Digital Tools and AI for Contemporary Mathematical Productivity
Date:
Talk at Brazilian Center for Geometry (CBG), Campinas, SP, Brazil
From reducing friction to changing the research frontier.
There are now two AI stories in mathematics, and a serious workflow needs both without confusing them: a productivity story — search, summarize, code, format, organize, communicate — and a discovery story — conjecture, compute, prove, verify, generalize. The first promise of AI was to remove friction from research; the newer question is whether it can also change the mathematics itself.
The talk opens by tracing how quickly the frontier moved: formal silver at the IMO with AlphaProof and AlphaGeometry 2 (2024), natural-language gold (2025), research agents such as AlphaEvolve, Aletheia and First Proof (2025–26), and then genuine open problems in 2026 — the refutation of the grid-optimality intuition for planar unit distances, and a counterexample to the Jacobian conjecture for \(n \geq 3\). These are a threshold, not the end of the story: a few verified examples establish possibility, but not yet reliability, breadth, taste, or autonomous theory-building. The question is no longer “can a language model do mathematics?” but “which mathematical workflows make it capable — and trustworthy?”
The second half is practical. Everyday workflows remain the most reliable gain, so tasks should be chosen by leverage against the cost of a wrong answer: literature triage and formatting can be delegated freely, while computations, proofs, and citation claims must be verified heavily. I discuss using AI as a funnel rather than a substitute when reading, accelerating revision rather than authorship when writing, delegating form while retaining semantic control in LaTeX and slides, and building a durable stack — files, version control, reproducible environments, a reference manager — before adding an AI layer, since AI amplifies the quality of the system beneath it, including its disorder.
Throughout, the recurring themes are that prompting is specification, that one should ask for adversarial work rather than agreement so that uncertainty becomes legible, that verification should scale with mathematical stakes, and that provenance — model, prompts, code, seeds, failed attempts, verification steps — is part of the mathematics. The closing argument is that the durable insight is about attention: AI can expand the number of ideas tried, connections noticed, and objects checked, long before it becomes an autonomous mathematician. The goal is not to automate judgment, but to spend more of it where it matters.
Outline
- Two AI stories in mathematics: productivity and discovery
- How the frontier moved, 2024–2026, and what the case studies do and do not show
- Benchmarks as evidence, not research
- Choosing tasks by leverage and cost of error
- Reading, writing, LaTeX and talks: delegating form, retaining intent
- Prompting as specification; adversarial prompting
- Verification proportional to stakes; provenance and privacy
- A seven-day experiment and a minimum viable system for a mathematician
- What should remain unmistakably human
The talk closes with an invitation to MathIA AmLat, a community for researchers, students and educators interested in the uses, limits and implications of AI in mathematical research, education and collaboration.