Selecting a connected subnetwork enriched in individually important vertices is an approach commonly used in many areas of bioinformatics, including analysis of gene expression data, mutations, metabolomic profiles and others. It can be formulated as a recovery of an active module from which an experimental signal is generated. Commonly, methods for solving this problem result in a single subne...

A vertex ranking of a graph is an assignment of ranks (or colors) to the vertices of the graph, in such a way that any simple path connecting two vertices of equal rank, must contain a vertex of a higher rank. In this paper we study a relaxation of this notion, in which the requirement above should only hold for paths of some bounded length l for some fixed l. For instance, already the case l =...

Structured low-rank approximation is the problem of minimizing a weighted Frobenius distance to a given matrix among all matrices of fixed rank in a linear space of matrices. We study the critical points of this optimization problem using algebraic geometry. A particular focus lies on Hankel matrices, Sylvester matrices and generic linear spaces.

We study the generic rank of 3-tensors using results from matrices and algebraic geometry. We state a conjecture about the exact values of the generic rank of 3-tensors over the complex numbers. We also discuss generic ranks over the real numbers. 2000 Mathematics Subject Classification. 14A25, 14P10, 15A69.

A matroid-like structure defined on a convex geometry, called a cg-matroid, is defined by S. Fujishige, G. A. Koshevoy, and Y. Sano in [9]. A cg-matroid whose rank function is naturally defined is called a strict cg-matroid. In this paper, we give characterizations of strict cg-matroids by their rank functions.

We show that, if q is a prime power at most 5, then every rank-r matroid with no U2,q+2-minor has no more lines than a rank-r projective geometry over GF(q). We also give examples showing that for every other prime power this bound does not hold.

We describe applications of Koszul cohomology to the BrillNoether theory of rank 2 vector bundles. Among other things, we show that in every genus g > 10, there exist curves invalidating Mercat’s Conjecture for rank 2 bundles. On the other hand, we prove that Mercat’s Conjecture holds for general curves of bounded genus, and its failure locus is a Koszul divisor in the moduli space of curves. W...