Understanding macroscale functionality of metal halide perovskites in terms of nanoscale heterogeneities

Hybrid metal halide perovskites have shown an unprecedented rise as semiconductor building blocks for solar energy conversion and light-emitting applications. Currently, the field moves empirically towards more and more complex chemical compositions, including mixed halide quadruple cation compounds that allow optical properties to be tuned and show promise for better stability. Despite tremendous progress in the field, there is a need for better understanding of mechanisms of efficiency loss and instabilities to facilitate rational optimization of composition. Starting from the device level and then diving into nanoscale properties, we highlight how structural and compositional heterogeneities affect macroscopic optoelectronic characteristics. Furthermore, we provide an overview of some of the advanced spectroscopy and imaging methods that are used to probe disorder and non-uniformities. A unique feature of hybrid halide perovskite compounds is the propensity for these heterogeneities to evolve in space and time under relatively mild illumination and applied electric fields, such as those found within active devices. This introduces an additional challenge for characterization and calls for application of complimentary probes that can aid in correlating the properties of local disorder with macroscopic function, with the ultimate goal of rationally tailoring synthesis towards optimal structures and compositions

Guaranteed Minimum-Rank Solutions of Linear Matrix Equations via Nuclear Norm Minimization

The affine rank minimization problem consists of finding a matrix of minimum rank that satisfies a given system of linear equality constraints. Such problems have appeared in the literature of a diverse set of fields including system identification and control, Euclidean embedding, and collaborative filtering. Although specific instances can often be solved with specialized algorithms, the general affine rank minimization problem is NP-hard. In this paper, we show that if a certain restricted isometry property holds for the linear transformation defining the constraints, the minimum rank solution can be recovered by solving a convex optimization problem, namely the minimization of the nuclear norm over the given affine space. We present several random ensembles of equations where the restricted isometry property holds with overwhelming probability. The techniques used in our analysis have strong parallels in the compressed sensing framework. We discuss how affine rank minimization generalizes this pre-existing concept and outline a dictionary relating concepts from cardinality minimization to those of rank minimization

The fusion algebra of bimodule categories

We establish an algebra-isomorphism between the complexified Grothendieck ring F of certain bimodule categories over a modular tensor category and the endomorphism algebra of appropriate morphism spaces of those bimodule categories. This provides a purely categorical proof of a conjecture by Ostrik concerning the structure of F. As a by-product we obtain a concrete expression for the structure constants of the Grothendieck ring of the bimodule category in terms of endomorphisms of the tensor unit of the underlying modular tensor category.Comment: 16 page

Surface group representations in SL2(C){\rm SL}_2({\mathbb C}) with finite mapping class orbits

Given an oriented surface of positive genus with finitely many punctures, we classify the finite orbits of the mapping class group action on the moduli space of semisimple complex special linear two dimensional representations of the fundamental group of the surface. For surfaces of genus at least two, such orbits correspond to homomorphisms with finite image. For genus one, they correspond to the finite or special dihedral representations. We also obtain an analogous result for bounded orbits in the moduli space.Comment: 30 pages, 5 figures, accepted for publication in Geometry & Topolog

Complexity of ITL model checking: some well-behaved fragments of the interval logic HS

Model checking has been successfully used in many computer science fields, including artificial intelligence, theoretical computer science, and databases. Most of the proposed solutions make use of classical, point-based temporal logics, while little work has been done in the interval temporal logic setting. Recently, a non-elementary model checking algorithm for Halpern and Shoham's modal logic of time intervals HS over finite Kripke structures (under the homogeneity assumption) and an EXPSPACE model checking procedure for two meaningful fragments of it have been proposed. In this paper, we show that more efficient model checking procedures can be developed for some expressive enough fragments of HS