Closing

Conclusions

What the framework buys us

The arc of the book runs from traditional neural networks, through the geometric framework of the 5 Gs, to convolution and attention as its two great families. The unifying claim is concrete: the differences between ANNs, CNNs, GNNs and Transformers reduce to three choices — the geometric domain, the symmetry group, and the connectivity structure. Equivariant message passing is the common implementation, and equivariant universality guarantees its expressive power.

Open problems

• The Weisfeiler–Leman barrier — standard message-passing networks inherit the expressive ceiling of the 1-WL test; surpassing it at polynomial cost is open.
• Rigidity versus adaptability — where to sit on the CNN–attention spectrum for a given domain has no general answer yet.
• Gauge equivariance and global topology — local gauge symmetry is elegant, but global topological obstructions make scalable implementations hard.
• Higher-order structures — architectures over simplicial and cellular complexes are still immature compared to graph networks.

Future directions

The book points toward hybrid CNN–attention architectures that interpolate between rigidity and adaptability, models over mixed-curvature and product geometries, sheaf neural networks, and generalised harmonic analysis beyond compact groups. The closing reflection is that geometry is not just a performance trick but an organising principle — a theory with predictive power for the principled design of future architectures, not merely a taxonomy of existing ones.