Research in Representation Theory and Automorphic Forms

本项目旨在利用分析、同调代数和几何的方法来研究实还原群的表示的结构和应用。特别是，首席研究人员计划继续他所说的在复数上的还原群的实形式之间的“转移”的发展。一项新的内容涉及确定显式重数公式(类似于Blattner公式)，用于将半单群的表示限制为对称子群(非紧)。该理论还为证明难以分析的表示的单一性提供了一种新的方法。这项工作使用了基本的不变量理论和计算密集的代数几何。对于后者，Matt Clegg和作者将完成Groebner数组的代码，该代码现在已经到位。这种硬件还帮助解决了组合学、表示论和量子计算中的几个问题。相对便宜的快速计算工具改变了“纯数学”的格局。事实上，它模糊了纯粹数学和应用数学之间的区别。数学家现在可以进行大量计算的实验，以帮助预测要通过数学推理证明的定理的形式。这项拟议的研究将利用这些工具。它还将致力于提高大规模计算的有用性，其输出可以是由数十亿字节长的文件描述的数学对象。虽然这一项目的基本研究目标超出了短期应用的范围，但预计方法论将导致实际应用。例如，在没有读取整个文件的能力的情况下，帮助确定描述复杂对象的巨大文件的哪些方面是最重要的方法。

AbstractWallachThis project aims to use the methods of analysis, homological algebra and geometry in order to study the structure and applications of representations of real reductive groups. In particular the principal investigator plans to continue his development of what he calls "transfer" between real forms of reductive groups over the complex numbers. A new ingredient involves determining explicit multiplicity formulas (analogous to Blattner's formula) for restriction of representations of semisimple groups to symmetric subgroups (noncompact). The theory also provides a new method for proving the unitarizability of representations that are difficult to analyze analytically. This work uses basic invariant theory and computationally intensive algebraic geometry. For the latter, Matt Clegg and the author will be completing the code for the Groebner array which is now in place. This hardware has also been helpful in solving several problems in combinatorics, representation theory, and quantum computing.The relatively inexpensive access to fast computational tools has changed the landscape of "pure mathematics". In fact, it has blurred the distinction between pure and applied mathematics. Mathematicians can now do experiments involving massive calculations to help predict the form of theorems to be proved by mathematical reasoning. This proposed research will make use of these tools. It will also work to enhance the usefulness of massive calculations whose output can be mathematical objects that are described by files that are billions of bytes long . Although the basic research aims of this project are outside the scope of short term applications the methodology is expected to lead to practical applications. For example, methods to help in deciding which aspects of an immense file that describes a complicated object are most important without having the ability to read the full file.