A functorial construction of moduli of sheaves

We show how natural functors from the category of coherent sheaves on a projective scheme to categories of Kronecker modules can be used to construct moduli spaces of semistable sheaves. This construction simplifies or clarifies technical aspects of existing constructions and yields new simpler definitions of theta functions, about which more complete results can be proved.Comment: 52 pp. Dedicated to the memory of Joseph Le Potier. To appear in Inventiones Mathematicae. Slight change in the definition of the Kronecker algebra in Secs 1 (p3) and 2.2 (p6), with corresponding small alterations elsewhere, to make the constructions work for non-reduced schemes. Section 6.5 rewritten. Remark 2.6 and new references adde

Quivers, Tilings, Branes and Rhombi

We describe a simple algorithm that computes the recently discovered brane tilings for a given generic toric singular Calabi-Yau threefold. This therefore gives AdS/CFT dual quiver gauge theories for D3-branes probing the given non-compact manifold. The algorithm solves a longstanding problem by computing superpotentials for these theories directly from the toric diagram of the singularity. We study the parameter space of a-maximization; this study is made possible by identifying the R-charges of bifundamental fields as angles in the brane tiling. We also study Seiberg duality from a new perspective.Comment: 36 pages, 40 figures, JHEP

Mixed Hodge polynomials of character varieties

We calculate the E-polynomials of certain twisted GL(n,C)-character varieties M_n of Riemann surfaces by counting points over finite fields using the character table of the finite group of Lie-type GL(n,F_q) and a theorem proved in the appendix by N. Katz. We deduce from this calculation several geometric results, for example, the value of the topological Euler characteristic of the associated PGL(n,C)-character variety. The calculation also leads to several conjectures about the cohomology of M_n: an explicit conjecture for its mixed Hodge polynomial; a conjectured curious Hard Lefschetz theorem and a conjecture relating the pure part to absolutely indecomposable representations of a certain quiver. We prove these conjectures for n = 2.Comment: with an appendix by Nicholas M. Katz; 57 pages. revised version: New definition for homogeneous weight in Definition 4.1.6, subsequent arguments modified. Some other minor changes. To appear in Invent. Mat

Anosov representations: Domains of discontinuity and applications

The notion of Anosov representations has been introduced by Labourie in his study of the Hitchin component for SL(n,R). Subsequently, Anosov representations have been studied mainly for surface groups, in particular in the context of higher Teichmueller spaces, and for lattices in SO(1,n). In this article we extend the notion of Anosov representations to representations of arbitrary word hyperbolic groups and start the systematic study of their geometric properties. In particular, given an Anosov representation of $\Gamma$ into G we explicitly construct open subsets of compact G-spaces, on which $\Gamma$ acts properly discontinuously and with compact quotient. As a consequence we show that higher Teichmueller spaces parametrize locally homogeneous geometric structures on compact manifolds. We also obtain applications regarding (non-standard) compact Clifford-Klein forms and compactifications of locally symmetric spaces of infinite volume.Comment: 63 pages, accepted for publication in Inventiones Mathematica

The design, construction, operation, and maintenance of the Replicated Modular Estuarine Mesocosm

Perhaps the most difficult job of the ecotoxicologist is extrapolating data calculated from laboratory experiments with high precision and accuracy into the real world of highly-dynamics aquatic environments. The establishment of baseline laboratory toxicity testing data for individual compounds and ecologically important and field studies serve as a precursor to ecosystem level studies needed for ecological risk assessment. The first stage in the field portion of risk assessment is the determination of actual environmental concentrations of the contaminant being studied and matching those concentrations with laboratory toxicity tests. Risk estimates can be produced via risk quotients that would determine the probability that adverse effects may occur. In this first stage of risk assessment, environmental realism is often not achieved. This is due, in part, to the fact that single-species laboratory toxicity tests, while highly controlled, do not account for the complex interactions (Chemical, physical, and biological) that take place in the natural environment. By controlling as many variables in the laboratory as possible, an experiment can be produced in such a fashion that real effects from a compound can be determined for a particular test organism. This type of approach obviously makes comparison with real world data most difficult. Conversely, field oriented studies fall short in the interpretation of ecological risk assessment because of low statistical power, lack of adequate replicaiton, and the enormous amount of time and money needed to perform such studies. Unlike a controlled laboratory bioassay, many other stressors other than the chemical compound in question affect organisms in the environment. These stressors range from natural occurrences (such as changes in temperature, salinity, and community interactions) to other confounding anthropogenic inputs. Therefore, an improved aquatic toxicity test that will enhance environmental realism and increase the accuracy of future ecotoxicological risk assessments is needed