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 $\mathfrak{gl}_N$ , to a slice between spherical Schubert varieties in the affine Grassmannian of $PGL_n$ (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 $\Gamma$. The induction and restriction functors allow us to define a categorical action of the corresponding symmetric Kac-Moody algebra $\mathfrak{g}_{\Gamma}$ on category $\mathcal O$ for these Coulomb branch algebras. When $\Gamma$ 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