IMA SUMMER SCHOOL ON MODERN APPLICATIONS OF REPRESENTATION THEORY (SUPPLEMENTARY FUNDING), July 20 - August 6, 2014

IMA 表征理论现代应用暑期学校（补充经费），2014 年 7 月 20 日至 8 月 6 日

• 批准号: 1642608
• 负责人: Imre Kondor
• 金额: $ 2.47万
• 依托单位: Pennsylvania State Univ University Park
• 依托单位国家: 美国
• 项目类别: Standard Grant
• 财政年份: 2015
• 资助国家: 美国
• 起止时间: 2015-12-01 至 2017-08-31
• 项目状态: 已结题
• 来源: https://www.nsf.gov/awardsearch/showAward?AWD_ID=1642608&HistoricalAwards=false
• 关键词: IMA; SUMMER; MODERN; APPLICATIONS; REPRESENTATION

A three-week long summer school for graduate students in "Modern Applications of Representation Theory" will take place at the University of Chicago from July 20 to August 6, 2014 as part of the Institute for Mathematics and Applications (IMA) Graduate Students Summer Programs. In mathematics symmetries are described by abstract objects called groups. Representation theory, originating in the work of mathematicians such as Issai Schur and Ferdinand Frobenius at the end of the 19th century, studies how the structure of groups can be captured by simpler, linear objects, in particular, matrices. Apart from making groups more concrete, it has long been realized that this is critical for understanding how groups act on other objects. For example, in physics one is interested in how the symmetry groups of nature, such as translations and rotations, act on physical systems. This is why, starting in the 1920's, the representation theory of unitary groups, in particular, has been central to the development of quantum mechanics and particle physics. Recently, surprising connections have also been uncovered between representation theory and problems of a seemingly very different character, such as finding the number of operations required on a computer to carry out arithmetic computations, aligning a large number of noisy images of the same molecule taken by a certain type of electron-microscope, and ranking problems in statistical machine learning. The aim of the summer school is to engage graduate students from across the sciences in research in these exciting new areas.In recent years representation theory has found new applications in a range of domains from cryo-electron microscopy, through machine learning, to holographic algorithms. The fact that the subject is usually taught for an audience in pure mathematics makes it challenging for graduate students in more applied disciplines to learn the material. Conversely, mathematicians are often not aware of the new, exciting applications of representation theory. This summer school attempts to bridge this gap by starting with a short introduction to representation theory, followed by a series of mini-courses on some of the most recent work on using representation theoretical ideas in imaging, signal processing, holographic algorithms, quantum computing, algebraic and geometric computational complexity theory, non-commutative Fourier transforms, and other areas. Each of these subjects is presented by one of the leading experts in the area. The website for the summer school can be found here: www.ima.umn.edu/event/index.php?event_id=PISG7.20-8.6.14

为期三周的研究生暑期学校“表示论的现代应用”将于 2014 年 7 月 20 日至 8 月 6 日在芝加哥大学举行，作为数学与应用研究所 (IMA) 研究生暑期项目的一部分。 在数学中，对称性由称为群的抽象对象来描述。 表示论起源于 19 世纪末 Issai Schur 和 Ferdinand Frobenius 等数学家的工作，研究如何通过更简单的线性对象（特别是矩阵）来捕获群的结构。 除了使群体更加具体之外，人们很早就意识到这对于理解群体如何作用于其他对象至关重要。 例如，在物理学中，人们对自然的对称群（例如平移和旋转）如何作用于物理系统感兴趣。 这就是为什么从 20 年代开始，酉群的表示论一直是量子力学和粒子物理学发展的核心。 最近，在表示论和看似性质截然不同的问题之间也发现了令人惊讶的联系，例如计算计算机进行算术计算所需的操作数量，对齐由计算机拍摄的同一分子的大量噪声图像。某种类型的电子显微镜，以及统计机器学习中的排名问题。 暑期学校的目的是让来自各个科学领域的研究生参与这些令人兴奋的新领域的研究。近年来，表示理论在从冷冻电子显微镜、机器学习到全息算法等一系列领域中找到了新的应用。 事实上，该学科通常是针对纯数学的受众教授的，这使得更多应用学科的研究生学习这些材料变得具有挑战性。相反，数学家往往不知道表示论令人兴奋的新应用。 这个暑期学校试图弥合这一差距，首先对表示理论进行简短介绍，然后开设一系列迷你课程，介绍在成像、信号处理、全息算法、量子计算等领域使用表示理论思想的一些最新工作。代数和几何计算复杂性理论、非交换傅立叶变换等领域。 每个主题均由该领域的一位领先专家介绍。 暑期学校的网站可以在这里找到：www.ima.umn.edu/event/index.php?event_id=PISG7.20-8.6.14