Graduate Student Seminar with JR Cruise on What is the Deal with Topology?

Despite it's ubiquity in modern condensed-matter, quantum-information, and high-energy physics, topology has yet to recieve the standardized introductory-level pedagogical treatment that we theorists find so helpful for topics like differential equations and analysis in Hilbert spaces. Inspite of this, most physicists are familiar with at least a few concepts in topology, i.e. those which may be expressed in the form of coffee-cups-are-donuts jokes; but where is the connection between coffee cups/donuts, and quantum computers? In my talk, we will explore the mathematical foundations of topology (I promise there will be enough memes to make this not horifically boring) from the perspective of physicists: to this end, we will use familiar examples of topology from quantum mechanics to figure out what the effect we are trying to generalize actually is, and once we have generalized it, see how far it can take us into novel applications. Finally, since I would like to tell you about my actual research, we will conclude with a discussion of (fractional) quantum Hall matter, one of the richest topological playgrounds in nature.