A∞A_\infty Algebras from Slightly Broken Higher Spin Symmetries

We define a class of $A_\infty$-algebras that are obtained by deformations of higher spin symmetries. While higher spin symmetries of a free CFT form an associative algebra, the slightly broken higher spin symmetries give rise to a minimal $A_\infty$-algebra extending the associative one. These $A_\infty$-algebras are related to non-commutative deformation quantization much as the unbroken higher spin symmetries result from the conventional deformation quantization. In the case of three dimensions there is an additional parameter that the $A_\infty$-structure depends on, which is to be related to the Chern-Simons level. The deformations corresponding to the bosonic and fermionic matter lead to the same $A_\infty$-algebra, thus manifesting the three-dimensional bosonization conjecture. In all other cases we consider, the $A_\infty$-deformation is determined by a generalized free field in one dimension lower.Comment: 48 pages, some pictures; typos fixed, presentation improve

Hyper-connectivity index for fuzzy graph with application

Connectivity concept is one of the most important parameters in fuzzy graphs (FGs). The stability of a FG is dependent on the strength of connectedness between each pair of vertices. Depending on “strength of connectedness between each pair of vertices” hyper-connectivity index for fuzzy graph (FHCI) is introduced and studied this index for various FGs like partial fuzzy subgraph, fuzzy subgraph, complete fuzzy graph, saturated cycle, isomorphic fuzzy graphs, etc. A relation of FHCI is established between fuzzy graph and partial fuzzy subgraph. Also, a relation between FHCI and connectivity index for fuzzy graph (FCI) is provided. In the end of the article, a decision-making problem is presented and solved it by using FHCI. Also, a comparison is provided among related indices on the result of application and shown that our method gives better result.Publisher's Versio

Non-commutative Geometry, Index Theory and Mathematical Physics

Non-commutative geometry today is a new but mature branch of mathematics shedding light on many other areas from number theory to operator algebras. In the 2018 meeting two of these connections were highlighted. For once, the applications to mathematical physics, in particular quantum field theory. Indeed, it was quantum theory which told us first that the world on small scales inherently is non-commutative. The second connection was to index theory with its applications in differential geometry. Here, non-commutative geometry provides the fine tools to obtain higher information

Noncommutative Geometry

Many of the different aspects of Noncommutative Geometry were represented in the talks. The list of topics that were covered includes in particular new insight into the geometry of a noncommutative torus, local index formulae in various situations, C*-algebras and dynamical systems associated with number theoretic structures, new methods in K-theory for noncommutative algebras as well as new progress in quantum field theory using concepts from noncommutative geometry

Transport in time-dependent dynamical systems: Finite-time coherent sets

We study the transport properties of nonautonomous chaotic dynamical systems over a finite time duration. We are particularly interested in those regions that remain coherent and relatively non-dispersive over finite periods of time, despite the chaotic nature of the system. We develop a novel probabilistic methodology based upon transfer operators that automatically detects maximally coherent sets. The approach is very simple to implement, requiring only singular vector computations of a matrix of transitions induced by the dynamics. We illustrate our new methodology on an idealized stratospheric flow and in two and three dimensional analyses of European Centre for Medium Range Weather Forecasting (ECMWF) reanalysis data