Studying advanced mathematics with AI

When you study a proof, the difficult part is often knowing which definition is in force, which earlier result a step uses, or where a hypothesis enters. Manifield keeps those pieces in a library and a dependency graph. Its optional AI assistant, Prismo, can read the same items while you work.

This workflow suits university mathematics, graduate study and reading a paper in your research area. Start with notes you can check against a course, a textbook or the original paper. You need a Mac with macOS 15 or later; the LaTeX graph tool also works in your browser.

#Start with a small set of notes

Follow the setup guide and open the included Baire category example, or create a project for the topic you are studying.

If you bring your own .tex file or PDF, choose Import into the library to ask the Organizer to propose separate definitions, lemmas and theorems. Review the statements and dependency links against the original before saving them. Importing this way uses an AI model. You can also write items by hand with New Item.

Keep your notation with the project. Two authors may use the same symbol for different objects, and a familiar theorem name does not fix its assumptions. A project has a notation file and LaTeX macros so that your definitions stay in view. The project folder explains the files.

#Read the dependency graph alongside the proof

Choose a theorem you want to understand. Open its item in the library and use the Dependency Graph to find the earlier results it cites. Read the immediate prerequisites first, then follow their prerequisites when a step still feels unexplained.

For example, the supplied closed graph theorem draft uses the bounded inverse theorem. That result uses the open mapping theorem, which in turn rests on earlier lemmas. Following those links gives you a reading order through the notes.

Treat the links as part of the working notes. Imported dependencies can be wrong or incomplete; check them against the actual proof. You can explore the idea without installing the app in Euclid's dependency graph.

#Ask questions that point to your actual definitions

Connect a supported provider in Settings → AI Providers, then try a focused question in Ask mode. Ask mode lets the assistant read and discuss your project.

For the included example, a useful starting prompt is:

Read thm-closed-graph and its immediate prerequisites. Explain why the graph is a Banach space and where completeness of the two original spaces is used. Cite the project items you use.

Other questions you can adapt to your own notes:

• Which hypothesis is used in this step? Point to the definition or earlier result.
• Explain this argument in the notation of my project, and identify any step the notes leave unwritten.
• What changes if this assumption is removed? Distinguish a counterexample from a reason the current proof stops working.

When an answer cites an item, open the link beside the conversation and read it. Check that the cited statement has the hypotheses the answer needs. Prismo can make mathematical mistakes, including mistakes in the way it applies a correct theorem.

#Work through one gap at a time

Before asking for a complete proof, identify the step you want to fill. In Propose mode, ask the assistant for a specific edit and inspect it before accepting.

The Baire example contains a draft of the closed graph theorem with a step labelled Still to write. The worked example explains that gap and the norm estimate needed to fill it. You can compare the explanation with the original files and then attempt the step yourself.

A useful question after seeing an argument is: “Which part of this estimate uses the definition of the norm, and which part uses a theorem?” That helps reveal circular reasoning, where a proof quietly assumes the conclusion it is trying to establish.

#Keep the evidence visible

An item marked Proof drafted still needs review. A computational check can provide evidence for examples, and an AI review can report Review found no errors. Human approval and formal verification have their own statuses, set by you. Manifield currently has no Lean integration.

Prismo can run small Python checks after you set up the Python environment. A counterexample can settle a proposed universal claim; a finite search that finds none leaves the general question open. The app keeps the script and output with the project so you can inspect what was actually tested.

Attach the books and papers you rely on as sources. The assistant can then cite their imported statements alongside your notes. You control access to their full text.

#What is free, and what uses a model

Manifield is free and open source. Reading and writing library items, browsing the graph, searching, and keeping notation work without AI. The browser tool for LaTeX dependency graphs is also free and processes files locally.

AI-assisted imports and conversations use your own provider account or a ChatGPT plan's included usage. Costs and limits depend on that provider or plan. Experimental research mode can consume many tokens; ordinary study can begin with a small project and focused questions.

Start with installation and the example project, or read what the assistant can do before connecting a model.