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We develop a theory of localization for braid group representations associated with objects in braided fusion categories and, more generally, to Yang–Baxter (YB) operators in monoidal categories. The essential problem is to determine when a family of braid representations can be uniformly modelled upon a tensor power of a fixed vector space in such a way that the braid group generators act “locally”. Although related to the notion of (quasi-)fiber functors for fusion categories, remarkably, such localizations can exist for representations associated with objects of non-integral dimension. We conjecture that such localizations exist precisely when the object in question has dimension the square-root of an integer and prove several key special cases of the conjecture. [ABSTRACT FROM PUBLISHER]

The rising severity and frequency of wildfires in recent years in the United States have raised numerous concerns regarding the improvement in wildfire emergency response management and decision-making systems, which require operational high temporal and spatial resolution monitoring capabilities. Satellites are one of the tools that can be used for wildfire monitoring. However, none of the currently available satellite systems provide both high temporal and spatial resolution. For example, GOES-17 geostationary satellite fire products have high temporal (1–5 min) but low spatial resolution (≥2 km), and VIIRS polar orbiter satellite fire products have low temporal (~12 h) but high spatial resolution (375 m). This work aims to leverage currently available satellite data sources, such as GOES and VIIRS, along with deep learning (DL) advances to achieve an operational high-resolution, both spatially and temporarily, wildfire monitoring tool. Specifically, this study considers the problem of increasing the spatial resolution of high temporal but low spatial resolution GOES-17 data products using low temporal but high spatial resolution VIIRS data products. The main idea is using an Autoencoder DL model to learn how to map GOES-17 geostationary low spatial resolution satellite images to VIIRS polar orbiter high spatial resolution satellite images. In this context, several loss functions and DL architectures are implemented and tested to predict both the fire area and the corresponding brightness temperature. These models are trained and tested on wildfire sites from 2019 to 2021 in the western U.S. The results indicate that DL models can improve the spatial resolution of GOES-17 images, leading to images that mimic the spatial resolution of VIIRS images. Combined with GOES-17 higher temporal resolution, the DL model can provide high-resolution near-real-time wildfire monitoring capability as well as semi-continuous wildfire progression maps. [ABSTRACT FROM AUTHOR]

We give a rigorous development of the construction of new braided fusion categories from a given category known as zesting. This method has been used in the past to provide categorifications of new fusion rule algebras, modular data, and minimal modular extensions of super-modular categories. Here we provide a complete obstruction theory and parameterization approach to the construction and illustrate its utility with several examples. [ABSTRACT FROM AUTHOR]

The necklace braid group NB n is the motion group of the n + 1 component necklace link L n in Euclidean R 3 . Here L n consists of n pairwise unlinked Euclidean circles each linked to an auxiliary circle. Partially motivated by physical considerations, we study representations of the necklace braid group NB n , especially those obtained as extensions of representations of the braid group B n and the loop braid group LB n . We show that any irreducible B n representation extends to NB n in a standard way. We also find some non-standard extensions of several well-known B n -representations such as the Burau and LKB representations. Moreover, we prove that any local representation of B n (i.e., coming from a braided vector space) can be extended to NB n , in contrast to the situation with LB n . We also discuss some directions for future study from categorical and physical perspectives. [ABSTRACT FROM AUTHOR]

Abstract We study novel invariants of modular categories that are beyond the modular data, with an eye towards a simple set of complete invariants for modular categories. Our focus is on the W -matrix—the quantum invariant of a colored framed Whitehead link from the associated TQFT of a modular category. We prove that the W -matrix and the set of punctured S -matrices are strictly beyond the modular data (S , T). Whether or not the triple (S , T , W) constitutes a complete invariant of modular categories remains an open question. [ABSTRACT FROM AUTHOR]