Minimal Representations and Theta Correspondence

这笔赠款将支持美国的与会者参加在奥地利维也纳的欧文·薛定谔研究所(ESI)举行的为期五天的研讨会，题为“最低限度的陈述和Theta通信”。该活动将于2022年4月11日至15日举行。50多年来，会议上讨论的最小表示主题在数学中发挥了中心和统一的作用，它们在量子力学和通信理论中得到了应用，在数学中也在数论中得到了应用。Theta对应关系使用最小表示法来连接不同的数学对象。最近，随着豪的二元性猜想的完整证据在几年前发表，theta对应的研究工作加快了步伐。该奖项将允许美国学者与国际研究人员合作，继续在全球舞台上研究极小表示和theta对应，并向材料介绍新一代数学家。极小表示可应用于自同构形式、调和分析，例如基本仿射空间上的傅里叶变换，以及数学物理，例如弦理论中的散射幅度。它们起源于亚普勒群的Weil表示，它们通过波函数与薛定谔的量子力学公式相联系。在数学中，最小表示的最深刻的应用是通过theta对应--即豪的二元性对应的推广。在朗兰兹计划和算术中，较新的“特殊”theta对应一直是至关重要的。这次会议是一次及时的活动，向与会者介绍了最近在最低限度的陈述和theta通信方面的最新突破。ESI研讨会将汇集来自世界各地的顶尖学者，展示这一数学领域的最新进展，这笔赠款将允许美国学者前往奥地利参加这一重要活动。有关会议的更多信息，请参阅https://www.esi.ac.at/events/e420.This奖项，该奖项反映了美国国家科学基金会的法定使命，并通过使用基金会的智力优势和更广泛的影响审查标准进行评估，被认为值得支持。

This grant will provide support for U.S. based participants to attend a five day workshop titled "Minimal Representations and Theta Correspondence," to be held at the Erwin Schrödinger Institute (ESI) in Vienna, Austria. The event will take place April 11-15 2022. The minimal representations topics discussed at the conference have played a central and unifying role in mathematics for over fifty years, where they find applications to quantum mechanics and communication theory, and within mathematics to number theory. Theta correspondences use minimal representations to connect disparate mathematical objects. Work on theta correspondences has accelerated recently, with the full proof of Howe's Duality Conjecture published just a few years ago. The award will allow U.S. scholars to continue working on minimal representations and theta correspondences on the global stage, with an international community of researchers, as well as introduce a new generation of mathematicians to the material.Minimal representations have applications to automorphic forms, harmonic analysis, for example, the Fourier transforms on basic affine spaces, and mathematical physics, for example, scattering amplitudes in string theory. They originate in the Weil representation of the metaplectic group, where they are connected to Schrödinger's formulation of quantum mechanics via wave functions. Within mathematics, the most profound applications of minimal representations are through theta correspondences -- a generalization of Howe's duality correspondence. Newer "exceptional" theta correspondences have been vital in the Langlands program and arithmetic. This conference is a timely event, bringing participants up-to-date on recent breakthroughs on minimal representations and theta correspondences. The ESI workshop will bring together the top scholars from around the world to showcase the latest progress in this field of mathematics and the grant will allow U.S. scholars to travel to Austria and attend this important event. For more information on the conference, see https://www.esi.ac.at/events/e420.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.