Spectral structures are omnipresent in mathematics and also in daily life:
An entity allows for a variety of different appearances, may it be light, color, opinions, eigenvalues of differential operators or prime ideals in a ring. The entity is only understood by an overall view on all of its appearances. The entity usually comes with an inner structure: Light may be infrared, visible, ultraviolet. Color may be light, dark, pastel, bright. Opinions, in contrast, are always divided. Topology provides a wide range of mathematical tools designed to describe and analyze the inner structure of an object, how the different appearances fit together to form an entity.
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 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.
Theoretical and applied mathematicians from different directions work at SFB in close intercommunion to harness the considerable potential of straddling research.