Shane Rankin - Graduate Notes

As part of the "Master's Phase" of the Ph.D. program here at UCR, we are required to take 4 graduate "core" sequences. During this time I've taken notes by hand, however, I also TeX these as well. On this page, I'll upload these notes as I update them, as well as list the professors I had taken these courses with (or sat in on for differential equations). No doubt that these are riddled with errors. These are written nearly as presented in lecture, with my own thoughts (and some choice quotes from professors) added. The only exception is the Part A differential equations notes. I did not sit in on this class, but Emerald Win (another graduate student at UCR) generously shared her handwritten notes which I then TeX'd. The notes assume undergraduate knowledge in math, but (hopefully) not too much background prerequisite knowledge. I also have my notes from some electives I've taken here (Commutative and Homological Algebra) as well as a course on Category Theory from UCLA.

Graduate Compendium

Full Text - Last Updated 7/16/25

These are the collection of all the below notes, as well as every exercise, midterm, and final that has been given in the two years I took these courses. There are no solutions in this document, however I have most of the solutions; If you ever want to know how to solve a problem- feel free to reach out.

Algebra

Notes: Last updated 6/8/22

Instructors: Part A - Dr. Wee Liang Gan, Part B - Dr. Jacob Greenstein, Part C - Dr. Carl Mautner

Qualifying Exam 2022

Complex Analysis

Notes: Last Updated 6/1/22

Instructors: Part A - Dr. Bun Wong, Part B - Dr. James Kelliher, Part C - Dr. Brian Collier

Qualifying Exam 2022

Real Analysis

Notes: Last Updated 6/20/23

Instructors: Part A - Dr. Qi Zhang , Part B - Dr. Zhenghe Zhang, Part C - Dr. Feng Xu

Topology