Geometrical structures in mathematical physics

这项拟议的研究旨在建立和加强数学和量子物理之间的联系。我们认为这是一个重要的方向，因为这种联系可以(就像它们过去所做的那样)在这两个领域带来新的见解。一般来说，在一个领域使用另一个领域的技术可能会带来看待问题的新方法，从而导致发现，如果一个人停留在一个封闭的研究领域，就不太容易做出这些发现。 一方面，我们计划利用我们对数学中一个重要的微分方程(Painlevé-6)的见解来揭示它的结构与数学和量子物理中出现的其他一些方程(Korteweg-de Vries和Schrödinger方程)的结构之间的相似之处。 另一方面，我们建议研究拓扑递归方法在量子力学中的应用。最近出现在数学中的拓扑递归方法表明，有关量子力学系统的信息可能被编码在一个相关的数学对象中，称为黎曼曲面。我们计划发现这种新的量子力学方法对物理学的影响。 相反，提案的另一部分建议使用量子场论的方法来列举数学对象，即具有一定性质的表面上的图形。这是可能的，因为我们在最近的工作中建立了这样的图和一个特定量子场论的费曼图之间的对应关系。 该提案的第四部分涉及被称为团簇代数的数学结构与量子场论中非常特殊的所谓BPS(Bogomolnyi-Prasad-Sommerfield)态之间的关系。事实证明，星系团代数理论中的某些数字计算了这种BPS状态的数量。物理学家和数学家都发现了这一事实，他们研究的是相同的问题，但采用了截然不同的方法。我们计划将一个领域的成果翻译成另一个领域的语言，并发现每个领域的结果对另一个领域的新影响。 参与这些项目的学生，除了所涉及的数学知识外，还将学习一些物理知识，这将给他们在数学物理研究方面提供一个良好的开端。

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.