--- name: expository description: > Write an expository paper: precise definitions, diagrams, examples, and known theorems with proofs in your own words. Arrange pedagogically so no idea is used before it is proved. This is your knowledge base — go work on it when stuck on a hard proof. Understanding the landscape is how breakthroughs happen. argument-hint: --- # Expository: Know What You Know You are writing an expository paper on: $ARGUMENTS This is not a research paper. This is a **pedagogical document for yourself** — a place where you organize everything that is known about a topic so that you truly understand it, and so that it is available to you when you need it during a proof. This document is alive. It grows as your understanding deepens. When you are stuck on a proof, come here and work on this instead. The act of writing up known results in your own words builds the understanding that unblocks you. ## Output Create the expository paper as a LaTeX file in the project's working directory (ask Robin where if unclear). This is a real document — not scratch notes. It should be readable, well-structured, and beautiful. ## Phase 1 — Definitions Write down EVERY definition relevant to the topic. For each definition: 1. **State it precisely.** Full formal definition, no shortcuts. 2. **Give the simplest nontrivial example.** If the definition is "LR tableau" then show one for a specific triple of partitions. Draw it. 3. **Give a non-example.** Show something that almost satisfies the definition but doesn't. What condition does it violate? Non-examples are often more informative than examples. 4. **Draw a diagram** if the object has visual structure (tableaux, puzzles, graphs, lattice paths — most combinatorial objects do). Use TikZ. 5. **Note equivalent definitions** if they exist, and state precisely under what conditions they are equivalent. Do not assume equivalence — verify it or cite it with a specific reference. ### Dependency order Arrange definitions so that **no definition uses a term that hasn't been defined yet**. This is a topological sort of your concepts. If you find a circular dependency, it reveals a structural insight about the topic — note it. ## Phase 2 — What We Know List every known result (theorem, lemma, proposition) relevant to the topic. For each: 1. **State it precisely.** Full hypotheses, full conclusion. 2. **Write the proof in your own words.** Not a citation. Not "see [KTW03, Thm 2.1]." Actually write the proof. If you cannot write the proof, you do not understand the result, and that gap will bite you during research. Mark it clearly: `% TODO: I don't understand this proof yet — need to work through it`. 3. **If the proof is long**, break it into lemmas. Each lemma gets its own proof. The same "no idea used before proved" rule applies within proofs. 4. **Give a concrete example** of the theorem in action. Walk through a specific instance with specific numbers/objects. Show the theorem being true, not just state that it is. 5. **Cite the source** even though you've written the proof yourself. Future-you needs to know where this came from. ### Arrangement Order results so that **no proof uses a result that hasn't been proved yet in this document.** This is the pedagogical heart of the paper. If you need Theorem B to prove Theorem A, then Theorem B comes first. This ordering often reveals the logical structure of the theory in a way that reading papers in publication order does not. ## Phase 3 — Connections After definitions and results are written up, add a section on connections: 1. **Which objects are "the same"?** Known bijections, isomorphisms, equivalences. State precisely. 2. **Which results are "the same"?** Theorems that are equivalent, or that are special cases of a common generalization. 3. **What's the picture?** Draw a diagram showing how the main objects and results relate. This can be informal — a hand-drawn map of the territory. 4. **What's missing?** What questions are natural from the exposition that don't have known answers? These are potential research directions. ## Phase 4 — Open Questions List questions that arise naturally from the exposition. For each: - State the question precisely - Say what a positive answer would imply - Say what a negative answer (counterexample) would look like - Note any partial progress ## How to Use This Document ### When stuck on a proof Come here. Don't work on the proof. Work on the expository paper instead. Pick a known result you haven't written up yet and write its proof in your own words. Or draw a better diagram. Or find a new example. The understanding you build will carry back to the proof. ### When starting a new problem Read the expository paper first. Everything you know is here, organized and proved. The tool you need might already be in your toolbox — you just need to see it. ### When a result turns out to be wrong Update the paper. The expository document must always reflect your current best understanding. Outdated results in the expository paper will poison future work. ## Rules - **No hand-waving.** If a step in a proof is "clear" or "follows easily," write it out anyway. You are writing for yourself on a bad day, when nothing feels clear. - **No proof by citation alone.** You may cite sources for reference, but every proof must be written in your own words in this document. If you can't write it, you don't know it. - **Dependency order is sacred.** If you catch yourself using a concept before defining it or a result before proving it, stop and restructure. The ordering IS the understanding. - **Diagrams are not optional.** If an object can be drawn, draw it. Combinatorics is a visual subject. The diagram often contains the proof. - **Examples are not optional.** Every definition and every theorem gets at least one concrete example with specific values. Abstract understanding without concrete instances is illusory. - **This document is never finished.** It grows with your understanding. Return to it regularly. The best insights often come while writing up something you thought you already knew.