Problems arising from theta correspondences

θ对应产生的问题

• 批准号: 1359774
• 负责人: Gordan Savin
• 金额: $ 18万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2014
• 资助国家: 美国
• 起止时间: 2014-09-01 至 2019-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1359774&HistoricalAwards=false
• 关键词: Problems; arising; theta; correspondences

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.

找到多项式的根是数学和数理论中最古老的问题之一。二次多项式根源的一种公式可以追溯到一千100年，它是印度数学家婆罗门（Brahmagupta）。 意大利文艺复兴时期的数学家解决了立方和四分之一的多项式方程。高度多项式直到现代才能抵抗，当时很明显，无法为其根编写一个简单的公式。取而代之的是，数学家意识到，可以通过查看其根源的排列来理解多项式。根的排列形成一个称为组的数学对象。该组是多项式难度的量度。在20世纪，对小组进行了广泛的研究，并继续研究。我们对群体的了解可以转化为对多项式的理解。这是该项目的主要对象。特别是，我们可以理解大型多项式，其相应的组形成一个称为G2的家族。 例如，这些是大的和非平凡的群体，最小的群体具有12096个元素。PI将研究试验性引起的转化组。在数学中，试验是指三个矢量空间之间的相互作用。也许最有趣的情况是，当三个空间具有尺寸8时。然后，试验的原理产生了几种特殊的数学结构：特殊的投射平面，欧几里得的常见公理虽然存在，而某些预期属性则没有，并且特殊的属性却没有。转换称为G2和D4。该组G2大约在一百年前被德国数学家发现，但由于其在物理和弦理论中的作用，最近引起了很多关注。这项工作的主要目的是将G2组的表示形式分类。 ，正如兰兰兹猜想所预测的。为此，我们研究了异常的Theta对应关系和双对，其中双对成员为G2。主要的新成分是G2的保护原则，有点类似于古典群体的保护原则，但是，D4组扮演了Kudla-Rallis的角色加倍技巧。因此，大部分工作都致力于研究D4及其表示。