480 research outputs found
A quantum Mirkovi\'c-Vybornov isomorphism
We present a quantization of an isomorphism of Mirkovi\'c and Vybornov which
relates the intersection of a Slodowy slice and a nilpotent orbit closure in
, to a slice between spherical Schubert varieties in the
affine Grassmannian of (with weights encoded by the Jordan types of the
nilpotent orbits). A quantization of the former variety is provided by a
parabolic W-algebra and of the latter by a truncated shifted Yangian. Building
on earlier work of Brundan and Kleshchev, we define an explicit isomorphism
between these non-commutative algebras, and show that its classical limit is a
variation of the original isomorphism of Mirkovi\'c and Vybornov. As a
corollary, we deduce that the W-algebra is free as a left (or right) module
over its Gelfand-Tsetlin subalgebra, as conjectured by Futorny, Molev, and
Ovsienko.Comment: v2: 48 pages. Major rewrite following referee comments. Added proof
of a conjecture of Futorny, Molev, and Ovsienko that the finite W-algebra is
free over its Gelfand-Tsetlin subalgebr
Highest weights for truncated shifted Yangians and product monomial crystals
Truncated shifted Yangians are a family of algebras which are natural
quantizations of slices in the affine Grassmannian. We study the highest weight
representations of these algebras. In particular, we conjecture that the
possible highest weights for these algebras are described by product monomial
crystals, certain natural subcrystals of Nakajima's monomials. We prove this
conjecture in type A. We also place our results in the context of symplectic
duality and prove a conjecture of Hikita in this situation.Comment: 57 page
Yangians and quantizations of slices in the affine Grassmannian
We study quantizations of transverse slices to Schubert varieties in the
affine Grassmannian. The quantization is constructed using quantum groups
called shifted Yangians --- these are subalgebras of the Yangian we introduce
which generalize the Brundan-Kleshchev shifted Yangian to arbitrary type.
Building on ideas of Gerasimov-Kharchev-Lebedev-Oblezin, we prove that a
quotient of the shifted Yangian quantizes a scheme supported on the transverse
slices, and we formulate a conjectural description of the defining ideal of
these slices which implies that the scheme is reduced. This conjecture also
implies the conjectural quantization of the Zastava spaces for PGL(n) of
Finkelberg-Rybnykov.Comment: 37 pages; v2, slightly strengthened Theorem 2.
Lie algebra actions on module categories for truncated shifted Yangians
We develop a theory of parabolic induction and restriction functors relating
modules over Coulomb branch algebras, in the sense of
Braverman-Finkelberg-Nakajima. Our functors generalize Bezrukavnikov-Etingof's
induction and restriction functors for Cherednik algebras, but their definition
uses different tools.
After this general definition, we focus on quiver gauge theories attached to
a quiver . The induction and restriction functors allow us to define a
categorical action of the corresponding symmetric Kac-Moody algebra
on category for these Coulomb branch
algebras. When is of Dynkin type, the Coulomb branch algebras are
truncated shifted Yangians and quantize generalized affine Grassmannian slices.
Thus, we regard our action as a categorification of the geometric Satake
correspondence.
To establish this categorical action, we define a new class of "flavoured"
KLRW algebras, which are similar to the diagrammatic algebras originally
constructed by the second author for the purpose of tensor product
categorification. We prove an equivalence between the category of
Gelfand-Tsetlin modules over a Coulomb branch algebra and the modules over a
flavoured KLRW algebra. This equivalence relates the categorical action by
induction and restriction functors to the usual categorical action on modules
over a KLRW algebra.Comment: 66 pages, version 2: many corrections, improved treatment of GK
dimension, 71 page
Nanofunctionalized zirconia and barium sulfate particles as bone cement additives
Zirconia (ZrO2) and barium sulfate (BaSO4) particles were introduced into a methyl methacrylate monomer (MMA) solution with polymethyl methacrylate (PMMA) beads during polymerization to develop the following novel bone cements: bone cements with unfunctionalized ZrO2 micron particles, bone cements with unfunctionalized ZrO2 nanoparticles, bone cements with ZrO2 nanoparticles functionalized with 3-(trimethoxysilyl)propyl methacrylate (TMS), bone cements with unfunctionalized BaSO4 micron particles, bone cements with unfunctionalized BaSO4 nanoparticles, and bone cements with BaSO4 nanoparticles functionalized with TMS. Results demonstrated that in vitro osteoblast (bone-forming cell) densities were greater on bone cements containing BaSO4 ceramic particles after four hours compared to control unmodified bone cements. Osteoblast densities were also greater on bone cements containing all of the ceramic particles after 24 hours compared to unmodified bone cements, particularly those bone cements containing nanofunctionalized ceramic particles. Bone cements containing ceramic particles demonstrated significantly altered mechanical properties; specifically, under tensile loading, plain bone cements and bone cements containing unfunctionalized ceramic particles exhibited brittle failure modes whereas bone cements containing nanofunctionalized ceramic particles exhibited plastic failure modes. Finally, all bone cements containing ceramic particles possessed greater radio-opacity than unmodified bone cements. In summary, the results of this study demonstrated a positive impact on the properties of traditional bone cements for orthopedic applications with the addition of unfunctionalized and TMS functionalized ceramic nanoparticles
#4 - Endosomal Proteins in Neurodevelopmental Disorders
Rett Syndrome is a neurodevelopmental disorder that primarily affects females and is detected at 6-18 months of age. Rett Syndrome results from a mutations in the methyl-CpG binding protein 2 gene (MECP2) which is found on the X chromosome. Its mutation results in impairment in cognitive, sensory, emotional, motor and autonomic functions. Schizophrenia is also a neurodevelopmental disorder with onset in adulthood, however there is no single genetic cause. Endosomal proteins have been implicated in both disorders through GWAS studies and animal models, suggesting a common molecular mechanism shared between these neurodevelopmental disorders. Previous research in our lab has demonstrated a disruption in endosomal trafficking in animal models of both disorders. This study explores the levels and localization of endosomal proteins in coronal brain sections for mice models for these neurodevelopmental disorders. Using immunohistochemistry, we will examine protein levels in the hippocampus and cortex for endosomal trafficking markers. We will also use whole-brain derived synaptosomes and western blots to examine the subcellular levels of these endosomal markers. Our data demonstrate a reduction of endosomal proteins in the hippocampus of mouse models of neurodevelopmental disorders. Future studies will include investigations into the affected cargo being mis-trafficked in these disorders
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