Mellin motives, periods and high energy physics

拟议的研究位于理论数学和粒子物理学的交叉点。它涉及到研究一类特殊的数，称为周期，其中π是最著名的例子。周期是数学中最常出现的量之一。它们的名字来源于它们在世纪的起源，当时伽利略研究了钟摆完成摆动所需的时间，开普勒也提出了关于行星绕太阳运行的同样问题。在现代数学中，周期的定义更为普遍;它们在几何和数论的经典领域的许多中心问题中发挥着重要作用。在拟议的研究中，我将努力发现和研究新的结构，满足时期。粒子物理学通过另一类称为费曼积分的数字发挥作用，物理学家费力地计算费曼积分，以便对粒子对撞机实验进行预测，例如在欧洲核子研究中心的大型强子对撞机中进行的实验。这些实验对于验证现有的粒子物理学理论或发现其中的启发性差异是必要的，并且之前已经导致了新的基本粒子的发现，如希格斯玻色子。因此，计算费曼积分的困难是该领域发展的主要障碍。幸运的是，费曼积分是周期的例子，这个项目涉及应用代数几何的工具来回答有关其结构的问题，特别是被称为宇宙伽罗瓦群的对象捕获的结构。这项工作将帮助物理学家计算费曼积分，扩大通过量子场论预测的物理过程的范围。此外，物理学家遇到的模式可以帮助揭示一般周期的结构，这反过来又可以用来解决纯数学中的问题。正是这种数学和物理学的相互作用，使研究期间提出的研究特别令人兴奋。

The proposed research lies at the intersection of theoretical mathematics and particle physics. It involves the study of a particular class of numbers called periods, of which pi is the most famous example. Periods are some of the most frequently occurring quantities in mathematics. Their name comes from their origin in the 17th century, when Galilei studied the amount of time it takes a pendulum to complete a swing, and Kepler asked the same question about a planet travelling around the Sun. In modern mathematics, periods are defined much more generally; they play an essential role in many central questions in the classical fields of geometry and number theory. In the proposed research I will work to uncover and study new structures which periods satisfy. Particle physics comes into play via another class of numbers, called Feynman integrals, which are laboriously computed by physicists in order to make predictions for particle collider experiments such as the ones performed in the Large Hadron Collider at CERN. These experiments are necessary for validating existing particle physics theories or finding illuminating discrepancies in them, and have previously led to the discovery of new elementary particles such as the Higgs boson. The difficulty in computing Feynman integrals thus presents a major obstacle for advancements in the field. Fortunately, Feynman integrals are examples of periods, and this project involves applying tools from algebraic geometry to answer questions about their structure, in particular the structure captured by an object called the Cosmic Galois group. This work will help physicists compute Feynman integrals, expanding the scope of physical processes which are possible to predict via quantum field theory. In addition, patterns encountered by physicists can help uncover structures that hold for periods in general, which, in turn, can be used to solve problems in pure mathematics. It is this interplay of mathematics and physics that makes the proposed research during the fellowship particularly exciting.