Small Representations and Applications

小型表示和应用

• 批准号: 0551846
• 负责人: Gordan Savin
• 金额: $ 12.86万
• 依托单位: University of Utah
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2006
• 资助国家: 美国
• 起止时间: 2006-06-01 至 2009-05-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0551846&HistoricalAwards=false
• 关键词: Small; Representations; Applications

Gordan Savin is continuing his work on algebraic aspects of analysis,with applications in number theory. The main tools of this research areminimal representations (discoveredby Kazhdan and Savin) which have been successful in dealing with certainaspects of Langlands conjectures.Indeed, several instances of Langlands conjectures, out ofreach of standard methods, can be obtained throughuse of minimal representations.Analysis, broadly defined, deals withfunctions satisfying certain differential equations. Analysis ofdifferential equations, in general, is a very hard problem. For example,differential equations coming from the dynamics of fluids are notoriouslydifficult. One possible reason for this is a lack of symmetries. On theother hand, if there are plenty of symmetries to work with, thenthe corresponding analysis is easier, since algebraic tools can be used.This research deals precisely with situationswhen a large group of symmetries is present. The main purposeof this research is to construct small representations ofgroups of Lie type. Smallness, roughly speaking, refers to the fact thatit is possible to combine together many symmetries acting on arelatively small space.Importance of having small representationscan be illustrated with the following example.One of the most famous (and most notorious) groups is the Reisz-FischerMonster. Yes, this group is huge and hence its name,but the size is not the main difficultyhere. The main problem is the lack of small representations.Indeed, the group of permutations of 60 letters - nobody would considerthis group a ``monster'' - is in fact bigger then the Riesz-FischerMonster. However, this group of permutations can be written down(i.e. represented) by means of60 by 60 matrices. The monster, on the other hand, requires use of(roughly) 2000 by 2000 matrices. It is our hope that the smallrepresentations will be a fruitful tool that can be used to answersome relevant questions in analysis and number theory.

戈丹·萨维因（Gordan Savin）继续在分析的代数方面继续他的工作，并在数字理论方面进行了应用。这项研究的主要工具（由Kazhdan和Savin发现）成功地处理了Langlands猜想的某些情况。实际上，可以通过最小的表示，兰兰兹猜想的几种范围猜想，而不是标准方法，可以通过标准方法来获得。广泛定义，处理满足某些微分方程的功能。通常，分化方程的分析是一个非常困难的问题。例如，众所周知，来自流体动力学的微分方程是缺乏的。造成这种情况的一个可能原因是缺乏对称性。在其他手上，如果可以使用大量的对称性，则可以使用相应的分析更容易，因为可以使用代数工具。这项研究正是存在大量对称性的情况。这项研究的主要目的是构建谎言类型的小组的小表现。大概的话，很小的是指可以将许多对称性结合在一起的事实，这些对称性在弧形较小的空间上。具有以下示例的重要性来说明一个较小的代表。 Fischermonster。是的，这个群体很大，因此名称，但规模并不是主要困难。主要的问题是缺乏小代表。实际上，60个字母的排列组 - 没有人会考虑这一组````怪物'' - 实际上比里斯·菲舍尔默斯特（Riesz -Fischermonster）更大。但是，这组排列可以通过60 x 60矩阵写下（即表示）。另一方面，怪物需要使用（大致）2000 x 2000矩阵。我们希望小陈述将是一种富有成果的工具，可以用来回答分析和数理论中的相关问题。