Research in Representation Theory and Automorphic Forms

表示论和自守形式研究

• 批准号: 0200305
• 负责人: Nolan Wallach
• 金额: $ 21.63万
• 依托单位: University of California-San Diego
• 依托单位国家: 美国
• 项目类别: Continuing Grant
• 财政年份: 2002
• 资助国家: 美国
• 起止时间: 2002-08-01 至 2005-07-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=0200305&HistoricalAwards=false
• 关键词: Research; Representation; Theory; Automorphic; Forms

AbstractWallachThis project will study several related problems in representation theory, non-commutative harmonic analysis and algebraic group theory. In representation theory it involves the construction, study and application of small unitary representations of real reductive groups. In harmonic analysis, it involves dropping the K-finite condition in Paley-Wiener theorems for rapidly decreasing functions on a real reductive group. This analysis will be applied to proving a version of the Casselman-Wallach theorem depending on parameters. This theorem will imply a meromorphic continuation of non-K-finite Eisenstein series. Related to these more analytic problems we will study the algebraic problem of determining graded multiplicities for the irreducible constituents of the action of a reductive algebraic group on an affine cone. The latter work also has applications to the analysis and to the study of measures of entanglement in quantum computing.Representation theory has its roots in nineteenth century invariant theory, early twentieth century quantum mechanics and mid-twentieth century number theory. In this first decade of the twenty first century the theory has returned to its roots. The nineteenth century invariant theory emphasized concrete questions on binary forms with algorithmic solutions. These problems have reemerged and are now being generalized to apply to quantum computation. Early quantum mechanics studied puzzling and weird measurements involving photons, electrons etc. These phenomena led to the Hilbert space approach to quantum mechanics. The philosophical debates of the early quantum mechanics have reemerged as quantum information technology. The Hilbert space approach also gave birth to representation theory, which has as one of its main applications in number theory. The Langlands program has established a goal for the twenty first century to establish a non-commutative class field theory (Wile's proof of Fermat's Last Theorem is actually proof of a special case of the Tanayama-Shimura conjecture which is a special case of the Langlands program). This project is in the interface of all of these exciting directions.

摘要Wallach该项目将研究表示论、非交换调和分析和代数群论中的几个相关问题。 在表示论中，它涉及实数还原群的小酉表示的构造、研究和应用。 在调和分析中，它涉及放弃佩利-维纳定理中的 K 有限条件，以实现实数约简群上的快速递减函数。 该分析将用于根据参数证明卡塞尔曼-瓦拉赫定理的一个版本。 该定理意味着非 K 有限爱森斯坦级数的亚纯延续。 与这些更多的分析问题相关，我们将研究确定仿射锥上还原代数群作用的不可约成分的分级重数的代数问题。 后者的工作也适用于量子计算中纠缠测量的分析和研究。表示论起源于十九世纪的不变理论、二十世纪初的量子力学和二十世纪中叶的数论。 在二十一世纪的头十年里，这一理论又回到了它的根源。 十九世纪的不变理论强调具有算法解决方案的二进制形式的具体问题。这些问题再次出现，并且现在被推广到量子计算。 早期的量子力学研究了涉及光子、电子等的令人费解和奇怪的测量。这些现象导致了量子力学的希尔伯特空间方法。早期量子力学的哲学争论以量子信息技术的形式重新出现。 希尔伯特空间方法还催生了表示论，它在数论中的主要应用之一。朗兰兹纲领确立了二十一世纪建立非交换类场论的目标（怀尔对费马大定理的证明实际上是田名山-志村猜想的一个特例的证明，而田名山-志村猜想是朗兰兹纲领的一个特例）。 该项目处于所有这些令人兴奋的方向的界面中。