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: