Conference: Amplituhedra, Cluster Algebras and Positive Geometry

2024年5月29-31日，在哈佛大学数学科学与应用中心(CMSA)举行了题为《Amplituhedra，簇代数与正几何》的会议。近年来，在粒子物理和宇宙学中，我们对量子可观测的理解发生了显著的范式转变，揭示了它们是从被称为正几何的底层新数学对象中出现的。会议将集中在振幅多面体上，这是正几何的第一个也是主要的例子，它描述了特定量子场论(QFT)中的粒子相互作用。我们的目标是探索振幅体和簇代数之间的联系，这是一种在数学和数学物理的各个领域都有广泛应用的数学理论。会议还将积极吸引和授权初级研究人员和女性，确保他们的完整存在和对会议的有影响力的贡献。更准确地说，物理学家Arkani-Hamed和Trnka在Lusztig和Postnikov关于正Grassman的工作的基础上，于2013年引入了振幅多面体作为一个几何对象，用于在N=4超杨Mills理论(SYM)中“解释”所谓的BCFW递推，用于计算散射幅度。同时，团簇代数--最初由Fomin和Zlevinsky用来研究全正性--在描述N=4 SYM散射振幅的奇异性方面起着至关重要的作用。因此，人们可以使用量子场论的思想来将团簇代数与正几何联系起来，特别是与振幅面联系起来。此外，QFT还可以用来发现正几何的新例子。我们的计划将汇集在相邻领域工作的广泛的数学家和物理学家，以在代数组合学和几何学中建立新的联系，并促进我们对散射幅度和QFT的物理理解。会议网站是https://cmsa.fas.harvard.edu/event/amplituhedra2024/This奖，反映了国家科学基金会的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

The conference "Amplituhedra, Cluster Algebras and Positive Geometry" will be held on May 29-31 2024 at the Harvard Center of Mathematical Sciences and Applications (CMSA). In recent years, a remarkable paradigm shift has occurred in our understanding of quantum observables in particle physics and cosmology, revealing their emergence from underlying novel mathematical objects known as positive geometries. The conference will center on the amplituhedron, the first and major example of a positive geometry, which describes particle interactions in a certain quantum field theory (QFT). We aim to explore connections between the amplituhedron and cluster algebras, a mathematical theory with broad applications across various areas of mathematics and mathematical physics. The conference will also actively engage and empower junior researchers and women, ensuring their integral presence and impactful contributions to the conference.More precisely, building on the work of Lusztig and Postnikov on the positive Grassmannian, the physicists Arkani-Hamed and Trnka introduced the amplituhedron in 2013 as a geometric object that "explains" the so-called BCFW recurrence for computing scattering amplitudes in N = 4 super Yang Mills theory (SYM). Simultaneously, cluster algebras – originally introduced by Fomin and Zelevinsky to study total positivity – have been revealed to have a crucial role in describing singularities of N = 4 SYM scattering amplitudes. Thus, one can use ideas from quantum field theory to connect cluster algebras to positive geometries, and in particular to the amplituhedron. Additionally, QFT can also be used to discover new examples of positive geometries. Our program will bring together a wide range of mathematicians and physicists working on adjacent areas both to draw new connections within algebraic combinatorics and geometry and to advance our physical understanding of scattering amplitudes and QFT. The conference website is https://cmsa.fas.harvard.edu/event/amplituhedra2024/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.