Advanced Mathematical Methods for Condensed Matter Physics

Catalog description

We will study some of the more advanced mathematical methods that have been or are currently being deployed to address interesting problems in condensed matter theory. Exactly what topics we will cover will depend on the particular interests and background of those who show up for the class. In this context, "Advanced" means not generally covered in existing courses on math methods or in any depth in quantum mechanics or statistical mechanics. "Mathematical methods" refers to results that are understood with a level of mathematical rigor that might not fully satisfy a mathematician, but will at least be recognizable as provable, if not fully proven. Some possible topics include uses of instantons, projective representations of familiar groups, homotopy theory, anomalies, dualities in field theory and statistical mechanics, and methods for characterizing the properties of gauge theories. (Gauge theories are neither new nor particularly advanced but tend not to be covered in condensed matter physics courses.) Additional topics that may be included - depending on instructor preparation - are basics of modern differential/algebraic topology and geometry (homology, cohomology, vector bundles, characteristic classes, and index theorems), K-theory (as used in classifying topological free fermion insulators), and category theory (relevant to the classification of symmetry-protected topological phases of matter).