Approach

AI as an instrument, not an oracle.

We believe artificial intelligence does not replace mathematical judgement. It widens the reach of our work, letting us test more ideas, notice new connections and check more objects. On this page we explain how we think about this work and how it happens in practice.

Two stories

Productivity and discovery

There are two ways to describe what AI can do for mathematics. We work with both, and the second is where we put most of our effort.

The productivity story

search · summarise · code · format · organise · communicate

AI can make everyday research lighter, helping with reading the literature, writing code and LaTeX, and proofreading. These tasks can be delegated safely when the cost of a mistake is low.

The discovery story

conjecture · compute · prove · verify · generalise

AI can also widen what is possible in mathematics, approximating objects with no known formula and revealing patterns and examples that would be hard to find by hand. This kind of use calls for careful verification at every step.

The loop

From question to verification, and back again

Our projects usually go through the same five stages. Each one leaves a concrete result, and the last one hands back a question sharper than the one we started with.

  1. Question

    It all starts with a precise mathematical question, such as an invariant we want to understand, an object to find, an equation to solve or a conjecture to test.

    statement and what is known

  2. Representation

    We translate the objects into data, such as weights, point samples on a manifold, graphs or coefficients, and choose what to measure, whether a loss function, a score or an invariant.

    data and functional

  3. Experiment

    We train models, run searches or compute approximations, with tools such as neural networks, PINN-style losses, symbolic regression, metaheuristics and language models.

    reproducible code

  4. Interpretation

    We go from the numbers back to mathematics, trying to understand which variables matter, which formula fits the data and which pattern suggests a conjecture.

    conjectures and candidates

  5. Verification

    We certify everything we can, using exact arithmetic, computer algebra, error estimates, formal proof or traditional proof, and we always make clear what is evidence and what is a theorem.

    result and provenance

Four modes

The four modes, with examples

This is a practical way to think about where AI can contribute to a problem. Many projects combine more than one mode.

02

Predict

Learn invariants from data and formulate conjectures

When a large database of examples exists, supervised models learn to predict invariants that are expensive to compute. Symbolic regression and model analysis turn good predictions into candidate formulas and conjectures.

In our work

04

Assist

Language models in day-to-day research

AI assistants and agents already make many parts of research easier, such as literature search, programming, writing in LaTeX and organising the work, and they are starting to take part in computations and proofs. Our rule is to always verify in proportion to what is at stake.

In our work

Principles

How we work

  1. Mathematics first

    We always start from the question, not from the model. AI comes into the work when it widens what we can compute, test or find.

  2. Evidence is not proof

    Numerical results suggest, and proofs establish. We always make clear which is which, and the rigour of verification matches what is at stake.

  3. Provenance is part of the result

    We record code, data, random seeds, models and prompts, and publish them together with the paper.

  4. Open by default

    We keep code and datasets public, so that others can reproduce, critique and build on our work.

  5. A partnership of equals

    Whoever brings the question leads the mathematics, and we help with the method. We agree on credit and expectations openly from the very first conversation.

Diagnosis

Is your problem a good fit?

Good signs

  • You can generate many examples, or a database, table or classification already exists.
  • There is a computable quantity that measures success, such as an invariant, an energy, an error or a functional.
  • You are looking for rare objects in a very large space, such as extremal examples, counterexamples or special configurations.
  • You need approximate solutions of PDEs or of variational problems with no explicit solution.
  • You suspect a pattern, but there is too much data to examine by hand.

Still challenging

  • Purely conceptual questions, with no examples that can be computed.
  • Problems where even a single example is out of computational reach.
  • Expecting AI to produce a complete proof on its own. For now, it mostly helps find the way.

If you are unsure, please write to us anyway. Helping to shape the question is part of our work.Propose a problem

Collaboration

How a proposal works

See a filled-in example proposal

  1. We receive your proposal

    As soon as you send the form, a confirmation appears on screen. The proposal and the PDF, if any, are available only to the lab team.

    immediate

  2. Standard format and preliminary score

    With the help of a language model (Claude), we organise the problem into a proposal in a standard format, which we will later send you by email. The model also gives it a preliminary score based on impact, feasibility, time, computational cost and available data, and the scores form a ranking. Since each standardized proposal consumes tokens, a paid and limited resource, CBG members have priority at this stage.

    automatic · CBG priority

  3. Reading and assessment by the team

    Our team reads the proposals carefully, following the order of the ranking. This stage is done by people and usually takes a few weeks.

    a few weeks

  4. We get in touch

    We reply by email with our decision and, when it makes sense, with an invitation to talk.

    by email

You don't need to know AI. You need a good question.

If you work in geometry or in any other area of mathematics and have a problem with many examples, a hard computation or a pattern you can't yet explain, let's talk.

Propose a problem