Research in Representation Theory & Automorphic Forms

这个项目涉及到三个看似无关的数学部分：真实的约化群的表示理论，自守形式的傅里叶系数和量子计算中的纠缠数学。这些主题的线索包括不变量理论、有限维表示理论、组合数学和群作用的代数几何。前两个学科在二十世纪数学的伟大胜利中发挥了重要作用。后一个主题是为世纪后半叶的计算做准备。要研究的表示论包括寻找新的方法来构造最难以捉摸的酉表示，我们在这个提议中称之为小的。傅立叶系数的分析涉及到寻找广义Whittaker模的“最一般”重数1定理。纠缠的工作涉及到寻找有用的纠缠度量，这些度量可以被实验物理学家用来尝试建造量子计算机。表示论起源于世纪的不变量理论，世纪早期的量子力学和世纪中期的数论。 在二十一世纪的第一个十年里，这一理论又回到了它的根源。 世纪的不变理论强调具体问题的二进制形式与算法的解决方案。这些问题再次出现，现在正在推广到量子计算。 早期的量子力学研究了光子、电子等令人困惑和奇怪的测量，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在量子信息技术中重新出现。 希尔伯特空间方法也催生了表示论，它是它在数论中的主要应用之一。朗兰兹纲领为21世纪确立了一个目标，即建立一个非对易类场论（怀尔对费马大定理的证明实际上是对Tanayama-Shimura猜想的一个特例的证明，而Tanayama-Shimura猜想是朗兰兹纲领的一个特例）。 这个项目是所有这些令人兴奋的方向的接口。

AbstractWallachThis project involves three seemingly unrelated parts of mathematics: the representationtheory of real reductive groups, the Fourier coefficients of automorphic forms and themathematics of entanglement in quantum computing. The threads that hold these subjectstogether involve the invariant theory, finite dimensional representation theory, combinatorics and the algebraic geometry of group actions. The first two subjects have played an important role in the great triumphs of mathematics in the twentieth century. The latter subject is in preparation for computing in the second half of this century. The representation theory to be studied involves finding new ways of constructing the most elusive unitary representations which we call small in this proposal. The analysis of Fourier coefficients involves the search for the "most general" multiplicity one theorem for generalized Whittaker modules. The work on entanglement involves finding useful measures of entanglement that can be used by experimental physicists in their attempt to build quantum computers.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.