Writings

This is a book draft which I keep updated on my GitHub. The "raison d'etre" of this manuscript is to introduce modular tensor categories, my favorite algebraic structure. Modular tensor categories are a beautiful and powerful mathematical tool for understanding topological phases of matter. None of the current sources in the literature strike my preferred balance between mathematical precision (giving rigorous definitions and proving theorems) and physical motivation (explaining why we care about these things in the first place).

This note gives an introduction to the quantum Hall transition, a universality class of localization-delocalization phase transitions. This note has a somewhat novel perspective on the matter, in the sense that the specific models that I introduce are not the usual ones used in the literature. I chose them to be pedagogical from the perspective of probability theory. In particular, the note does not rely on any knowledge of the quantum Hall effect.

This note gives an introduction to lattice classification problems in many-body quantum physics, with a focus on invertible phases of matter. The novel feature of this note is that I use a non-standard formalism for lattice systems, in terms of compatible families of reduced density matrices on every finite region of space. The point of this formalism is that it is elementary to state (since everything is finite dimensional) but can host deep mathematical questions which are only present in the thermodynamic limit.

This note gives a little taste of what "arithmetic fracture" is all about. It's a wonderful technique from homotopy theory, which can be used to prove powerful theorems about compact Lie groups. This note states some theorems that were proved using this technique (which are interesting in their own right) and then moves on to giving an exposition of the arithmetic fracture itself.

These slides give a little introduction to fusion categories. The slides are messy. The first slide has a succinct definition of fusion categories using linear algebra, which people who want to learn the subject but aren't comfortable with categories might appreciate.

This note gives an exposition of the proof of the ADE classification result for matrices with non-negative integer coefficients whose spectral radius is less than 2. The point of this note isn't so much the theorem - it's the proof, which is wonderful and highlights the underlying truth behind deeper ADE classification results in Lie theory, singularity theory, and elsewhere.