Title: Mathematics of Topological Quantum Computing,
Abstract: In topological quantum computing, information is encoded in "knotted" quantum
states of topological phases of matter, thus being locked into topology to
prevent decay. Topological precision has been confirmed in quantum Hall liquids
by experiments to an accuracy of $10^{-10}$, and harnessed to stabilize quantum
memory. In this survey, we discuss the conceptual development of this
interdisciplinary field at the juncture of mathematics, physics and computer
science. Our focus is on computing and physical motivations, basic mathematical
notions and results, open problems and future directions related to and/or
inspired by topological quantum computing.