Paper Companions

Paper companions are informal commentaries intended to sit alongside a research paper. They are not seminar talks, they are not alternative expositions of the proofs and they do not replace the written proof. Rather, they are a record of how I understand the mathematics: the internal structure of the argument, the dependencies between its parts, and the points where the real difficulty lies.

The aim of a paper companion is to share how the mathematics lives in my head. Rather than presenting a polished narrative, it tries to make visible the internal picture that guides the argument, how I organise the ideas mentally, what I pay attention to, and how I judge what matters and what does not. This is the level at which I think about the paper day to day, but which rarely appears explicitly in the written text.

The tone is deliberately informal. These recordings are closer to an office conversation with someone reading the paper than to a polished presentation. A certain amount of roughness is intentional. The paper itself remains the authoritative record of correctness; the companion records perspective, judgement, and understanding.