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

RUI：三维流形中的结：量子拓扑、双曲几何和应用

• 批准号: 1906323
• 负责人: Helen Wong
• 金额: $ 22.93万
• 依托单位: Claremont McKenna College
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2019
• 资助国家: 美国
• 起止时间: 2019-06-01 至 2023-11-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1906323&HistoricalAwards=false
• 关键词: RUI; Knots; Three; Dimensional; Manifolds

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.

该项目分为两个不同的研究领域，涉及一个称为拓扑的数学领域，该领域研究物体即使在连续扭曲或变形时也保持不变的属性。 一个方向与量子物理学有关，另一个方向与分子生物学有关。 2016年，物理学家因将拓扑应用于凝聚态物理研究而获得诺贝尔奖，其底层数学框架被称为拓扑量子场论（TQFT）。 该项目的第一部分重点关注 TQFT 的拓扑结构以及有关它们的猜想。 PI 旨在进一步增进对该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相关，例如拓扑量子计算机的理论基础和开发。 该项目的第二部分是关于蛋白质的拓扑结构，蛋白质足够长且灵活，可以表现出打结或连接。 据信，这种拓扑特征会影响蛋白质的功能，而蛋白质的功能由其三维位置决定。 然而，人们对蛋白质如何折叠成打结状态知之甚少，本项目从拓扑学角度分析蛋白质折叠理论。 特别是，打结蛋白与帕金森氏症等神经退行性疾病有关，并且在用于生物修复的细菌中发现。对分子打结机制的更好理解可能会带来针对影响特定生物功能的拓扑特征的新方法。 该奖项还支持本科生参与这项研究。具体来说，量子拓扑的研究主要围绕曲面的考夫曼括号绞线代数，特别是其表示。 绞纱代数与量子构造有关，例如琼斯多项式和 Witten-Reshetikhin-Turaev 拓扑量子场论，以及双曲几何构造，特别是 SL(2,C) 字符变体。 该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。 出于类似的目标，该项目还研究了包括弧在内的绞纱代数的最新推广。 在第二条研究中，拓扑技术将用于分析来自实验室和计算机模拟实验的有关打结蛋白质的证据，以便开发关于蛋白质如何折叠成打结结构的新理论。 然后可以将理论折叠途径与广泛可用的结构数据进行比较，以确定特定蛋白质家族最可能的折叠途径。 因此，在为所有打结蛋白的折叠途径提供有价值的见解的同时，这项研究还旨在简化研究特定打结蛋白的分子生物学家的分析。该奖项反映了 NSF 的法定使命，并通过使用基金会的智力评估进行评估，被认为值得支持。优点和更广泛的影响审查标准。

项目成果

Representations of the Kauffman bracket skein algebra III: closed surfaces and naturality

考夫曼括号绞线代数 III 的表示：闭曲面和自然性

• DOI: 10.4171/qt/125
• 发表时间: 2019
• 期刊: Quantum Topology
• 影响因子: 1.1
• 作者: Bonahon, Francis;Wong, Helen
• 通讯作者: Wong, Helen

The Roger–Yang skein algebra and the decorated Teichmüller space

Roger-Yang 绞线代数和装饰 Teichmüller 空间

• DOI: 10.4171/qt/150
• 发表时间: 2021
• 期刊: Quantum Topology
• 影响因子: 1.1
• 作者: Moon, Han-Bom;Wong, Helen
• 通讯作者: Wong, Helen

The search for leakage-free entangling Fibonacci braiding gates

• DOI: 10.1088/1751-8121/ab488e
• 发表时间: 2019-11-08
• 期刊: JOURNAL OF PHYSICS A-MATHEMATICAL AND THEORETICAL
• 影响因子: 2.1
• 作者: Cui, Shawn X.;Tian, Kevin T.;Wong, Helen M.
• 通讯作者: Wong, Helen M.

Topological descriptions of protein folding

• DOI: 10.1073/pnas.1808312116
• 发表时间: 2019-05-07
• 期刊: PROCEEDINGS OF THE NATIONAL ACADEMY OF SCIENCES OF THE UNITED STATES OF AMERICA
• 影响因子: 11.1
• 作者: Flapan, Erica;He, Adam;Wong, Helen
• 通讯作者: Wong, Helen