This book contains some written versions of survey lectures presented at the conference on the legacy of Hermann Weyl organised in Bielefeld in 2006. Their topics cover a broad area of mathematics that was influenced by the work of H. Weyl. The enormous impact of his work in representation theory and harmonic analysis is illustrated in papers by R. Goodman (algebraic group versions of results in harmonic analysis on compact symmetric spaces and the horospherical Radon transform), E. van den Ban (harmonic analysis on groups and non-compact symmetric spaces), R. Howe, E.-C. Tan and J. F. Willenbring (development of classical invariant theory and its applications to associated pairs of classical symmetric pairs, reciprocity algebras and branching rules) and J. C. Jantzen (generalisations of the Weyl character formula to the setting of semi-simple algebraic groups over algebraically closed fields with prime characteristics, Kac-Moody algebras and quantum groups).

Topics from analysis are discussed by W. N. Everitt and H. Kalf (generalisations of Sturm-Liouville theory and their relations to quantum physics), M. J. Pflaum (the Weyl quantisation, deformations quantisation and the algebraic index theorem), A. M. Hansson and A. Laptev (sharp spectral inequalities for the Heisenberg Laplacian with Dirichlet boundary conditions) and D. W. Stroock (generalisations of the Weyl potential theoretical approach to hypoellipticity proofs). The influence of Weyl’s ideas on various themes in number theory can be found in papers by U. Hamendstädt (equidistribution for quadratic differentials, interplay among number theory, geometry and dynamical systems), W. Müller (generalisations of the Weyl law on asymptotic distribution of eigenvalues of the Laplace operator and relations to authomorphic forms on locally symmetric spaces), Ch. Deninger (relations among dynamical systems on foliated manifolds, transversal index theory and arithmetic geometry) and R. M. Weiss (the structure of affine buildings). The book ends with an historical sketch by P. Roquette on relations between E. Noether and H. Weyl in the period 1915-1935. The book offers an interesting overview of the impact of H. Weyl’s work on contemporary mathematics.