CAREER: Geometric and Topological Methods in Shape Analysis, with Applications in Molecular Biology

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Shape analysis, in particular shape characterization and matching, is a fundamental problem appearing in a broad range of research fields. In particular, in molecular biology, it is generally believed that the functionalities of proteins are largely determined by their three dimensional structures. Hence understanding molecular functionality, a task essential to fundamental biological problems such as protein folding and drug design, depends on precise analysis of molecular structures. However, while much success has been achieved in molecular sequence analysis, success on the structural side is more limited, to a large degree due to a lack of accurate and efficient characterization and matching algorithms. To address these challenges, this project focuses on shape characterization and matching using geometric and topological methods, with driving applications coming from molecular shape analysis. The geometric shapes investigated in this project include not only standard objects such as curves and surfaces, but also other complex shapes, such as the union-of-balls representation. The project studies a broad range of fundamental issues involved in their analysis, such as comparing multiple shapes, efficient searching in structural databases, matching with flexibility, and developing mathematically justified methods to describe features for both static and deformed shapes. It investigates the mathematical structure behind these problems, and develops practical algorithms that are also theoretically sound. The main tools involved in this research are computational geometry and computational topology, which form a natural platform for analyzing shapes. Finally, by developing effective computational frameworks for manipulating and processing various geometric shapes, this project provides an important step towards large-scale molecular structural analysis, which is essential to understanding life at the molecular level. At the same time, this multi-disciplinary project helps to broaden the scope of theoretically sound computational methods for real-life problems, as well as to further bridge computer science, mathematics, and structural biology.