The human genome consists of over 3 billion nucleotides that have an average distance of 3.4 Angstroms between each base, which equates to over two meters of DNA contained within the 125 $\mu m^3$ volume diploid cell nuclei. The dense compaction of chromatin by the supercoiling of DNA forms distinct architectural modules called topologically associated domains (TADs), which keep protein-coding genes, noncoding RNAs and epigenetic regulatory elements in close nuclear space. It has recently been shown that these conserved chromatin structures may contribute to tissue-specific gene expression through the encapsulation of genes and cis-regulatory elements, and mutations that affect TADs can lead to developmental disorders and some forms of cancer. At the population-level, genomic structural variation contributes more to cumulative genetic difference than any other class of mutation, yet much remains to be studied as to how structural variation affects TADs. Here, we study the functional effects of structural variants (SVs) through the analysis of chromatin topology and gene activity for three trio families sampled from genetically diverse populations from the Human Genome Structural Variation Consortium. We then leverage clinically-relevant recurrent genomic rearrangements in acute lymphoblastic leukemia and propose a machine learning approach to identify the rare Philadelphia-like subtype based on the gene activities within lymphoblastoid chromatin domains. This analysis has found that TADs may improve our understanding of how SVs contribute to diverse gene expression patterns in health and disease.