This week's Mathematical Sciences Department colloquium is given by Dr. Paul Bruillard at Texas A&M University, Dept. of Mathematics.
A physical system is said to be in topological phase if at low energies and long wavelength the physical observables are invariant under smooth deformations. These physical systems have applications in a wide range of disciplines, especially in quantum information science. Quantum computers based on such systems are topologically protected from decoherence. This fault-tolerance removes the need for the expensive error--correcting codes required by the qubit model. Topological phases of matter can be studied through their algebraic manifestations, modular categories. Thus, a complete classification of these categories would provide a taxonomy of admissible topological phases.
In this talk, we will review the connections between topological phases of matter and modular categories and discuss the classification program for such categories. We will then consider recent foundational work on the classification program as well as deep connections between these categories and number theory which make classification a tractable problem. In particular, we will consider how contemporary number theoretic techniques can be used to prove rank finiteness of modular categories. This analysis leads to intriguing primality conditions on modular categories, which are related to classical questions regarding the infinitude of Sophie Germain and Fermat primes. Time permitting, we will discuss how these techniques can be practically applied to extend the rank classification of modular categories.
Coffee will be served in FO 2.610F at 1:30 PM.
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