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

这个奖项围绕着两个核心的数学问题。第一种来自数学物理和离散几何，被称为“维格纳结晶”，以物理学家尤金·维格纳（Eugene Wigner） 1934年里程碑式的论文命名。在这里，我们希望给出一个严格的证明，来解释为什么电子（或者更一般地说，某些分子）会把自己排列成一个结晶状的蜂窝状网格。这与球体填充问题有关（该问题要求精确地找到大小相等、不重叠的球体填充空间的密度），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“统一对偶问题”的一个重要例子，它更一般地要求确定保持距离的对称群的所有可能表现形式（称为“统一表示”）。James Arthur的猜想预测了奇异的、特殊类型的酉表示，这是特别有趣的，因为它们似乎与数论（在许多情况下也与弦理论）有关。该奖项的大部分资金将用于支持研究这些问题的研究生。提出的Wigner结晶的攻击计划是证明六边形晶格在平面上是普遍最优的，因为它使任何距离平方的完全单调函数的势能最小。这样的结果被PI和合作者在8维和24维上证明了，并且被认为（通过Mircea Petrache和Sylvia Serfaty的工作）暗示了维格纳结晶。这将需要从经典的模块形式中开发更多的机制，这是8维和24维的关键成分，因为问题在二维中是完全不同的。亚瑟的猜想也将通过自同构形式进行攻击，以使用爱森斯坦级数构建亚瑟表征的实现（如果可能的话）。其他工具将是统一算法，它现在已经作为李群图谱软件包的一部分实现，以及Adams-Barbasch-Vogan关于Arthur包结构的重要结果。该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的知识价值和更广泛的影响审查标准进行评估，被认为值得支持。

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.