Project Summary Rearrangements of genomic segments or structural variation (SVs), include changes (increase/decrease) of copy number, inversions, translocations, and other mechanisms that change or rearrange the DNA content of a cell. Complex SVs can mediate many constitutional diseases; highly pathological germline rearrangements can damage the viability of the embryo; and somatic rearrangements can increase the pathology of many diseases, including cancer. The genomic footprint of complex SVs is a changed karyotype, defined by a collection of sequences of oriented genomic intervals whose coordinates are drawn from a reference genome, so that every sequence corresponds to a haploid or marker chromosome. The goal of this renewal proposal is to develop tools to elucidate the karyotype of a donor. In Aim 1, we will develop methods to reconstruct focal amplifications using long-read technology. In Aim 2, we will develop methods to elucidate genome scale karyotypes for constitutional disorders using optical genome maps. Highly rearranged regions will be clarified using a mix of experiments and computation in Aim 3. Finally, in Aim 4, we will extend the notion of a karyotype to predict the 3-dimensional conformation of ecDNA using Hi-C. The development of our aims will require novel graph theoretic and combinatorial optimization methods applied to new sequencing technologies. Their successful implementation will provide novel algorithms and software tools for improved karyotyping, deeper insights into gene regulation and DNA repair for focal amplifications, and better understanding of disease pathology mediated by complex variations.