Extending our earlier work on Lax-type shocks of systems of conservation laws (see [GM+1], [GM+2], [GM+4]), we establish existence and stability of curved multidimensional shock fronts in the vanishing viscosity limit for general Lax- or undercompressive-type shock waves of nonconservative hyperbolic systems with parabolic regularization. The hyperbolic equations may be of variable multiplicity, and the parabolic regularization may be of “real,” or partially parabolic, type. We prove an existence result for inviscid nonconservative shocks which extends a one-dimensional result of X to multidimensional shocks. Lin [L] proved by quite different methods. In addition, we construct families of smooth viscous shocks converging to a given inviscid shock as viscosity goes to zero, thereby justifying the small viscosity limit for multidimensional nonconservative shocks.

In our previous work on shocks, we made use of conservative form, especially in parts of the low-frequency analysis. Thus, most of the new analysis of this article is concentrated in this area. By adopting the more general nonconservative viewpoint, we are able to shed new light on both the viscous and inviscid theories. For example, we can now provide a clearer geometric motivation for the low-frequency analysis in the viscous case. Also, we show that one may, in the treatment of inviscid stability of nonclassical and/or nonconservative shocks, remove an apparently restrictive technical assumption made by Mokrane [Mo] and Coulombel [C] in their work on, respectively, shock-type nonconservative boundary problems and conservative undercompressive shocks. Another advantage of the nonconservative perspective is that Lax and undercompressive shocks can be treated by exactly the same analysis.

The trace set of a Fuchsian group $\Gamma$ encodes the set of lengths of closed geodesics in the surface $\Gamma \backslash \mathbb{H}$. Luo and Sarnak [3] showed that the trace set of a cofinite arithmetic Fuchsian group satisfies the bounded clustering (BC) property. Sarnak [5] then conjectured that the BC property actually characterizes arithmetic Fuchsian groups. Schmutz [6] stated the even stronger conjecture that a cofinite Fuchsian group is arithmetic if its trace set has linear growth. He proposed a proof of this conjecture in the case when the group $\Gamma$ contains at least one parabolic element, but unfortunately, this proof contains a gap. In this article, we point out this gap, and we prove Sarnak's conjecture under the assumption that the Fuchsian group $\Gamma$ contains parabolic elements.

We consider Voronoi's reduction theory of positive definite quadratic forms, which is based on Delone subdivision. We extend it to forms and Delone subdivisions having a prescribed symmetry group. Even more generally, the theory is developed for forms that are restricted to a linear subspace in the space of quadratic forms. We apply the new theory to complete the classification of totally real, thin algebraic number fields which was recently initiated by Bayer-Fluckiger [BF] and Bayer-Fluckiger and Nebe [BFN]. Moreover, we apply it to construct new best-known sphere coverings in dimensions $9,\dots ,15$.

Perhaps the most important problem in representation theory in the 1970s and early 1980s was the determination of the multiplicity of composition factors in a Verma module. This problem was settled by the proof of the Kazhdan-Lusztig conjecture, which states that the multiplicities may be computed via Kazhdan-Lusztig polynomials. In this article, we introduce signed Kazhdan-Lusztig polynomials, a variation of Kazhdan-Lusztig polynomials which encodes signature information in addition to composition factor multiplicities and Jantzen filtration levels. Careful consideration of Gabber and Joseph's proof of Kazhdan and Lusztig's recursive formula for computing Kazhdan-Lusztig polynomials and an application of Jantzen's determinant formula lead to a recursive formula for the signed Kazhdan-Lusztig polynomials. We use these polynomials to compute the signature of an invariant Hermitian form on an irreducible highest-weight module. Such a formula has applications to unitarity testing.