Hyperplane arrangement cohomology and monomials in the exterior algebra

We show that if is the complement of a complex hyperplane arrangement, then the homology of has linear free resolution as a module over the exterior algebra on the first cohomology of . We study invariants of that can be deduced from this resolution. A key ingredient is a result of Aramova, Avramov, and Herzog (2000) on resolutions of monomial ideals in the exterior algebra. We give a new conceptual proof of this result.

     

Cohomology of the Orlik–Solomon Algebras and Local Systems

The paper provides a combinatorial method to decide when the space of local systems with nonvanishing first cohomology on the complement to an arrangement of lines in a complex projective plane has as an irreducible component a subgroup of positive dimension. Partial classification of arrangements having such a component of positive dimension and a comparison theorem for cohomology of Orlik–Solomon algebra and cohomology of local systems are given. The methods are based on Vinberg–Kac classification of generalized Cartan matrices and study of pencils of algebraic curves defined by mentioned positive dimensional components.     

Combinatorial covers and vanishing of cohomology

We use a Mayer–Vietoris-like spectral sequence to establish vanishing results for the cohomology of complements of linear and elliptic hyperplane arrangements, as part of a more general framework involving duality and abelian duality properties of spaces and groups. In the process, we consider cohomology of local systems with a general, Cohen–Macaulay-type condition. As a result, we recover known vanishing theorems for rank-1 local systems as well as group ring coefficients and obtain new generalizations.     

Probing nanoscale solids at thermal extremes

We report a novel nanoscale thermal platform compatible with extreme temperature operation and real-time high-resolution transmission electron microscopy. Applied to multiwall carbon nanotubes, we find atomic-scale stability to 3200 K, demonstrating that carbon nanotubes are more robust than graphite or diamond. Even at these thermal extremes, nanotubes maintain 10% of their peak thermal conductivity and support electrical current densities approximately 2 x 10{8} A/cm{2}. We also apply this platform to determine the diameter dependence of the melting temperature of gold nanocrystals down to three nanometers.     

Modules of splines on polyhedral complexes

The results of this paper were partially presented at the Oberwolfach Conference organized by W. Dahmen and A. Dress in October of 1990