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

形状分析，尤其是形状的表征和匹配，是出现在广泛研究领域中的一个基本问题。尤其在分子生物学中，人们普遍认为蛋白质的功能在很大程度上是由其三维结构决定的。因此，了解分子功能是一项对蛋白质折叠和药物设计等基本生物学问题至关重要的任务，依赖于对分子结构的精确分析。然而，尽管在分子序列分析方面已经取得了很大的成功，但在结构方面的成功却更加有限，这在很大程度上是由于缺乏准确和有效的表征和匹配算法。为了应对这些挑战，该项目专注于使用几何和拓扑方法进行形状表征和匹配，其驱动应用来自分子形状分析。本项目中研究的几何形状不仅包括曲线、曲面等标准对象，还包括其他复杂形状，如球并表示。该项目研究了它们分析中涉及的广泛的基本问题，例如比较多个形状、在结构数据库中的有效搜索、与灵活性的匹配，以及开发数学上合理的方法来描述静态和变形形状的特征。它研究了这些问题背后的数学结构，并开发了理论上也是合理的实用算法。本研究涉及的主要工具是计算几何和计算拓扑学，它们构成了分析形状的天然平台。最后，通过开发有效的计算框架来操纵和处理各种几何形状，该项目向大规模分子结构分析迈出了重要的一步，这是在分子水平上理解生命所必需的。同时，这个多学科的项目有助于拓宽现实问题理论上合理的计算方法的范围，并进一步架起计算机科学、数学和结构生物学的桥梁。

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.