Writing style (doc / postdoc)

Audience: PhD students and postdocs — readers who already know standard undergrad material and want precise, citable, research-usable summaries.

Not the primary audience: first courses in calculus or “SOH-CAH-TOA” pedagogy. Undergrad-named pages exist only as compressed recall, written in the language you would use in a prelim or a paper’s preliminaries.

Voice

Prefer Avoid
Named theorems, sharp hypotheses Motivational stories
“iff”, universal/existential quantifiers Hand-wavy “always”
Standard notation from papers (e.g. \(C_c^\infty\), \(\operatorname{Spec}\), \(R\)-mod) Schoolroom vocabulary alone
Counterexamples and edge cases Three near-identical numerical drills
Literature pointers (one classic ref) Textbook marketing

Page shape

  1. Job — research role of the object (one line)
  2. Core — definitions, functoriality/universality when relevant, main theorems (tables)
  3. See it — one non-toy example or computation that appears in papers
  4. Check — 1–2 hard self-checks (proof-sketch or counterexample), not arithmetic drills
  5. Pitfalls — standard confusion at the research boundary
  6. Refs / Next — Bourbaki/Stacks/standard monographs where useful + internal links

Depth targets

Layer Expectation
“Recall” pages (arithmetic, school algebra, basic trig) ≤ 40 lines; definitions as used later; zero pedagogy theater
Core grad tools (measure, FA, AA, alg top, PDE, complex) Compact but graduate: hypotheses visible; proof ideas one sentence
Domain indexes Map + dependency order for a reading program

Educational moves (research)

  • Hypothesis sensitivity: what fails if completeness / Noetherian / second-countable is dropped
  • Functoriality: induced maps, natural transformations
  • Dictionary: analyst ↔ algebraist ↔ geometer names for the same fact
  • Scale: finite-dim vs infinite-dim; smooth vs continuous; étale vs analytic

Assume the reader can fill \(\varepsilon\)–\(\delta\) or diagram chases; give the statement and the obstruction.