RUI: Knots in Three-Dimensional Manifolds: Quantum Topology, Hyperbolic Geometry, and Applications

这个项目被分成两个不同的研究领域，涉及一个被称为拓扑学的数学领域，它研究即使在连续扭曲或变形时保持不变的对象的属性。一个方向与量子物理有关，另一个方向与分子生物学有关。2016年，物理学家因将拓扑学应用于凝聚态物理研究而获得诺贝尔奖，其基础数学框架被称为拓扑量子场论(TQFT)。该项目的第一部分集中于由TQFT构造拓扑以及关于它们的猜想。PI的目的是进一步加深对这门学科的数学和理论物理方面之间联系的基本理解。这项工作可能与实际应用有关，例如拓扑量子计算机的理论基础和发展。该项目的第二部分是关于蛋白质的拓扑结构，这些蛋白质足够长和灵活，足以显示打结或连接。人们认为，这种拓扑特征会影响蛋白质的功能，而蛋白质的功能受其三维位置的支配。然而，人们对蛋白质如何折叠成打结状态知之甚少，这个项目从拓扑的角度分析了蛋白质折叠的理论。特别是，打结蛋白与帕金森氏症等神经退行性疾病有关，并在用于生物修复的细菌中发现；更好地了解分子打结机制可能会导致针对影响特定生物功能的拓扑特征的新方法。该奖项还支持参与这项研究的本科生。具体地说，量子拓扑学的研究集中在曲面的Kauffman括号斜线代数，特别是它的表示。Skein代数与量子结构有关，例如Jones多项式和Witten-Reshetikhin-Turaev拓扑量子场论，以及双曲几何结构，特别是SL(2，C)特征标集。这项研究将探索这种关系，并利用它来更好地理解几何拓扑中的其他不变量。出于类似的目的，该项目还研究了包含圆弧的skein代数的最新推广。在第二条研究线中，拓扑学的技术将被用来分析实验室和计算机模拟实验中关于打结蛋白质的证据，以开发关于蛋白质如何折叠成打结构型的新理论。然后，可以将理论折叠路径与广泛可用的结构数据进行比较，以便确定特定蛋白质家族最可能的折叠路径。因此，在为所有打结蛋白质的折叠路径提供有价值的见解的同时，这项研究旨在简化分子生物学家研究特定打结蛋白质的分析。这一奖项反映了NSF的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This project is split into two different areas of research concerning a field of mathematics called topology, which studies the properties of objects that remain the same even when they are twisted or deformed continuously. One direction relates to quantum physics, and the other to molecular biology. In 2016, physicists won the Nobel Prize for applying topology to research in condensed matter physics, and the underlying mathematical framework is called a topological quantum field theory (TQFT). The first part of the project focuses on topological constructions from TQFTs and conjectures about them. The PI aims to further advance the basic understanding of the connections between the mathematical and the theoretical physical sides of the subject. This work may be relevant to practical applications, such as the theoretical foundations and development of a topological quantum computer. The second part of the project is about the topology of proteins, which are long and flexible enough to exhibit knotting or linking. It is believed that such topological characteristics affect a protein's functionality, which is governed by its three-dimensional placement. However, little is known about how the proteins fold into a knotted state, and this project analyzes theories of protein folding from a topological viewpoint. In particular, knotted proteins are implicated in neurodegenerative disorders like Parkinson's and are found in bacteria used for bioremediation; a better understanding of the molecular knotting mechanism may lead to novel ways to target topological characteristics which affect specific biological functions. The award also supports undergraduate students participating in this research. Specifically, the research in quantum topology centers around the Kauffman bracket skein algebra of a surface, especially its representations. The skein algebra is related to quantum constructions, such as the Jones polynomial and the Witten-Reshetikhin-Turaev topological quantum field theory, as well as hyperbolic geometric constructions, particularly the SL(2,C)-character variety. The research will explore this relationship, and to exploit it for better understanding other invariants in geometric topology. With similar aims, the project also investigates recent generalizations of the skein algebra that includes arcs. In the second line of research, techniques from topology will be used to analyze evidence from laboratory and computer simulation experiments about knotted proteins, in order to develop new theories for how proteins might fold into a knotted configuration. The theoretical folding pathways can then be compared against widely available structural data in order to identify the most likely folding pathways for specific families of proteins. Thus, while providing valuable insights into folding pathways for all knotted proteins, this research aims to simplify the analysis for molecular biologists studying specific knotted proteins as well.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.