SFB/TR12 Symmetries and Universality in Mesoscopic Systems

Mesoscopic systems, situated at the boundary between the quantum and classical worlds, exhibit spectral and transport properties that obey universal laws controlled by symmetry. On mesoscopic scales, classical chaos leads to structural stability, and the dynamics of the relevant observables is influenced by disorder; quantum mechanical coherence has not yet been obliterated by thermal and dissipative effects; and interactions between the constitutive degrees of freedom need not be hidden in mean fields.

Guided by the concept of symmetry classes and its recently discovered extension, it is the purpose of the SFB/TR to explore the domain of mesoscopic phenomena for fermionic and bosonic systems, and to firmly establish the mechanisms underlying universality. The mathematical objects basic to this endeavor are symmetric spaces of compact and non-compact type and their generalizations in the category of supermanifolds, whose geometry, analysis and field theory is to be developed. To make the connection with the quasiclassical physics of short wave lengths, our research proposal complements the spectral analysis of deterministic and stochastic operators by investigations of (i) the geometry of the associated dynamical systems, (ii) the semiclassical asymptotics emerging from trace formulas, and (iii) limit phenomena in the representation theory of Lie groups. Random matrix theory will play an important role in the stochastic modeling of universal behavior.