### abstract

- A second quantum revolution in and around the construction of a useful quantum computer has been advancing dramatically in the last few years. Topological phases of matter, the importance of which has been recognized by scientific awards that include the 2016 Nobel prize in physics, exhibit many-body quantum entanglement. This makes such materials prime candidates for use in a quantum computer. Topological quantum computation is maturing at the forefront of the second quantum revolution as a primary application of topological phases of matter. The theoretical foundation for the second quantum revolution remains under development, but it appears clear that algebras and their representations will play a role analogous to that played by group theory in the first quantum revolution. This focused research group aims to formulate the theoretical foundations of topological quantum computation, leading to an eventual theoretical foundation for the second quantum revolution. It is anticipated that the results of the research will guide and accelerate the construction of a topological quantum computer. A working topological quantum computer will fundamentally transform the landscape of information science and technology. The project includes participation by graduate students and postdoctoral associates in the interdisciplinary research.The goal of topological quantum computation is the construction of a useful quantum computer based on braiding anyons. The hardware of an anyonic quantum computer will be a topological phase of matter that harbors non-abelian anyons. A physical system is in a topological phase if at low energies some physical quantities are topologically invariant. Topological properties are non-local, yet can manifest themselves through local geometric properties. The success of topological quantum computation hinges on controlling topological phases and understanding their computational power. This research addresses the mathematical, physical, and computational aspects of topological quantum computation. The projects include classification of super-modular categories, vector-valued modular forms for modular categories, extension of modular categories to three dimensions, simulation of conformal field theories, topological quantum computation with gapped boundaries and symmetry defects, and universality of topological computing models. The research has potential impacts ranging from new understanding of vertex operator algebras to the development of useful quantum computers. One specific goal is a structure theory of modular categories analogous to that of finite groups. Such a theory would lead to a structure theory of two-dimensional topological phases of matter.