Perverse Sheaves on affine Grassmannians and Langlands Duality

This is an expanded version of the text ``Perverse Sheaves on Loop Grassmannians and Langlands Duality'', AG/9703010. The main new result is a topological realization of algebraic representations of reductive groups over arbitrary rings. We outline a proof of a geometric version of the Satake isomorphism. Given a connected, complex algebraic reductive group G we show that the tensor category of representations of the Langlands dual group is naturally equivalent to a certain category of perverse sheaves on the loop Grassmannian of G. The above result has been announced by Ginsburg in and some of the arguments are borrowed from his approach. However, we use a more "natural" commutativity constraint for the convolution product, due to Drinfeld. Secondly, we give a direct proof that the global cohomology functor is exact and decompose this cohomology functor into a direct sum of weights. The geometry underlying our arguments leads to a construction of a canonical basis of Weyl modules given by algebraic cycles and an explicit construction of the group algebra of the dual group in terms of the affine Grassmannian. We completely avoid the use of the decomposition theorem which makes our techniques applicable to perverse sheaves with coefficients over an arbitrary commutative ring. We deduce the classical Satake isomorphism using the affine Grassmannian defined over a finite field. This note contains indications of proofs of some of the results. The details will appear elsewhere.Comment: Amstex, 12 page

Linear Koszul Duality II - Coherent sheaves on perfect sheaves

In this paper we continue the study (initiated in a previous article) of linear Koszul duality, a geometric version of the standard duality between modules over symmetric and exterior algebras. We construct this duality in a very general setting, and prove its compatibility with morphisms of vector bundles and base change.Comment: Final version, to appear in JLMS. The numbering differs from the published version, and is the one used in our papers [MR2] and [MR3] from the bibliograph

Intersection cohomology of Drinfeld's compactifications

Let $X$ be a smooth complete curve, $G$ be a reductive group and $P\subset G$ a parabolic. Following Drinfeld, one defines a compactification \widetilde{\on{Bun}}_P of the moduli stack of $P$-bundles on $X$. The present paper is concerned with the explicit description of the Intersection Cohomology sheaf of \widetilde{\on{Bun}}_P. The description is given in terms of the combinatorics of the Langlands dual Lie algebra $\check{\mathfrak g}$.Comment: An erratum adde

Modules over the small quantum group and semi-infinite flag manifold

We develop a theory of perverse sheaves on the semi-infinite flag manifold $G((t))/N((t))\cdot T[[t]]$, and show that the subcategory of Iwahori-monodromy perverse sheaves is equivalent to the regular block of the category of representations of the small quantum group at an even root of unity

Knot homology via derived categories of coherent sheaves II, sl(m) case

Using derived categories of equivariant coherent sheaves we construct a knot homology theory which categorifies the quantum sl(m) knot polynomial. Our knot homology naturally satisfies the categorified MOY relations and is conjecturally isomorphic to Khovanov-Rozansky homology. Our construction is motivated by the geometric Satake correspondence and is related to Manolescu's by homological mirror symmetry.Comment: 51 pages, 9 figure

Categorical geometric skew Howe duality

We categorify the R-matrix isomorphism between tensor products of minuscule representations of U_q(sl(n)) by constructing an equivalence between the derived categories of coherent sheaves on the corresponding convolution products in the affine Grassmannian. The main step in the construction is a categorification of representations of U_q(sl(2)) which are related to representations of U_q(sl(n)) by quantum skew Howe duality. The resulting equivalence is part of the program of algebro-geometric categorification of Reshitikhin-Turaev tangle invariants developed by the first two authors.Comment: 31 page

The London theory of the crossing-vortex lattice in highly anisotropic layered superconductors

A novel description of Josephson vortices (JVs) crossed by the pancake vortices (PVs) is proposed on the basis of the anisotropic London theory. The field distribution of a JV and its energy have been calculated for both dense ($a\lambda_J$) PV lattices with distance $a$ between PVs, and the nonlinear JV core size $\lambda_J$. It is shown that the ``shifted'' PV lattice (PVs displaced mainly along JVs in the crossing vortex lattice structure), formed in high out-of-plane magnetic fields transforms into the PV lattice ``trapped'' by the JV sublattice at a certain field, lower than $\Phi_0/\gamma^2s^2$, where $\Phi_0$ is the flux quantum, $\gamma$ is the anisotropy parameter and $s$ is the distance between CuO$_2$ planes. With further decreasing $B_z$, the free energy of the crossing vortex lattice structure (PV and JV sublattices coexist separately) can exceed the free energy of the tilted lattice (common PV-JV vortex structure) in the case of $\gamma s<\lambda_{ab}$ with the in-plane penetration depth $\lambda_{ab}$ if the low ($B_x<\gamma\Phi_0/\lambda_{ab}^2$) or high ($B_x\gtrsim \Phi_0/\gamma s^2$) in-plane magnetic field is applied. It means that the crossing vortex structure is realized in the intermediate field orientations, while the tilted vortex lattice can exist if the magnetic field is aligned near the $c$-axis and the $ab$-plane as well. In the intermediate in-plane fields $\gamma\Phi_0/\lambda_{ab}^2\lesssim B_x \lesssim \Phi_0/\gamma s^2$, the crossing vortex structure with the ``trapped'' PV sublattice seems to settle in until the lock-in transition occurs since this structure has the lower energy with respect to the tilted vortex structure in the magnetic field ${\vec H}$ oriented near the $ab$-plane.Comment: 15 pages, 6 figures, accepted for publication in PR

Characterization of red mud/metakaolin-based geopolymers as modified by Ca(OH)2

Geopolymers are an emerging class of materials that offer an alternative to the Portland cement as the binder of structural concrete. One of the advantages is that the primary source of their production is waste alumosilicate materials from different industries. One of the key issues in geopolymer synthesis is the low level of mechanical properties due to porosity as well as the high activity of conductivity carriers. It can often lead to limited application possibilities, so the objective is to obtain an enhanced strength as well as decreased cracking tendency through microstructure modification. The introduction of Ca(OH)2, under certain pH conditions could lead to the filling-the-pores process and improving the mechanical properties. The aim was to understand the role that calcium plays in the geopolymer synthesis, and to define which reaction prevails under the synthesis conditions: formation of geopolymer gel or calcium silicate hydrate that contains aluminum substitution (CASH). The synthesis was performed with different raw materials (with or without red mud) and different alkalinity conditions. Ca(OH)2 was the obligatory supplement to both of the mixtures. Different techniques were performed for the testing of reaction products, as well as to define the microstructural changes as the generator of improved mechanical properties and changed electrical conductivity. The characteristics of the geopolymer's macrostructure were defined by means of an SEM analysis. Compressive strength and electrical conductivity are among the investigated product's properties. X-ray diffraction (XRD) and Fourier transform infra-red spectroscopy (FTIR) were used for the identification of various crystalline phases and an amorphous phase

Azotobacter chroococcum F8/2: a multitasking bacterial strain in sugar beet biopriming

This study assesses the effects of Azotobacter biopriming on the early development of sugar beet. Azotobacter chroococcum F8/2 was screened for plant growth promoting characteristics and biopriming effects were estimated through germination parameters and the structural changes of the root tissues. A. chroococcum F8/2 was characterized as a contributor to nitrogen, iron, and potassium availability, as well as a producer of auxin and 1-aminocyclopropane-1-carboxilic acid deaminase. Applied biopriming had reduced mean germination time by 34.44% and increased vigor I by 90.99% compared to control. Volatile blend comprised 47.67% ethanol, 32.01% 2-methyl-propanol, 17.32% 3-methyl-1-butanol, and a trace of 2,3-butanedione. Root micromorphological analysis of bioprimed sugar beet revealed a considerable increase in primary, secondary xylem area, and vessels size. Obtained results determine A. chroococcum F8/2 as a successful biopriming agent, and active participant in nutrient availability and hormonal status modulation affecting root vascular tissue. © 2022 The Author(s). Published by Informa UK Limited, trading as Taylor & Francis Group

The hypertoric intersection cohomology ring

We present a functorial computation of the equivariant intersection cohomology of a hypertoric variety, and endow it with a natural ring structure. When the hyperplane arrangement associated with the hypertoric variety is unimodular, we show that this ring structure is induced by a ring structure on the equivariant intersection cohomology sheaf in the equivariant derived category. The computation is given in terms of a localization functor which takes equivariant sheaves on a sufficiently nice stratified space to sheaves on a poset.Comment: Significant revisions in Section 5, with several corrected proof