Problems arising from theta correspondences

求多项式的根是数学和数论中最古老的问题之一。一个求二次多项式根的公式可以追溯到1500年前的印度数学家婆罗门笈多（Brahmagupta）。意大利文艺复兴时期的数学家解决了三次和四次多项式方程。高次多项式一直抵抗到现代，当人们清楚地认识到不可能写出一个简单的公式来求它们的根时。相反，数学家意识到，多项式可以通过观察其根的排列来理解。根的排列形成一个数学对象，称为群。群是多项式难度的度量。群体在20世纪被广泛研究，并在今天继续被研究。我们对群的知识反过来可以转化为对多项式的理解。这是本项目的主要目标。特别是，我们可以理解大的多项式，其对应的群形成一个叫做G2的族。这些是大而非平凡的群，例如，最小的群有12096个元素。PI将研究由试验产生的转换组。在数学中，三角性是指三个向量空间之间的相互作用。也许最有趣的情况是当三个空间的维度为8时。然后，三性原理产生了几个特殊的数学结构：特殊的投影平面，其中欧几里得的通常公理成立，而一些预期的性质却不成立，以及称为G2和D4的特殊变换群。G2群大约在100年前由德国数学家发现，但最近由于它在物理学和弦理论中的作用而引起了广泛关注。这项工作的主要目的是对群G2在局部域上的表示进行分类，正如朗兰兹猜想所预测的那样。为此，我们研究了特殊的对应和对偶，其中对偶中的一个成员是G2。关键的新成分是G2的守恒原理，有点类似于经典群的守恒原理，然而，D4群扮演了Kudla-Rallis加倍技巧的角色。因此，大部分工作都致力于研究D4及其表示。

Finding roots of polynomials is one of the oldest problems in mathematics and number theory. A formula for roots of a quadratic polynomial goes back fifteen hundred years to Brahmagupta, an Indian mathematician. Cubic and quartic polynomial equations were solved by Italian renaissance mathematicians. Higher degree polynomials resisted until modern times, when it become clear that it is not possible to write a simple formula for their roots. Instead, mathematicians realized, polynomials can be understood by looking at permutations of their roots. Permutations of roots form a mathematical object called a group. The group is a measure of the difficulty of a polynomial. Groups were studied extensively in the 20th century and continue to be studied today. Our knowledge of groups can be in turn translated into understanding of polynomials. This is the main object of this project. In particular, we can understand large polynomials whose corresponding groups form a family called G2. These are large and non-trivial groups, for example, the smallest has 12096 elements.The PI will study groups of transformations arising from triality. In mathematics, triality refers to an interaction among three vector spaces. Perhaps the most interesting case is when the three spaces have dimension 8. Then the principle of triality gives rise to several exceptional mathematical structures: the exceptional projective plane, where the usual axioms of Euclid hold while some expected properties do not, and exceptional groups of transformations called G2 and D4. The group G2 was discovered by german mathematicians about one hundred years ago, but has recently attracted much attention due to its role in physics and string theory, in particular.The main object of this work is to classify representations of the group G2 over local fields, as predicted by Langlands conjectures. To that end we study exceptional theta correspondences and dual pairs where one member of the dual pair is G2. The key new ingredient is a conservation principle for G2, somewhat analogous to the conservation principle for classical groups, however, with the group D4 playing the role of the Kudla-Rallis doubling trick. Thus much of the work is devoted to study of D4 and its representations.