Notes from a PhD student, researching differential geometry and its applications to physics.
Across the last few years, I have found a good source of academic inspiration to be something somewhat niche: finding mathematician or physicist's own expositions which document their research journey. For example, I came across a digitisation of robert Wayne Thomason's notebooks, some 73 of them, which cover years of notes for his research. I also recently came across Scott Young's series of 'ultralearning' documentation. This was something I found quite intersting, not neccessarily for the ultralearning aprt, but more because of the incredible documentation of notes and exams across a year of learning.
In this light, I thought it would be interesting to document the notes which I take for my own maths research, largely on differential geometry and its applications to theoretical physics. Perhaps this will be intersting to others, especially people who are wondering what exactly a 'life in the day of' a mathematician looks like. Hint, it's a lot of coffee.
Began with the foundational book by Joyce on holonomy and calibrated geometry. I’m trying to get a strong conceptual feel for what the holonomy group is, what a calibration and calibrated submanifold is - then leading to the link between the two. I have also consulted Lotay's fantastic lecture notes on minimal submanifolds and calibrated geometry. I'm aiming to start by making basic notes and answering simple questions myself. Then, when I have assessed that I've learned enough, I'm going to write up a Feynman-technique-style exposition.
Search for references which have a clear use for aiding my research. This could be past theses, books, research papers... Upload each to Zotoro so I have the appropriate references which I can upload as Bibtex as and when needed. Annotate each key reference! That is, either annotate on iPad in Goodnotes or files, or print out and annotate by hand. Mark and highlight key sections or equations or theorems which are important. Log any important proofs into a journal. Make sure the annotated PDFs are kept together neatly for navigation.