. In the same vein, investigations of integral structures are most successfully treated by viewing them from different perspectives, often require the usage of most advanced mathematical methods, and frequently lead to unexpected connections.This point is illustrated by the classification of wallpaper groups, i.e., discrete groups of isometries of the plane that contain two linearly independent translations. As intricate double-periodic arabesques, we meet them in the medieval Alhambra palace in Granada. It is a classical fact that there are precisely 17 wallpaper groups. This result has a geometric aspect, as it provides the number of flat compact orbifold surfaces; and it also has an interpretation within representation theory: it is part of the classification of hereditary categories over the field of real numbers.As integral structures necessitate a combined approach from different mathematical sub-disciplines, we will embark on a broad research programme reaching from algebraic geometry to analysis on manifolds, from geometric group theory and algebraic combinatorics to representation theory of associative algebras. With joint forces from the participating universities, we intend to answer major questions in the algebraic and analytic theory of automorphic forms, categorical representation theory and algebraic geometry, as well as classical and p-adic harmonic analysis on symmetric spaces.

• DFG Programme: CRC/Transregios

Current projects

• A01 - The structure of (almost) lattices – algebra, analysis, and arithmetic (Project Heads Alfes-Neumann, Claudia ;  Baake, Michael ;  Voll, Christopher )
• A02 - Algebraic and arithmetic aspects of aperiodicity (Project Heads Baake, Michael ;  Klüners, Jürgen )
• A03 - Codes and designs (Project Heads Baumeister, Barbara ;  Rösler, Margit ;  Schmidt, Kai-Uwe )
• A04 - Combinatorial Euler products (Project Heads Blomer, Valentin ;  Klüners, Jürgen ;  Voll, Christopher )
• A05 - Affine Kac–Moody groups: analysis, algebra, and arithmetic (Project Heads Burban, Igor ;  Bux, Kai-Uwe ;  Glöckner, Helge )
• A06 - Zeta functions of integral quiver representations (Project Heads Crawley-Boevey, William ;  Voll, Christopher )
• A07 - Matroids, codes, and their q-analogs (Project Heads Kühne, Lukas ;  Schmidt, Kai-Uwe )
• B01 - Theta lifts and equidistribution (Project Heads Alfes-Neumann, Claudia ;  Blomer, Valentin )
• B02 - Spectral theory in higher rank and infinite volume (Project Heads Blomer, Valentin ;  Weich, Tobias )
• B03 - Spherical harmonic analysis of affine buildings and Macdonald theory (Project Heads Bux, Kai-Uwe ;  Hilgert, Joachim ;  Rösler, Margit )
• B04 - Geodesic flows and Weyl chamber flows on affine buildings (Project Heads Bux, Kai-Uwe ;  Hilgert, Joachim ;  Weich, Tobias )
• B05 - p-adic L-functions, L-invariants and the cohomology of arithmetic groups (Project Heads Januszewski, Fabian ;  Spieß, Michael )
• B06 - Equivariant cohomology and Shimura varieties (Project Head Spieß, Michael )
• C01 - Hyper-Kähler varieties and moduli spaces (Project Heads Barros, Ignacio ;  Vial, Ph.D., Charles )
• C02 - Hereditary categories, reflection groups, and non-commutative curves (Project Heads Baumeister, Barbara ;  Burban, Igor ;  Crawley-Boevey, William )
• C03 - Tame patterns in the representation theory of reductive Lie groups and arithmetic geometry (Project Heads Burban, Igor ;  Crawley-Boevey, William ;  Januszewski, Fabian )
• C04 - Counting points on quiver Grassmannians (Project Heads Franzen, Hans ;  Sauter, Julia )
• C06 - Stratifying derived categories over arbitrary bases (Project Heads Krause, Henning ;  Lau, Eike )
• C07 - Derived-splinters and full exceptional collections (Project Heads Krause, Henning ;  Lau, Eike ;  Vial, Ph.D., Charles )
• C08 - Cohomological structures of hyper-Kähler varieties (Project Heads Lau, Eike ;  Vial, Ph.D., Charles )
• Z - Central tasks of the Collaborative Research Centre (Project Head Bux, Kai-Uwe )

• Applicant Institution: Universität Bielefeld
• Co-Applicant Institution: Universität Paderborn
• Participating University: Rheinische Friedrich-Wilhelms-Universität Bonn
• Spokesperson: Professor Dr. Kai-Uwe Bux