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

为期三周的暑期学校研究生在“现代应用表示理论”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）研究生暑期课程的一部分。 在数学中，对称性是用称为群的抽象对象来描述的。 表示论起源于世纪末的数学家Issai Schur和Ferdinand Frobenius的工作，研究如何用更简单的线性对象（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界的对称群，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，幺正群的表示论特别成为量子力学和粒子物理发展的核心。 最近，表征理论与看似完全不同的问题之间的惊人联系也被发现了，例如找到计算机上进行算术计算所需的操作数量，对齐由某种类型的电子显微镜拍摄的同一分子的大量噪声图像，以及统计机器学习中的排名问题。 暑期学校的目的是吸引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，表征理论在一系列领域中找到了新的应用，从低温电子显微镜，通过机器学习，到全息算法。 事实上，这门课通常是为纯数学的观众讲授的，这使得更多应用学科的研究生学习材料变得具有挑战性。相反，数学家们往往不知道表示论的新的、令人兴奋的应用。 这个暑期学校试图弥合这一差距，从一个简短的介绍表示理论开始，然后是一系列的迷你课程，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的一些最新工作上使用表示理论的想法。 每一个主题都由该领域的顶尖专家提出。 暑期学校的网站可以在这里找到：www.ima.umn.edu/event/index.php?活动_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