Geometrical structures in mathematical physics

数学物理中的几何结构

• 批准号: RGPIN-2018-05413
• 负责人: Shramchenko, Vasilisa
• 金额: $ 1.68万
• 依托单位: Université de Sherbrooke
• 依托单位国家: 加拿大
• 项目类别: Discovery Grants Program - Individual
• 财政年份: 2019
• 资助国家: 加拿大
• 起止时间: 2019-01-01 至 2020-12-31
• 项目状态: 已结题
• 来源: https://www.nserc-crsng.gc.ca/ase-oro/Details-Detailles_eng.asp?id=679264
• 关键词: Geometrical; structures; mathematical; physics

The proposed research aims at establishing and strengthening connections between mathematics and quantum physics. We believe this to be an important direction as such connections can bring (as they have done in the past) new insights in both fields. In general, using techniques of one field in another might lead to new ways of looking at problems and thus to discoveries that would be less easy to make if one stays within a closed area of research. ******On one hand, we are planning to use our insight on one differential equation important in mathematics (Painlevé-6) to reveal similarities between its structure and the structure of some other equations arising in mathematical and quantum physics (Korteweg-de Vries and Schrödinger equations). ******On the other hand, we propose to study the topological recursion approach in application to quantum mechanics. Topological recursion method, having recently appeared in mathematics, suggests that information about quantum mechanical systems might be encoded in an associated mathematical object called Riemann surface. We plan to discover consequences for physics of this new approach to quantum mechanics. ******Another part of the proposal suggests, on the contrary, to use methods of quantum field theories to enumerate mathematical objects, namely graphs on surfaces with certain properties. This is possible due to the correspondence established in our recent work between such graphs and Feynman diagrams of one particular quantum field theory. ******The fourth part of the proposal concerns the relationship between mathematical structures called cluster algebras and the very special so-called BPS (Bogomolnyi-Prasad-Sommerfield) states in quantum field theories. It turned out that certain numbers in cluster algebra theory count the number of such BPS states. This fact has been discovered by both physicists and mathematicians studying the same questions but using quite different approaches. We are planning to translate results of one area into the language of the other, and discover new implications of results of each field for the other. ******Students working on these projects, in addition to the mathematics involved, will also learn some physics, which will give them an excellent start in research on mathematical physics. **

拟议的研究旨在建立和加强数学与量子物理学之间的联系。我们认为这是一个重要的方向，因为这种联系可以在两个领域中带来新见解（过去）。通常，在另一个领域中，使用一个领域的技术可能会导致新的方法来查看问题，从而发现，如果一个人留在封闭的研究领域，那将不太容易做到。 *****一方面，我们计划对数学中重要的一个微分方程（pachlevé-6）使用我们的见解，以揭示其结构和在数学和量子物理学中引起的其他一些方程的相似之处（Korteweg-de vries和Schrödinger方程）。另一方面，我们建议研究量子力学应用中的拓扑递归方法。最近出现在数学中的拓扑递归法表明，有关量子机械系统的信息可能是在称为Riemann Surface的相关数学对象中编码的。我们计划发现这种新方法的物理学的后果。 *****提案的另一部分建议使用量子场理论的方法来枚举数学对象，即在具有某些属性的表面上绘制数学对象。这是由于我们最近在一个特定的量子场理论的此类图和Feynman图之间建立的对应关系。 *****该提案的第四部分涉及量子场理论中称为群集代数的数学结构与非常特殊的所谓BPS（Bogomolnyi-prasad-Sommerfield）状态之间的关系。事实证明，集群代数理论中的某些数字计算了此类BPS状态的数量。这一事实是由物理学家和数学家都发现的，但是使用完全不同的方法研究了相同的问题。我们计划将一个领域的结果转化为另一种语言，并发现每个领域对彼此的结果的新含义。 ****从事这些项目的学生除了涉及的数学外，还将学习一些物理学，这将为他们提供数学物理学研究的良好开端。 **