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。第一个来自数学物理和离散几何，以物理学家尤金·维格纳1934年发表的里程碑式的论文而闻名，被称为“维格纳结晶”。在这里，人们希望给出一个严格的证据来解释为什么电子(或者更笼统地说，某些分子)自己排列在一个结晶的蜂窝状网格中。这与球体包装问题有关(该问题要求找出大小相等、不重叠的球可以准确地填充空间的密度)，PI和合作者计划使用数论中的工具来解决这个问题。第二个问题是“么正对偶问题”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的化身(称为“么正表示”)。詹姆斯·阿瑟的猜想预测了奇异的、特殊类型的么正表示，这是特别令人感兴趣的，因为它们似乎与数论有关(在许多情况下也与弦理论有关)。该奖项的大部分资金将支持研究生研究这些问题的各个方面。维格纳结晶提出的攻击计划是为了证明六角形晶格在平面上是普遍最优的，因为它最小化了距离平方的任何完全单调函数的势能。这样的结果被PI和合作者在8维和24维空间中证明了，并且(通过Mircea Petrache和Sylvia Serfaty的工作)被认为暗示了Wigner结晶。这将需要从经典的模块化形式开发更多的机械，这些形式是8维和24维的关键成分，因为在两个维度上的问题截然不同。亚瑟猜想也将通过自同构形式被攻击，以构造亚瑟表示的实现(在可能的情况下)使用艾森斯坦级数。其他工具将是统一性算法，现已作为利群图集软件包的一部分实施，以及Adams-Barbasch-Vogan关于Arthur包结构的重要成果。该奖项反映了NSF的法定使命，并已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。

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.