Automorphic Forms, Crystallization in the Plane, and Arthur’s Unitarity Conjecture

自守形式、平面结晶和亚瑟幺正猜想

• 批准号: 2101841
• 负责人: Stephen Miller
• 金额: $ 18万
• 依托单位: Rutgers University New Brunswick
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2021
• 资助国家: 美国
• 起止时间: 2021-07-15 至 2024-06-30
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=2101841&HistoricalAwards=false
• 关键词: Automorphic; Forms; Crystallization; Plane; Arthur

This award centers around two central mathematical problems. The first comes from mathematical physics and discrete geometry, and is known as “Wigner crystallization” after the physicist Eugene Wigner’s landmark 1934 paper. Here one wishes to give a rigorous proof explaining why electrons (or, more generally, certain molecules) arrange themselves in a crystalline, honeycomb-shaped grid. This is related to the sphere packing problem (which asks to find exactly how densely space can be filled by equal-sized, non-overlapping balls), and the PI and collaborators plan to use tools from number theory to attack the problem. The second problem is an important case of the “Unitary dual problem”, which more generally asks to determine all the possible incarnations of a symmetry group which preserve distances (known as “unitary representations”). Conjectures of James Arthur predict exotic, special types of unitary representations, which are of special interest because they appear to be related to number theory (and also to string theory, in many cases). The bulk of the funding from the award will support graduate students working on aspects of these problems.The proposed plan of attack for Wigner crystallization is to prove that the hexagonal lattice is universally optimal in the plane, in that it minimizes potential energy for any completely monotonic function of distance-squared. Such a result was proven by the PI and collaborators in 8 and 24 dimensions, and is known (by work of Mircea Petrache and Sylvia Serfaty) to imply Wigner crystallization. This will require developing more machinery from classical modular forms, which were the key ingredient in 8 and 24 dimensions, since the problem is quite different in two dimensions. Arthur’s conjecture will also be attacked via automorphic forms, to construct realizations of Arthur representations (when possible) using Eisenstein series. Other tools will be the unitarity algorithm which has now been implemented as part of the Atlas of Lie Groups software package, as well as important results of Adams-Barbasch-Vogan on the structure of Arthur packets.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.

该奖项围绕两个中心数学问题。 第一种来自数学物理学和离散几何学，在物理学家尤金维格纳1934年里程碑式的论文之后被称为“维格纳结晶”。 在这里，人们希望给出一个严格的证明，解释为什么电子（或者更一般地说，某些分子）会以结晶的、蜂窝状的网格排列。 这与球体填充问题有关（该问题要求精确地找到空间可以被大小相等，不重叠的球填充的密度），PI和合作者计划使用数论工具来解决这个问题。 第二个问题是“酉对偶问题”的一个重要例子，它更一般地要求确定一个对称群的所有可能的化身，这些对称群保持距离（称为“酉表示”）。 詹姆斯·亚瑟的猜想预言了奇异的、特殊类型的酉表示，它们之所以特别有趣，是因为它们似乎与数论有关（在许多情况下也与弦论有关）。 该奖项的大部分资金将用于支持研究生对这些问题的研究。维格纳结晶的拟议攻击计划是证明六方晶格在平面上是普遍最优的，因为它使距离平方的任何完全单调函数的势能最小化。 这个结果被PI和合作者在8维和24维中证明，并且被米尔恰·彼得拉什和西尔维亚·塞尔法蒂的工作所知，暗示着维格纳结晶。 这将需要从经典的模块形式中开发更多的机器，这些模块形式是8维和24维的关键成分，因为问题在二维中是完全不同的。 亚瑟猜想也将通过自守形式进行攻击，使用爱森斯坦级数构造亚瑟表示的实现（如果可能）。 其他工具将是酉算法，现在已实施的一部分，地图册的李群软件包，以及重要成果的亚当斯-Barbasch-Vogan的结构上的亚瑟packets.This奖项反映了NSF的法定使命，并已被认为是值得的支持，通过评估使用基金会的智力价值和更广泛的影响审查标准。