CBG – AI & Geometry Lab

From ruler and compass to neural networks.

We are the artificial intelligence lab of the Brazilian Center for Geometry. We use machine learning to approximate, predict and find geometric objects, and we help mathematicians across Brazil do the same in their own research.

Propose a problem How we work
Gradient descent of the Willmore energy W over tori of revolution, with points coloured by mean curvature. The ratio c = R/r converges to √2, which corresponds to the Clifford torus, with W = 2π². More details in E. Hirst, H. N. Sá Earp, T. S. R. Silva, “Minimising Willmore energy via neural flow”, arXiv:2604.04321 (2026).

A lab of CBG

Part of the Brazilian Center for Geometry

GeometrIA is the artificial intelligence lab of the Brazilian Center for Geometry (CBG), a Research, Innovation and Dissemination Center (CEPID) funded by FAPESP and hosted at IMECC/UNICAMP. CBG brings together researchers from several institutions in four thematic verticals, and the lab works where each of them meets AI.

  • Differential Geometry
  • Algebraic Geometry
  • Applications of Geometry
  • Education, Outreach and Innovation
About CBG

What we do

Research, bridge and community

01Research

We develop machine-learning methods for geometry, such as metrics and differential forms approximated by neural networks, topological invariants learned from data, and heuristic searches for rare combinatorial objects.

See publications

02Bridge

We talk with mathematicians who don't work with AI to find out, together, whether these tools can help their research and how. You bring the question, and we help design the experiment.

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03Community

We have a channel on Zulip, MathIA AmLat, where Latin American mathematicians discuss and follow news on AI and mathematics. Zulip is a free, open-source chat app organised by topic, like a real-time forum.

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Four modes

How AI enters mathematics

In practice, almost every AI-for-mathematics project fits one or more of these four modes, and all of them appear in our own work.

01

Approximate

Neural networks as approximators of geometric objects

Many geometric constructions are solutions of PDEs or minimisers of functionals with no closed form. Networks trained with losses that encode the equation itself (PINN-style) produce numerical approximations of metrics, forms and surfaces.

G2 3-forms on Calabi–Yau links · neural Willmore flow · black-hole metrics

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.

properties of finite groups learned from 131,406 Cayley graphs

03

Search

Find rare objects in huge spaces

Extremal examples, counterexamples and special configurations are often hidden in enormous combinatorial spaces. Metaheuristics guided by a well-chosen functional find candidates, which are then certified in exact arithmetic.

a new brane tiling with 26 fields · 6,146 certified free line arrangements

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.

CBG talk on digital tools and AI for mathematical productivity

Approach

A loop, not a black box

Every collaboration follows the same loop, going from the mathematical question to verification and then back to a sharper question.

  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

…and the loop starts again, now with a sharper question.

Read our approach

Research

Recent work

All publications

Approximate2026

Minimising Willmore energy via neural flow

E. Hirst, H. N. Sá Earp, T. S. R. Silva

The neural Willmore flow recovers the round sphere (genus 0) and the Clifford torus (genus 1), and opens a new approach to the open genus-2 case.

arXiv:2604.04321

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.

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