Important problems in the sciences, from biology to physics, belong to high-dimensional spaces. Yet high-dimensional space is unfathomably big, making generic high-dimensional data impossible to store on any classical computer. How can we solve such problems?

I will highlight three key problems of high dimensions: the curse of dimensionality, the challenge of continuous variables, and the problem of loops, and discuss how each of these problems is now falling to a blend of old and new ideas, including tensor networks, Arabic numerals, and belief propagation. Among the applications are machine learning techniques with applications in quantum physics and robotics, compressed methods for solving differential equations, and techniques for simulating quantum computers.