The geometry and combinatorics of Springer fibers

This survey paper describes Springer fibers, which are used in one of the earliest examples of a geometric representation. We will compare and contrast them with Schubert varieties, another family of subvarieties of the flag variety that play an important role in representation theory and combinatorics, but whose geometry is in many respects simpler. The end of the paper describes a way that Springer fibers and Schubert varieties are related, as well as open questions.Comment: 18 page

Dynamic Euler Diagram Drawing

In this paper we describe a method to lay out a graph enhanced Euler diagram so that it looks similar to a previously drawn graph enhanced Euler diagram. This task is non-trivial when the underlying structures of the diagrams differ. In particular, if a structural change is made to an existing drawn diagram, our work enables the presentation of the new diagram with minor disruption to the user's mental map. As the new diagram can be generated from an abstract representation, its initial embedding may be very different from that of the original. We have developed comparison measures for Euler diagrams, integrated into a multicriteria optimizer, and applied a force model for associated graphs that attempts to move nodes towards their positions in the original layout. To further enhance the usability of the system, the transition between diagrams can be animated

eXamine: a Cytoscape app for exploring annotated modules in networks

Background. Biological networks have growing importance for the interpretation of high-throughput "omics" data. Statistical and combinatorial methods allow to obtain mechanistic insights through the extraction of smaller subnetwork modules. Further enrichment analyses provide set-based annotations of these modules. Results. We present eXamine, a set-oriented visual analysis approach for annotated modules that displays set membership as contours on top of a node-link layout. Our approach extends upon Self Organizing Maps to simultaneously lay out nodes, links, and set contours. Conclusions. We implemented eXamine as a freely available Cytoscape app. Using eXamine we study a module that is activated by the virally-encoded G-protein coupled receptor US28 and formulate a novel hypothesis about its functioning

eulerForce: Force-directed Layout for Euler Diagrams

Euler diagrams use closed curves to represent sets and their relationships. They facilitate set analysis, as humans tend to perceive distinct regions when closed curves are drawn on a plane. However, current automatic methods often produce diagrams with irregular, non-smooth curves that are not easily distinguishable. Other methods restrict the shape of the curve to for instance a circle, but such methods cannot draw an Euler diagram with exactly the required curve intersections for any set relations. In this paper, we present eulerForce, as the first method to adopt a force-directed approach to improve the layout and the curves of Euler diagrams generated by current methods. The layouts are improved in quick time. Our evaluation of eulerForce indicates the benefits of a force-directed approach to generate comprehensible Euler diagrams for any set relations in relatively fast time

Spin-density-wave instabilities in the organic conductor (TMTSF)_2ClO_4: Role of anion ordering

We study the spin-density-wave instabilities in the quasi-one-dimensional conductor (TMTSF)_2ClO_4. The orientational order of the anions ClO_4 doubles the unit cell and leads to the presence of two electrnic bands at the Fermi level. From the Ginzburg-Landau expansion of the free energy, we determine the low-temperature phase diagram as a function of the strength of the Coulomb potential due to the anions. Upon increasing the anion potential, we first find a SDW phase corresponding to an interband pairing. This SDW phase is rapidly supressed, the metallic phase being then stable down to zero temperature. The SDW instability is restored when the anion potential becomes of the order of the hopping amplitude. The metal-SDW transition corresponds to an intraband pairing which leaves half of the Fermi surface metallic. At lower temperature, a second transition, corresponding to the other intraband pairing, takes place and opens a gap on the whole Fermi surface. We discuss the consequences of our results for the experimental phase diagram of (TMTSF)_2ClO_4 at high magnetic field.Comment: 13 pages, 10 figures, Version 2 with minor correction

Nestings of rational homogeneous varieties

In this paper we study the existence of sections of universal bundles on rational homogeneous varieties -- called nestings -- classifying them completely in the case in which the Lie algebra of the automorphism group of the variety is simple of classical type. In particular we show that, under this hypothesis, nestings do not exist unless there exists a proper algebraic subgroup of the automorphism group acting transitively on the base variety.Comment: Major revision of the exposition. To appear in Tranformation Group

Shapes of topological RNA structures

A topological RNA structure is derived from a diagram and its shape is obtained by collapsing the stacks of the structure into single arcs and by removing any arcs of length one. Shapes contain key topological, information and for fixed topological genus there exist only finitely many such shapes. We shall express topological RNA structures as unicellular maps, i.e. graphs together with a cyclic ordering of their half-edges. In this paper we prove a bijection of shapes of topological RNA structures. We furthermore derive a linear time algorithm generating shapes of fixed topological genus. We derive explicit expressions for the coefficients of the generating polynomial of these shapes and the generating function of RNA structures of genus $g$. Furthermore we outline how shapes can be used in order to extract essential information of RNA structure databases.Comment: 27 pages, 11 figures, 2 tables. arXiv admin note: text overlap with arXiv:1304.739