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

作为数学与应用研究所（IMA）研究生暑期项目的一部分，芝加哥大学将于2014年7月20日至8月6日为“表征理论的现代应用”研究生开设为期三周的暑期学校。在数学中，对称是用称为群的抽象对象来描述的。表示理论起源于19世纪末数学家伊赛·舒尔（Issai Schur）和费迪南德·弗罗贝纽斯（Ferdinand Frobenius）等人的工作，研究如何用更简单的线性对象（特别是矩阵）来捕捉群的结构。除了使群体更加具体之外，人们早就意识到，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，人们感兴趣的是自然的对称群，如平移和旋转，如何作用于物理系统。这就是为什么从20世纪20年代开始，酉群的表示理论成为量子力学和粒子物理学发展的核心。最近，表征理论和一个看似非常不同的问题之间也发现了令人惊讶的联系，比如计算计算机进行算术计算所需的操作次数，对某种类型的电子显微镜拍摄的同一分子的大量噪声图像进行对齐，以及统计机器学习中的问题排序。暑期学校的目的是吸引来自各个科学领域的研究生在这些令人兴奋的新领域进行研究。近年来，表征理论在低温电子显微镜、机器学习和全息算法等一系列领域得到了新的应用。事实上，这门学科通常是为纯数学领域的观众教授的，这使得更多应用学科的研究生学习这些材料具有挑战性。相反，数学家往往不知道新的，令人兴奋的应用表示理论。这个暑期学校试图通过对表示理论的简短介绍来弥合这一差距，随后是一系列关于在成像、信号处理、全息算法、量子计算、代数和几何计算复杂性理论、非交换傅立叶变换和其他领域中使用表示理论思想的最新工作的迷你课程。每一个主题都是由该领域的一位主要专家介绍的。暑期学校的网站在这里：www.ima.umn.edu/event/index.php?event_id=PISG7.20-8.6.14

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