Estudiantes de matemáticas: teoremas, demostraciones y notación que se mantiene consistente

Math papers are notation plus proof structure. LaTeX is the default because both get painful in Word once the argument runs longer than a page. The compiler is not the hard part. Consistency is: theorem numbers that jump when you insert a lemma, a letter V that means three different things, and proofs that leave the QED mark floating on the next line.

This is a practical setup for coursework, arXiv notes, and thesis chapters. Steal what you need; ignore the rest until it hurts.

amsthm early, before the first lemma

Load amsthm before you invent ad-hoc bold “Theorem.” headings:

\usepackage{amsthm}
\theoremstyle{plain}
\newtheorem{theorem}{Theorem}[section]
\newtheorem{lemma}[theorem]{Lemma}
\newtheorem{proposition}[theorem]{Proposition}
\newtheorem{corollary}[theorem]{Corollary}
\theoremstyle{definition}
\newtheorem{definition}{Definition}[section]
\newtheorem{example}{Example}[section]
\theoremstyle{remark}
\newtheorem*{remark}{Remark}

Numbering by section keeps references stable when you insert material above a result. Sharing a counter ([theorem]) for lemmas and propositions is common in pure math; applied papers sometimes number everything independently. Pick one and stick to it for the whole document.

Lesson: theorem and proof blocks.

Proofs that end cleanly

Use the proof environment so the end-of-proof mark lands in the right place:

\begin{proof}
  ...
\end{proof}

If the last line is a displayed equation, end with \qedhere so the box does not orphan on the following line. Nested proofs (claim inside a proof) work better as claim environments with their own short proofs than as indented prose that never closes.

For homework where the instructor wants hand-marked steps, still use LaTeX for the clean submission PDF, but keep rough work in a separate note file so you do not delete the only correct line while rewriting.

Macro hygiene

Define symbols once:

\newcommand{\R}{\mathbb{R}}
\newcommand{\N}{\mathbb{N}}
\newcommand{\eps}{\varepsilon}
\DeclareMathOperator{\Hom}{Hom}
\DeclareMathOperator{\Spec}{Spec}

If two notions share a letter, disambiguate with macros (\VectSpace vs \RandomVar) instead of reusing V with different fonts. Coauthors will not guess which font means which object. When a notation sheet grows past a screen, put it in notation.tex and \input it so everyone edits one file.

AI tools love inventing new symbols. If you use one, paste your macro block into the prompt and demand it only use defined names. The same discipline applies without AI: new symbol → add to sheet → then type.

Align for long derivations

align and split keep multi-line calculations readable. Number only lines you will cite with \eqref. A wall of numbered equations is harder to follow than a short narrative with two tagged lines.

\begin{align}
  f(x)
  &= g(x) + h(x) \label{eq:decomp} \\
  &= \cdots
\end{align}

For a single multi-line equation that should carry one number, use split inside equation. For cases, cases or aligned beats a home-made array of pipes.

Cross-references that survive reordering

Never write “by Theorem 2.3” as bare text. Use \ref or, with cleveref, \cref{thm:main}. Labels go after the caption or theorem title line, not before. If two chapters share labels like thm:main, prefix by chapter (ch3-thm:main) or you will get silent wrong targets when you merge files.

Project-wide go-to-definition and live broken-ref checks help on multi-file theses. That is normal research-editor territory now, not a luxury.

Multi-file notes and theses

Even a long homework set is easier with a root file and section inputs. Theses almost always want \include per chapter and \includeonly while you draft one chapter at a time. Keep figures under figures/ch3/ so reordering chapters does not break paths.

Full thesis compiles get slow. Draft figure mode, externalize heavy TikZ, and compile only the chapter you are editing when you can. See split chapter files and speed up compilation.

When the PDF is the hand-in

Coursework still needs the boring shell: margins per the syllabus, page numbers, title, and sometimes a statement of academic integrity. The quick reference covers that so you can spend attention on the proof.

Keep the source. A marked PDF without .tex is a dead end when the next problem set reuses the same preamble. A local folder with Git history (or a research editor that checkpoints compiles for you) beats final_v3_really.pdf on the desktop.

Before you submit

Make sure theorems and lemmas share a coherent numbering scheme, every symbol in the body is defined in the macro sheet, and proofs use proof / \qedhere where needed. Number only the equations you cite. All cross-refs should be \ref/\cref, not hard-coded numbers. Open the PDF once: fonts embed, margins match the brief.

Homework vs paper

For homework, clarity beats package cleverness. For a paper or thesis chapter, invest in the theorem setup and macro sheet once, then reuse. Do not rebuild numbering mid-project because a blog post suggested a different style.

Sharing with coauthors

Send a PDF for reading and a repo (or zip of sources) for editing. If a coauthor only returns scanned markup, apply it the same day and commit. Stale paper stacks are how two people “fix” the same lemma differently.

After that setup work, LaTeX mostly disappears and you can think about the math.