Research Math Notes
These are notes related to research level mathematics, mostly Operator Algebras/Functional Analysis.
Lecture Notes from Madrid: These are notes from my mini-course for the trimester program on ``L2-invariants and their analogues in positive characteristic."
Rough Intro to Sofic Entropy: A short and friendly introduction to sofic entropy, relating it to Shannon's coding theorem. Only rough proofs of statements are given, but filling in the gaps in the rough proofs makes for inciteful exercises.
Fuglede-Kadison Determinants Revisited. A revisited version of my Fuglede-Kadison determinant result. The main change is a large improvement to the organization of the paper, and there are a few simplifications to the proof using some of the more modern machinery of the subject.
Hyperlinear wreath sofic is hyerlinear: In this note, I explain how one can modify the methods in my paper with Andrew Sale "The wreath product of two sofic groups is sofic" to show that the the wreath product of a hyperlinear grop with a sofic group is sofic.
Types of von Neumann algebras and conditional expectations. These notes address whether a von Neumann algebra is Type I,II,III if it the image of a faithful normal conditional expectation of a von Neumann algebra of type I,II,III. The properties of being diffuse or atomic are addressed in this context. I was inspired to think about this issue after hearing Stefaan Vaes' lectures in Institue Henri Poincare at the von Neumann algebras and Ergodic Theory of Group Actions conference.
Measurable Operators.These are some notes on the space of measurable operators affiliated to a von Neumann algebra. The primary goal is to address the theory of noncommutative L^{p}-spaces. I try to keep the proofs as close to the commutative case as possible. In particular the L^{p}-space is defined directly as a certain family of measurable operators, and a direct proof is given for L^{p}-spaces being a complete linear space.
Normed Ideals in B(H): Some notes I wrote for a participating seminar talk at UCLA, giving a harmonic analysis style approach to basic facts about normed ideals in Bounded operators on a Hilbert space.
Direct Integrals.These are some notes I made on direct integral theory.
Disintegration of Measures. Some notes I made on disintegration of measure. This notion is useful to understand for measurable equivalence relations.
Past UCLA Analysis Qual Problems. The Analysis Qualifying Exam at UCLA can be quite difficult. These are solutions to what are, in my opinion, some of the hardest past qualifying exam solutions. In particular, included are entire solutions to the Spring 2009 and Fall 2009 qualifying exams, as well as solutions to all but two problems in the Spring 2011 exam (these two are standard theorems and are proved in many textbooks). These three analysis exams are particularly infamous,(with a sum total of four graduate students passing) and this is why I decided to give essentially entire solutions to these. Let me know if you have simpler solutions!
Double Duals: Some notes I made on double duals of C*-algebras. The proofs are all short. I give some quick applications to removing nonunital technicalities related to completely positive maps (this is in fact the universal solution to such issues), as well as existence of quasicentral approximate identities.
Notes from other mathematicians:
Notes froms GOALS on C*-algebras, von Neumann algebras, and some background material.
Jesse Peterson's lecture notes on Operator Algebras.