The purpose of this thesis is to examine the impact of environmental, educational, and household chaos on the development of an atypically developing child. The study evaluates the contribution and change of parental stress during the COVID-19 pandemic and school closures. Following Case Study methodology, this is an in-depth evaluation of one mother and child's experience, including data analysis of the child's social, emotional, and cognitive development before and after chaos. The results of this case study identified the development of a child before and after chaotic events occurred. Parental stress was also identified as a contributing factor in the results of this study. The results implicate that school systems are a vital piece to the support, management, and growth of healthy child development in atypical children. Results further suggest that parental stress and chaos play a key role in the healthy social-emotional development of a child. Results suggest that cognitive development benefits from smaller group interactions, rather than large, classroom sized, interactions.

This dissertation is devoted to investigation of quantum chaos in the Sachdev-Ye-Kitaev (SYK) and supersymmetric SYK models. First, a unified minimal scheme is developed to classify quantum chaos in the SYK and supersymmetric SYK models and also work out the structure of the energy levels in one periodic table. The SYK with even q-body or supersymmetric SYK with odd q-body interaction, with N even or odd number of sites, are put on an equal footing in the minimal Hilbert space; N (mod 8), q (mod 4) double Bott periodicity, and a reflection relation are identified. Then, exact diagonalizations are performed to study both the bulk energy level statistics and hard-edge behaviors. Excellent agreements between the exact diagonalization results and the symmetry classifications are demonstrated. This compact and systematic method can be transformed to map out more complicated periodic tables of SYK models with more degrees of freedom, tensor models, or symmetry protected topological phases.

No description available.

No description available.

No description available.

No description available.

No description available.

No description available.

No description available.

No description available.