Exponential growth of colored HOMFLY-PT homology

We define reduced colored sl(N) link homologies and use deformation spectral sequences to characterize their dependence on color and rank. We then define reduced colored HOMFLY-PT homologies and prove that they arise as large N limits of sl(N) homologies. Together, these results allow proofs of many aspects of the physically conjectured structure of the family of type A link homologies. In particular, we verify a conjecture of Gorsky, Gukov and Sto\v{s}i\'c about the growth of colored HOMFLY-PT homologies

Rational links and DT invariants of quivers

We prove that the generating functions for the colored HOMFLY-PT polynomials of rational links are specializations of the generating functions of the motivic Donaldson-Thomas invariants of appropriate quivers that we naturally associate with these links. This shows that the conjectural links-quivers correspondence of Kucharski-Reineke-Sto\v{s}i\'c-Su{\l}kowski as well as the LMOV conjecture hold for rational links. Along the way, we extend the links-quivers correspondence to tangles and, thus, explore elements of a skein theory for motivic Donaldson-Thomas invariants

\mathfrak{gl}_2 foams and the Khovanov homotopy type

The Blanchet link homology theory is an oriented model of Khovanov homology, functorial over the integers with respect to link cobordisms. We formulate a stable homotopy refinement of the Blanchet theory, based on a comparison of the Blanchet and Khovanov chain complexes associated to link diagrams. The construction of the stable homotopy type relies on the signed Burnside category approach of Sarkar-Scaduto-Stoffregen

Derived traces of Soergel categories

We study two kinds of categorical traces of (monoidal) dg categories, with particular interest in categories of Soergel bimodules. First, we explicitly compute the usual Hochschild homology, or derived vertical trace, of the category of Soergel bimodules in arbitrary types. Secondly, we introduce the notion of derived horizontal trace of a monoidal dg category and compute the derived horizontal trace of Soergel bimodules in type A. As an application we obtain a derived annular Khovanov-Rozansky link invariant with an action of full twist insertion, and thus a categorification of the HOMFLY-PT skein module of the solid torus

Algèbre diagrammatique et catégorification

Nous proposons une illustration diagrammatique abordable du concept de catégorification qui s’est développé au cours des vingt dernières années

Hausdorff dimension of operator semistable L\'evy processes

Let $X=\{X(t)\}_{t\geq0}$ be an operator semistable L\'evy process in \rd with exponent $E$, where $E$ is an invertible linear operator on \rd and $X$ is semi-selfsimilar with respect to $E$. By refining arguments given in Meerschaert and Xiao \cite{MX} for the special case of an operator stable (selfsimilar) L\'evy process, for an arbitrary Borel set B\subseteq\rr_+ we determine the Hausdorff dimension of the partial range $X(B)$ in terms of the real parts of the eigenvalues of $E$ and the Hausdorff dimension of $B$.Comment: 23 page

Features of the normal choriocapillaris with OCT-angiography: Density estimation and textural properties

The main objective of our work is to perform an in depth analysis of the structural features of normal choriocapillaris imaged with OCT Angiography. Specifically, we provide an optimal radius for a circular Region of Interest (ROI) to obtain a stable estimate of the subfoveal choriocapillaris density and characterize its textural properties using Markov Random Fields. On each binarized image of the choriocapillaris OCT Angiography we performed simulated measurements of the subfoveal choriocapillaris densities with circular Regions of Interest (ROIs) of different radii and with small random displacements from the center of the Foveal Avascular Zone (FAZ). We then calculated the variability of the density measure with different ROI radii. We then characterized the textural features of choriocapillaris binary images by estimating the parameters of an Ising model. For each image we calculated the Optimal Radius (OR) as the minimum ROI radius required to obtain a standard deviation in the simulation below 0.01. The density measured with the individual OR was 0.52 ± 0.07 (mean ± STD). Similar density values (0.51 ± 0.07) were obtained using a fixed ROI radius of 450 μm. The Ising model yielded two parameter estimates (β = 0.34 ± 0.03; γ = 0.003 ± 0.012; mean ± STD), characterizing pixel clustering and white pixel density respectively. Using the estimated parameters to synthetize new random textures via simulation we obtained a good reproduction of the original choriocapillaris structural features and density. In conclusion, we developed an extensive characterization of the normal subfoveal choriocapillaris that might be used for flow analysis and applied to the investigation pathological alterations

Proof of concept, randomized, placebo-controlled study of the effect of simvastatin on the course of age-related macular degeneration

BACKGROUND: HMG Co-A reductase inhibitors are ubiquitous in our community yet their potential role in age-related macular degeneration (AMD) remains to be determined. METHODOLOGY/PRINCIPAL FINDINGS: OBJECTIVES: To evaluate the effect of simvastatin on AMD progression and the effect modification by polymorphism in apolipoprotein E (ApoE) and complement factor H (CFH) genes. DESIGN: A proof of concept double-masked randomized controlled study. PARTICIPANTS: 114 participants aged 53 to 91 years, with either bilateral intermediate AMD or unilateral non-advanced AMD (with advanced AMD in fellow eye), BCVA ≥ 20/60 in at least one eye, and a normal lipid profile. INTERVENTION: Simvastatin 40 mg/day or placebo, allocated 1:1. MAIN OUTCOME MEASURES: Progression of AMD either to advanced AMD or in severity of non-advanced AMD. Results. The cumulative AMD progression rates were 70% in the placebo and 54% in the simvastatin group. Intent to treat multivariable logistic regression analysis, adjusted for age, sex, smoking and baseline AMD severity, showed a significant 2-fold decrease in the risk of progression in the simvastatin group: OR 0.43 (0.18-0.99), p = 0.047. Post-hoc analysis stratified by baseline AMD severity showed no benefit from treatment in those who had advanced AMD in the fellow eye before enrolment: OR 0.97 (0.27-3.52), p = 0.96, after adjusting for age, sex and smoking. However, there was a significant reduction in the risk of progression in the bilateral intermediate AMD group compared to placebo [adjusted OR 0.23 (0.07-0.75), p = 0.015]. The most prominent effect was observed amongst those who had the CC (Y402H) at risk genotype of the CFH gene [OR 0.08 (0.02-0.45), p = 0.004]. No evidence of harm from simvastatin intervention was detected. CONCLUSION/SIGNIFICANCE: Simvastatin may slow progression of non-advanced AMD, especially for those with the at risk CFH genotype CC (Y402H). Further exploration of the potential use of statins for AMD, with emphasis on genetic subgroups, is warranted. TRIAL REGISTRATION: Australian New Zealand Clinical Trial Registry (ANZCTR) ACTRN1260500032065

The role of the carotenoids, lutein and zeaxanthin, in protecting against age-related macular degeneration: A review based on controversial evidence

PURPOSE: A review of the role of the carotenoids, lutein and zeaxanthin, and their function in altering the pathogenesis of age-related macular degeneration (AMD). METHODS: Medline and Embase search. RESULTS: Recent evidence introduces the possibility that lutein and zeaxanthin, carotenoids found in a variety of fruits and vegetables may protect against the common eye disease of macular degeneration. This potential and the lack to slow the progression of macular degeneration, has fueled high public interest in the health benefits of these carotenoids and prompted their inclusion in various supplements. The body of evidence supporting a role in this disease ranges from basic studies in experimental animals to various other clinical and epidemiological studies. Whilst some epidemiological studies suggest a beneficial role for carotenoids in the prevention of AMD, others are found to be unrelated to it. Results of some clinical studies indicate that the risk for AMD is reduced when levels of the carotenoids are elevated in the serum or diet, but this correlation is not observed in other studies. Published data concerning the toxicity of the carotenoids or the optimum dosage of these supplements is lacking. CONCLUSION: An intake of dietary supplied nutrients rich in the carotenoids, lutein and zeaxanthin, appears to be beneficial in protecting retinal tissues, but this is not proven. Until scientifically sound knowledge is available we recommend for patients judged to be at risk for AMD to: alter their diet to more dark green leafy vegetables, wear UV protective lenses and a hat when outdoors. Future investigations on the role of nutrition, light exposure, genetics, and combinations of photodynamic therapy with intravitreal steroid (triamcinolone-acetonide) injections hold potential for future treatment possibilities

Derived Traces of Soergel Categories

We study two kinds of categorical traces of (monoidal) dg categories, with particular interest in categories of Soergel bimodules. First, we explicitly compute the usual Hochschild homology, or derived vertical trace, of the category of Soergel bimodules in arbitrary types. Secondly, we introduce the notion of derived horizontal trace of a monoidal dg category and compute the derived horizontal trace of Soergel bimodules in type A. As an application we obtain a derived annular Khovanov-Rozansky link invariant with an action of full twist insertion, and thus a categorification of the HOMFLY-PT skein module of the solid torus