Research in Representation Theory and Automorphic Forms

AbstractWallach该项目旨在利用分析、同调代数和几何的方法来研究实数还原群表示的结构和应用。特别是，首席研究员计划继续开发他所谓的复数还原群真实形式之间的“转移”。一个新的成分涉及确定显式多重性公式（类似于 Blattner 公式），以将半简单群的表示限制为对称子群（非紧）。该理论还提供了一种新方法来证明难以分析分析的表示的统一性。这项工作使用基本的不变理论和计算密集型代数几何。对于后者，Matt Clegg 和作者将完成现已到位的 Groebner 阵列的代码。该硬件还有助于解决组合学、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”的面貌。事实上，它模糊了纯数学和应用数学之间的区别。数学家现在可以进行涉及大量计算的实验，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。它还将致力于增强大规模计算的实用性，这些计算的输出可以是由数十亿字节长的文件描述的数学对象。尽管该项目的基础研究目标超出了短期应用的范围，但预计该方法将带来实际应用。 例如，在无法读取完整文件的情况下，帮助确定描述复杂对象的巨大文件的哪些方面最重要的方法。

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.