SFB 701:  Spectral Structures and Topological Methods in Mathematics

• Subject Area: Mathematics
• Term: from 2005 to 2017
• Project identifier: Deutsche Forschungsgemeinschaft (DFG) - Project number 15111527

Mathematics develops in contact with problems either posed by mathematical thoughts or in applying mathematical methods in science and society. During the last decades, the advances in pure and applied mathematics often appeared to be rather divergently diferentiating into various subfields. Nevertheless, they offer very promising perspectives if combined in new initiatives. New results with the potential of connecting both areas are rare, but when they occur, they are of tremendous impact and provide fundamental insight. Moreover, apparently different areas of mathematics reveal remarkable connections at closer inspection. This concerns their methods as well as their aims. Some of these connections will be studied in the Collaborative Research Centre.
Spectral structures are omnipresent in mathematics and in natural sciences. Topological methods allow us to describe invariant properties of mathematical objects under classes of deformations. In mathematics both are connected via global topological invariants expressed in terms of spectral data. Many significant developments in mathematics are connected with spectral structures and topological methods: New concepts of mathematical physics recently have had a significant impact in theoretical mathematics. We specifically mention the Seiberg-Witten invariants from topology, universal spectral, distributions from quantum physics and their appearance in number theory, the application of concepts from quantum field theory to the theory of moduli spaces in algebraic topology, as well as quantum groups. Conversely, modern methods developed in theoretical mathematics, in particular in topology and number theory, have proved useful not only in theoretical physics but also in other applications of mathematics, such as fluid dynamics, crystallography, and materials science.
Spectral structures and topological methods are basic unifying concepts for scientists in different fields of theoretical and applied mathematics who cooperate in the Collaborative Research Centre.

• DFG Programme: Collaborative Research Centres

Completed projects

• A01 - Spectral theory of aperiodic order (Project Heads Baake, Michael ;  Huck, Christian )
• A02 - Numerical Analysis of High-Dimensional Transfer Operators (Project Head Beyn, Wolf-Jürgen )
• A03 - Stochastic dynamics and bifurcations (Project Head Gentz, Barbara )
• A04 - Asymptotics of spectral distributions (Project Heads Götze, Friedrich ;  Kösters, Holger )
• A05 - Stochastic evolutions in continuum (Project Head Kondratiev, Yuri G. )
• A06 - Analysis and stochastic processes on metric measure spaces (Project Head Grigoryan, Alexander )
• A08 - Fine properties of long-range operators and processes (Project Head Kaßmann, Rolf Moritz )
• A09 - Dynamics and asymptotic behaviour of stochastic evolution systems (Project Heads Beyn, Wolf-Jürgen ;  Gentz, Barbara ;  Röckner, Michael )
• A10 - Nonlocal operators (Project Heads Grigoryan, Alexander ;  Kaßmann, Rolf Moritz ;  Kondratiev, Yuri G. )
• B01 - Diophantine inequalities, groups, and lattices (Project Heads Abels, Herbert ;  Bux, Kai-Uwe ;  Götze, Friedrich )
• B02 - Combinatorial and topological structure of aperiodic tilings (Project Heads Baake, Michael ;  Gähler, Franz )
• B03 - Numerical analysis of equivariant evolution equations (Project Head Beyn, Wolf-Jürgen )
• B04 - Kolmogorov operators and SPDE (Project Head Röckner, Michael )
• B05 - Algebraic varieties, cohomology, Abelian varieties (Project Heads Lau, Eike ;  Zink, Thomas )
• B06 - Invariant harmonic analysis and Selberg zeta functions (Project Head Hoffmann, Werner )
• B07 - Analysis von Diskretisierungsverfahren für nichtlineare Evolutionsgleichungen (Project Head Emmrich, Etienne )
• B08 - Initial value problems for nonlinear dispersive equations at critical regularity (Project Head Herr, Sebastian )
• C01 - Gauge theoretical methods in manifold theory (Project Head Bauer, Stefan )
• C02 - Linear Algebraic Groups over Arbitrary Fields (Project Heads Rehmann, Ulf ;  Rost, Markus )
• C03 - Topological and spectral structures in representation theory (Project Heads Krause, Henning ;  Ringel, Claus Michael )
• C04 - Milnor Conjecture, Galois Cohomology and Algebraic Cobordism (Project Head Rost, Markus )
• C05 - p-adic Symmetric Spaces, p-adic Uniformisation and L-functions (Project Heads Spieß, Michael ;  Zink, Thomas )
• C06 - Groups, Buildings, and Model Theory (Project Head Tent, Katrin )
• C07 - Automorphic representations and their local factors (Project Heads Hoffmann, Werner ;  Nickel, Andreas ;  Paskunas, Vytautas ;  Spieß, Michael )
• C08 - Finiteness Properties of Infinite Discrete Groups (Project Head Bux, Kai-Uwe )
• C10 - Local cohomology in representation theory (Project Heads Krause, Henning ;  Voll, Christopher )
• C11 - Algebraic and analytic aspects of holomorphic Lagrangian fibrations (Project Heads Haydys, Andriy ;  Rollenske, Sönke )
• C12 - Representation growth of arithmetic groups (Project Head Voll, Christopher )
• C13 - The geometry and combinatorics of groups (Project Heads Baumeister, Barbara ;  Bux, Kai-Uwe )
• Z - Central tasks (Project Head Götze, Friedrich )

• Applicant Institution: Universität Bielefeld
• Spokesperson: Professor Dr. Friedrich Götze