This program is primarily for graduate students of IMA Participating Institutions (PIs). Support for students at other U.S. universities may be available, conditional upon the outcomes of pending grant applications. In order to participate, students must complete the online application form, provide a personal statement, and submit 1) a letter of nomination from the PI chair (for students from an IMA PI) or (2) a recommendation letter (for students from institutions that are not an IMA PI).

A main portion of this program consists of a three-week summer school for graduate students to be held in Summer 2014 on the campus of the University of Chicago. It will focus on modern applications of representation theory discovered largely within approximately the last 10 years.

Specifically, we intend to cover applications of representation theory in algebraic and geometric computational complexity, cryo-electron imaging, digital signal processing, holographic algorithms and quantum computing, machine learning and pattern recognition, and a few other specialized topics.

The instruction in the summer school comprises several week-long lecture series supplemented by a foundational tutorial and two days of short hour-long lectures on specialized topics. The objectives are to (i) quickly review the basic materials (tutorial); (ii) focus on the developments of the last 10 years (several long lecture series); and (iii) provide a glimpse of the state of current research and open problems (a number of short talks).

Most of the materials intended to be covered in (ii) and (iii) are at this point not easily accessible — not covered in any textbooks, courses, or even survey articles — they are only available in the form of original research papers or preprints. It is our hope that the summer school will bring these materials to graduate students, postdocs, and in general, any nonspecialist.

Main Lectures

• Introduction to representation theory
• Introduction to representation theory of tensors
• Representation theory in cryoelectron microscopy
• Representation theory in computational complexity
• Representation theory in digital signal processing
• Representation theory in fast matrix multiplication
• Representation theory in machine learning
• Representation theory and tensor networks

Special Lectures

• Alpha permanents and random processes
• Mutually unbiased bases
• Representation theory in compressive sensing
• Representation theory in identity management
• Representation theory in phylogetics
• Representation theory in statistics