Let C be a complex affine reduced curve, and denote by H1(C) its first truncated cohomology group, i.e. the quotient of all regular differential 1-forms by exact 1-forms. First we introduce a nonnegative invariant μ(C, x) that measures the complexity of the singularity of C at the point x, and we establish the following formula: dim H(C) = dim H1(C) + ∑

We introduce the notion of a " non-commutative crepant " resolution of a singularity and show that it exists in certain cases. We also give some evidence for an extension of a conjecture by Bondal and Orlov, stating that different crepant resolutions of a Gorenstein singularity have the same derived category.

The stress singularity near the Mode III interface end of isotropic and orthotropic bi-material was studied. Based on arbitrary angle boundary conditions, by solving a class of harmonic equations we discussed the root of the characteristic equation with the help of complex function method of material fracture. The expression and variation of the singularity index λ of the tip of interfacial cra...