A comprehensive description of human genomes is essential for understanding human evolution and relationships between modern populations. However, most published literature focuses on local alignment comparison of several genes rather than the complete evolutionary record of individual genomes. Combining with data from the 1,000 Genomes Project, we successfully reconstructed 2,504 individual genomes and propose Divided Natural Vector method to analyze the distribution of nucleotides in the genomes. Comparisons based on autosomes, sex chromosomes and mitochondrial genomes reveal the genetic relationships between populations, and different inheritance pattern leads to different phylogenetic results. Results based on mitochondrial genomes confirm the “out-of-Africa” hypothesis and assert that humans, at least females, most likely originated in eastern Africa. The reconstructed genomes are stored on our server and can be further used for any genome-scale analysis of humans (http://yaulab.math.tsinghua.edu.cn/2022_1000genomesprojectdata/). This project provides the complete genomes of thousands of individuals and lays the groundwork for genome-level analyses of the genetic relationships between populations and the origin of humans.

Graph theory is important in information theory. We introduce a quantization process on graphs and apply the quantized graphs in quantum information. The quon language provides a mathematical theory to study such quantized graphs in a general framework. We give a new method to construct graphical quantum error correcting codes on quantized graphs and characterize all optimal ones. We establish a further connection to geometric group theory and construct quantum low-density parity-check stabilizer codes on the Cayley graphs of groups. Their logical qubits can be encoded by the ground states of newly constructed exactly solvable models with translation-invariant local Hamiltonians. Moreover, the Hamiltonian is gapped in the large limit when the underlying group is infinite.

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Shape analysis is a central problem in the field of computer vision. In 2D shape analysis, classification and recognition of objects from their observed silhouettes are extremely crucial but difficult. It usually involves an efficient representation of 2D shape space with a metric, so that its mathematical structure can be used for further analysis. Although the study of 2D simply-connected shapes has been subject to a corpus of literatures, the analysis of multiply-connected shapes is comparatively less studied. In this work, we propose a representation for general 2D multiply-connected domains with arbitrary topologies using conformal welding. A metric can be defined on the proposed representation space, which gives a metric to measure dissimilarities between objects. The main idea is to map the exterior and interior of the domain conformally to unit disks and circle domains (unit disk with several inner disks removed), using holomorphic 1-forms. A set of diffeomorphisms of the unit circle S1 can be obtained, which together with the conformal modules are used to define the shape signature. A shape distance between shape signatures can be defined to measure dissimilarities between shapes. We prove theoretically that the proposed shape signature uniquely determines the multiply-connected objects under suitable normalization. We also introduce a reconstruction algorithm to obtain shapes from their signatures. This completes our framework and allows us to move back and forth between shapes and signatures. With that, a morphing algorithm between shapes can be developed through the interpolation of the Beltrami coefficients associated with the signatures. Experiments have been carried out on shapes extracted from real images. Results demonstrate the efficacy of our proposed algorithm as a stable shape representation scheme.