Manifolds of isospectral arrow matrices

An arrow matrix is a matrix with zeroes outside the main diagonal, first row, and first column. We consider the space $M_{St_n,\lambda}$ of Hermitian arrow $(n+1)\times (n+1)$-matrices with fixed simple spectrum $\lambda$. We prove that this space is a smooth $2n$-manifold, and its smooth structure is independent on the spectrum. Next, this manifold carries the locally standard torus action: we describe the topology and combinatorics of its orbit space. If $n\geqslant 3$, the orbit space $M_{St_n,\lambda}/T^n$ is not a polytope, hence this manifold is not quasitoric. However, there is a natural permutation action on $M_{St_n,\lambda}$ which induces the combined action of a semidirect product $T^n\rtimes\Sigma_n$. The orbit space of this large action is a simple polytope. The structure of this polytope is described in the paper. In case $n=3$, the space $M_{St_3,\lambda}/T^3$ is a solid torus with boundary subdivided into hexagons in a regular way. This description allows to compute the cohomology ring and equivariant cohomology ring of the 6-dimensional manifold $M_{St_3,\lambda}$ using the general theory developed by the first author. This theory is also applied to a certain $6$-dimensional manifold called the twin of $M_{St_3,\lambda}$. The twin carries a half-dimensional torus action and has nontrivial tangent and normal bundles.Comment: 29 pages, 8 figure

All Abelian Quotient C.I.-Singularities Admit Projective Crepant Resolutions in All Dimensions

For Gorenstein quotient spaces $C^d/G$, a direct generalization of the classical McKay correspondence in dimensions $d\geq 4$ would primarily demand the existence of projective, crepant desingularizations. Since this turned out to be not always possible, Reid asked about special classes of such quotient spaces which would satisfy the above property. We prove that the underlying spaces of all Gorenstein abelian quotient singularities, which are embeddable as complete intersections of hypersurfaces in an affine space, have torus-equivariant projective crepant resolutions in all dimensions. We use techniques from toric and discrete geometry.Comment: revised version of MPI-preprint 97/4, 35 pages, 13 figures, latex2e-file (preprint.tex), macro packages and eps-file

Shortest Path versus Multi-Hub Routing in Networks with Uncertain Demand

We study a class of robust network design problems motivated by the need to scale core networks to meet increasingly dynamic capacity demands. Past work has focused on designing the network to support all hose matrices (all matrices not exceeding marginal bounds at the nodes). This model may be too conservative if additional information on traffic patterns is available. Another extreme is the fixed demand model, where one designs the network to support peak point-to-point demands. We introduce a capped hose model to explore a broader range of traffic matrices which includes the above two as special cases. It is known that optimal designs for the hose model are always determined by single-hub routing, and for the fixed- demand model are based on shortest-path routing. We shed light on the wider space of capped hose matrices in order to see which traffic models are more shortest path-like as opposed to hub-like. To address the space in between, we use hierarchical multi-hub routing templates, a generalization of hub and tree routing. In particular, we show that by adding peak capacities into the hose model, the single-hub tree-routing template is no longer cost-effective. This initiates the study of a class of robust network design (RND) problems restricted to these templates. Our empirical analysis is based on a heuristic for this new hierarchical RND problem. We also propose that it is possible to define a routing indicator that accounts for the strengths of the marginals and peak demands and use this information to choose the appropriate routing template. We benchmark our approach against other well-known routing templates, using representative carrier networks and a variety of different capped hose traffic demands, parameterized by the relative importance of their marginals as opposed to their point-to-point peak demands