Quantizations of conical symplectic resolutions I: local and global structure

We re-examine some topics in representation theory of Lie algebras and Springer theory in a more general context, viewing the universal enveloping algebra as an example of the section ring of a quantization of a conical symplectic resolution. While some modification from this classical context is necessary, many familiar features survive. These include a version of the Beilinson-Bernstein localization theorem, a theory of Harish-Chandra bimodules and their relationship to convolution operators on cohomology, and a discrete group action on the derived category of representations, generalizing the braid group action on category O via twisting functors. Our primary goal is to apply these results to other quantized symplectic resolutions, including quiver varieties and hypertoric varieties. This provides a new context for known results about Lie algebras, Cherednik algebras, finite W-algebras, and hypertoric enveloping algebras, while also pointing to the study of new algebras arising from more general resolutions.Comment: 74 pages; v4: minor changes based on referee comments; v5: minor adjustment in numbering to match published versio

Hypertoric category O

We study the representation theory of the invariant subalgebra of the Weyl algebra under a torus action, which we call a "hypertoric enveloping algebra." We define an analogue of BGG category O for this algebra, and identify it with a certain category of sheaves on a hypertoric variety. We prove that a regular block of this category is highest weight and Koszul, identify its Koszul dual, compute its center, and study its cell structure. We also consider a collection of derived auto-equivalences analogous to the shuffling and twisting functors for BGG category O.Comment: 65 pages, TikZ figures (PDF is recommended; DVI will not display correctly on all computers); v3: switched terminology for twisting and shuffling; final version; v4: small correction in definition of standard module

Gale duality and Koszul duality

Given an affine hyperplane arrangement with some additional structure, we define two finite-dimensional, noncommutative algebras, both of which are motivated by the geometry of hypertoric varieties. We show that these algebras are Koszul dual to each other, and that the roles of the two algebras are reversed by Gale duality. We also study the centers and representation categories of our algebras, which are in many ways analogous to integral blocks of category O.Comment: 55 pages; v2 contains significant revisions to proofs and to some of the results. Section 7 has been deleted; that material will be incorporated into a later paper by the same author

Smooth matter and source size in microlensing simulations of gravitationally lensed quasars

Several gravitationally lensed quasars are observed with anomalous magnifications in pairs of images that straddle a critical curve. Simple theoretical arguments suggest that the magnification of these images should be approximately equivalent, whereas one image is observed to be significantly demagnified. Microlensing provides a possible explanation for this discrepancy. There are two key parameters when modelling this effect. The first, the fraction of smooth matter in the lens at the image positions, has been explored by Schechter and Wambsganss (2002). They have shown that the anomalous flux ratio observed in the lensed quasar MG 0414+0534 is a priori a factor of 5 more likely if the assumed smooth matter content in the lens model is increased from 0% to 93%. The second parameter, the size of the emission region, is explored in this paper, and shown to be more significant. We find that the broadening of the magnification probability distributions due to smooth matter content is washed out for source sizes that are predicted by standard models for quasars. We apply our model to the anomalous lensed quasar MG 0414+0534, and find a 95% upper limit of 2.62 x 10^(16) h^(-1/2) (M/Msun)^(1/2) cm on the radius of the I-band emission region. The smooth matter percentage in the lens is unconstrained.Comment: 6 pages, 6 figures. To be published in MNRA

Localization algebras and deformations of Koszul algebras

We show that the center of a flat graded deformation of a standard Koszul algebra behaves in many ways like the torus-equivariant cohomology ring of an algebraic variety with finite fixed-point set. In particular, the center acts by characters on the deformed standard modules, providing a "localization map." We construct a universal graded deformation, and show that the spectrum of its center is supported on a certain arrangement of hyperplanes which is orthogonal to the arrangement coming the Koszul dual algebra. This is an algebraic version of a duality discovered by Goresky and MacPherson between the equivariant cohomology rings of partial flag varieties and Springer fibers; we recover and generalize their result by showing that the center of the universal deformation for the ring governing a block of parabolic category $\mathcal{O}$ for $\mathfrak{gl}_n$ is isomorphic to the equivariant cohomology of a Spaltenstein variety. We also identify the center of the deformed version of the "category $\mathcal{O}$" of a hyperplane arrangement (defined by the authors in a previous paper) with the equivariant cohomology of a hypertoric variety.Comment: 39 pages; v3: final versio

Computer vision ring

Visually-impaired people have difficulty reading handwritten or printed documents that are not written in Braille. Totally blind people cannot read handwritten or printed documents that are not in Braille. A large fraction of written language encountered in daily life or travel still does not appear in Braille, thereby denying visually impaired and completely blind people access to such information. The present disclosure describes techniques to scan one’s surroundings using a low-profile camera, to identify objects and/or written language using an object or character recognition system, and to convey scanned information aurally to a user. Machine learning and inference models are used to automatically perform the tasks of object identification, character recognition, etc. Such a system can be used by visually impaired and totally blind people, and also by individuals without visual impairment — such as travelers who want to translate written text found in their surroundings, e.g., signboards, menu items, etc

Microlensing Constraints on Broad Absorption and Emission Line Flows in the Quasar H1413+117

We present new integral field spectroscopy of the gravitationally lensed broad absorption line (BAL) quasar H1413+117, covering the ultraviolet to visible rest-frame spectral range. We observe strong microlensing signatures in lensed image D, and we use this microlensing to simultaneously constrain both the broad emission and broad absorption line gas. By modeling the lens system over the range of probable lensing galaxy redshifts and using on a new argument based on the wavelength-independence of the broad line lensing magnifications, we determine that there is no significant broad line emission from smaller than ~20 light days. We also perform spectral decomposition to derive the intrinsic broad emission line (BEL) and continuum spectrum, subject to BAL absorption. We also reconstruct the intrinsic BAL absorption profile, whose features allow us to constrain outflow kinematics in the context of a disk-wind model. We find a very sharp, blueshifted onset of absorption of 1,500 km/s in both C IV and N V that may correspond to an inner edge of a disk-wind's radial outflow. The lower ionization Si IV and Al III have higher-velocity absorption onsets, consistent with a decreasing ionization parameter with radius in an accelerating outflow. There is evidence of strong absorption in the BEL component which indicates a high covering factor for absorption over two orders of magnitude in outflow radius.Comment: 29 pages, 8 figure

A Novel Gonadotropin-Releasing Hormone 1 (Gnrh1) Enhancer-Derived Noncoding RNA Regulates Gnrh1 Gene Expression in GnRH Neuronal Cell Models.

Gonadotropin-releasing hormone (GnRH), a neuropeptide released from a small population of neurons in the hypothalamus, is the central mediator of the hypothalamic-pituitary-gonadal axis, and is required for normal reproductive development and function. Evolutionarily conserved regulatory elements in the mouse, rat, and human Gnrh1 gene include three enhancers and the proximal promoter, which confer Gnrh1 gene expression specifically in GnRH neurons. In immortalized mouse hypothalamic GnRH (GT1-7) neurons, which show pulsatile GnRH release in culture, RNA sequencing and RT-qPCR revealed that expression of a novel long noncoding RNA at Gnrh1 enhancer 1 correlates with high levels of GnRH mRNA expression. In GT1-7 neurons, which contain a transgene carrying 3 kb of the rat Gnrh1 regulatory region, both the mouse and rat Gnrh1 enhancer-derived noncoding RNAs (GnRH-E1 RNAs) are expressed. We investigated the characteristics and function of the endogenous mouse GnRH-E1 RNA. Strand-specific RT-PCR analysis of GnRH-E1 RNA in GT1-7 cells revealed GnRH-E1 RNAs that are transcribed in the sense and antisense directions from distinct 5' start sites, are 3' polyadenylated, and are over 2 kb in length. These RNAs are localized in the nucleus and have a half-life of over 8 hours. In GT1-7 neurons, siRNA knockdown of mouse GnRH-E1 RNA resulted in a significant decrease in the expression of the Gnrh1 primary transcript and Gnrh1 mRNA. Over-expression of either the sense or antisense mouse GnRH-E1 RNA in immature, migratory GnRH (GN11) neurons, which do not express either GnRH-E1 RNA or GnRH mRNA, induced the transcriptional activity of co-transfected rat Gnrh1 gene regulatory elements, where the induction requires the presence of the rat Gnrh1 promoter. Together, these data indicate that GnRH-E1 RNA is an inducer of Gnrh1 gene expression. GnRH-E1 RNA may play an important role in the development and maturation of GnRH neurons