AI Learned Geometry. Next It Needs Topology.

I predict that by the end of 2028, topology will be recognized as one of AI’s major new architectural variables. In at least one structurally demanding domain, a competitive system will learn a changing internal map of which parts connect, which parts touch, and how larger groups form. That structural map will constrain geometry and be revised by it.

Attention (central to most generative AI) can compare nearly every element, yet still miss global structure: what belongs together, which boundaries matter, and whether the result forms one coherent whole. Research portfolios set now must test that gap. Waiting risks dependence on systems that reproduce metric detail while violating global constraints. Investing without a hard benchmark risks burning scarce compute on elegant mathematics.

Attention has a fixed frame

Most current generative AI is metric first. It represents information as lists of numbers and learns by adjusting distances, angles, weighted similarities, and other continuous relationships to reduce error. Probability then assigns likelihoods to outputs within that learned space.

Attention expanded this approach from mostly nearby interactions to global, input dependent geometry. Under full attention, every token can sit on a nearly complete graph and receive a changing weight from every other token. Those weights can express rich relationships, but the underlying relational support is largely fixed. Masks and sparse attention can alter it, usually through design choices or separate rules. Transformers don’t literally lack topology. They generally don’t learn it as a semantic state that changes alongside the representation.

Structure, realization, and likelihood

Topology asks which parts connect, which groups form components, where cycles and boundaries exist, and which structural equivalences survive geometric deformation. It distinguishes a change in measurements from a change in structural class. Geometry specifies the concrete realization, including positions, distances, and angles. Probability says which realizations are likely.

Dominant architectures tend to embed structure implicitly inside geometry and probability. I expect future systems to factor these three languages more explicitly. Topology would describe allowable structural forms, geometry would realize them, and probability would rank the resulting possibilities.

Why the pressure is building

The first pressure comes from uneven global coherence. A model can use global attention and still produce precise details that don’t satisfy a global structural constraint. More context helps comparison, but comparison alone doesn’t ensure that all local choices belong to one consistent form.

A concrete example is the fatal 2016 Tesla Autopilot crash in Florida. Tesla said neither Autopilot nor the driver noticed the white side of a tractor trailer against a brightly lit sky, so the brake was not applied ¹. The system was managing lane position and speed while failing to distinguish the truck from the visual background. This was not a topology experiment, and topology is not a proven fix. It illustrates the narrower distinction between handling local relationships and maintaining a coherent model of the whole scene.

That gap creates a role for a different kind of compression. Geometry preserves measured relationships. Topology can preserve connectivity, components, cycles, and structural equivalence while ignoring perturbations that don’t change those properties. This invariance is selective rather than universal, which is precisely why it complements geometric detail.

The demand grows as tasks involve relationships among groups and across scales. Pairwise similarity can say that two entities are strongly related. It doesn’t necessarily capture when a group should be treated as one structural unit. In code, this could concern a dependency boundary. In molecular or physical representations, it could concern whether relations form an allowable configuration. Relational systems pose the same issue when group organization matters independently of strong individual links.

Computational integration is also becoming more plausible. A 2025 arXiv preprint proposes a neural network framework based on topology for images, graphs, and curved spaces, and reports improved performance especially in tasks that depend on local and hierarchical structure ². That doesn’t establish a new dominant architecture, but it shows that trainable topological machinery can operate across different data structures.

The largest opportunity follows from that integration: topology can become causally active rather than remain a diagnostic or auxiliary feature. The PFlow-T arXiv preprint describes a generative process that progressively removes topological features such as holes instead of adding random pixel noise. It reports better control over the number of components and holes, plus better handling of unfamiliar examples, on low resolution digit images ³. This is emerging evidence that topology can shape generation itself, although it remains far from a general result.

From topological features to topological state

Existing methods already touch the problem through graph neural networks, topology-aware measurements and training objectives, diffusion models informed by topology, and learned sparse graphs. Many still fix the topology in advance or use it as an extra signal. A meaningful topology first architecture would let structural state determine computation independently of fixed metric weights.

I encountered a narrow version of this distinction during my graduate physics research. An arXiv preprint reporting that work used Mapper, a method for grouping similar inputs and recording how the groups connect, to organize compressed versions of images before classifying them. On two standard image datasets, MNIST and FashionMNIST, it reported stronger classification performance under selected gradient based attacks than conventional image classifiers ⁴. The Mapper classifier did not expose a usable gradient, which made the selected gradient attacks ineffective ⁴. This was a modest proof point about preserving useful global information under perturbation. It wasn’t evidence that learned topology improves general reasoning or production robustness.

The missing architecture would make the proposed jump explicit:

CURRENT DOMINANT PARADIGM
Input → Continuous geometric representation → Weighted local or global interactions → Output

POSSIBLE NEXT PARADIGM
Input → Geometric state ⇄ Topological state → Output
        exact realization    allowable structure
              iterative coupling

In such a system, geometry would suggest possible connections, groups, or boundaries. The topological state would accept, reject, or reorganize them, then constrain the next geometric update. Probability would score the resulting realizations. This loop could run during inference or generation, allowing a structural hypothesis to change as the model encounters conflicts. Topology becomes causally active only when changing it alters computation beyond what changing metric weights would achieve.

The objections set the test

Geometry has been remarkably successful, and a sufficiently large geometric model may learn the necessary invariants implicitly. Explicit topology also discards geometric detail. The correct connectivity may be unknown, discrete structural changes are awkward for gradient descent, and searching over possible structures can be expensive. A mismatched topology can add complexity without helping the task.

The current evidence keeps those objections alive. PFlow-T is limited to selected topological targets and low resolution digit images ³. My Mapper work covers narrow datasets and an attack-specific mechanism that did not expose a usable gradient ⁴. Neither establishes lower hallucination rates, broader reasoning gains, production robustness, or favorable scaling economics. Limited benchmarks also make it difficult to tell whether an improvement comes from topology itself or another architectural choice.

That is why I’d fund a bounded research track now with a strict continuation criterion. Compare a model that updates topology during inference or generation against both a geometry only baseline and an otherwise identical system with frozen topology. Match compute, define the structural coherence or unfamiliar-example measure before training, and require an ablation showing that the changing structural state caused the gain.

By the end of 2028, my prediction earns support only if that test produces repeatable gains in at least one structurally demanding domain. If it doesn’t, topology should remain a useful specialist tool. If it does, AI architecture will have gained a complementary structural language: geometry for exact realization, topology for allowable form, and probability for what comes next.