RUI: Pure and Applied Knot Theory: Skeins, Hyperbolic Volumes, and Biopolymers

RUI：纯结理论和应用结理论：绞纱、双曲体积和生物聚合物

• 批准号: 2305414
• 负责人: Helen Wong
• 金额: $ 28.14万
• 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: 2023
• 资助国家: 美国
• 起止时间: 2023-07-15 至 2026-06-30
• 项目状态: 未结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2305414&HistoricalAwards=false
• 关键词: RUI; Pure; Applied; Knot; Theory

Knot theory is the mathematical study of entanglement of loops up to continuous deformation. One can create a knot by taking an entangled string and connecting the endpoints, and two knots are equivalent if one can continuously deform one to the other, for example by bending, stretching, and passing strands inside and through others, but without cutting or breaking the string in any way. This project considers both theoretical problems and applications of mathematical knot theory. One group of problems studies a family of invariants of knots related to quantum field theory from physics. More specifically, the project seeks to understand how the quantum invariant of a knot detects geometric properties of the knot and the 3-dimensional spaces that can be associated with it. The mathematical techniques from this research has potential applications to mathematical physics and theoretical topological quantum computing. Another group of problems concerns applications to the study of knotted proteins and other biopolymers, some of which are known to be associated to various diseases. The project uses knot theory techniques to develop a model that can be used to quantify and to relate local topological complexity with biophysical processes. The model can also be used to potentially design synthetic biopolymers with special biophysical properties. The project includes a number of research problems suitable for collaboration with undergraduate students, as well as outreach and dissemination activities that seek to increase interest in mathematics more generally. The PI has successfully involved undergraduate students in similar research in the past and will continue to advise and encourage students to continue careers in mathematics and related areas. The research is split into three parts, two seek to connect quantum topology with hyperbolic geometry and one applies knot theory to molecular biology. One project concerns a version of the Volume Conjecture based the theory of the Kauffman bracket skein algebra from quantum topology and its relationship to the Teichmuller space of a surface from hyperbolic geometry. A second project studies algebraic and geometric properties of a generalization of the Kauffman bracket algebra which is related to the decorated Teichmuller space of a surface with punctures. A third project involves a collaboration with a biophysicist to study local entanglements that are held tightly in place by molecular forces in biopolymers. The proposed knot-theoretic model would give a description of such local entanglements, allowing one to quantify and measure changes in the local topological complexity of biopolymers in experiments.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.

结理论是对环缠结直至连续变形的数学研究。人们可以通过一根缠结的绳子并连接端点来打一个结，如果一个结可以不断地使一个结变形到另一个结，那么两个结是等效的，例如通过弯曲、拉伸以及将线穿过其他线，但不会切割或断裂以任何方式的字符串。 该项目考虑数学结理论的理论问题和应用。一组问题研究与物理学量子场论相关的一系列纽结不变量。更具体地说，该项目旨在了解结的量子不变量如何检测结的几何特性以及与其相关的 3 维空间。这项研究的数学技术在数学物理和理论拓扑量子计算方面具有潜在的应用。另一组问题涉及打结蛋白和其他生物聚合物研究的应用，其中一些已知与各种疾病相关。该项目使用结理论技术开发一个模型，可用于量化局部拓扑复杂性并将其与生物物理过程联系起来。该模型还可用于设计具有特殊生物物理特性的合成生物聚合物。该项目包括许多适合与本科生合作的研究问题，以及旨在更广泛地提高数学兴趣的外展和传播活动。 PI 过去曾成功地让本科生参与类似的研究，并将继续建议和鼓励学生继续从事数学和相关领域的职业。 该研究分为三个部分，两个部分寻求将量子拓扑与双曲几何联系起来，一个将结理论应用于分子生物学。一个项目涉及体积猜想的一种版本，该版本基于量子拓扑中的考夫曼括号绞线代数理论及其与双曲几何表面的 Teichmuller 空间的关系。第二个项目研究考夫曼括号代数推广的代数和几何性质，该代数与带有穿孔的表面的装饰 Teichmuller 空间相关。第三个项目涉及与生物物理学家合作，研究生物聚合物中分子力紧紧固定的局部缠结。所提出的结理论模型将描述这种局部纠缠，使人们能够量化和测量实验中生物聚合物局部拓扑复杂性的变化。该奖项反映了 NSF 的法定使命，并通过使用基金会的评估进行评估，被认为值得支持。智力价值和更广泛的影响审查标准。