Everything Is A Math Wrapper, Part I: Topology For SEOs And AI/LLM Search Spaces

A few weeks ago I wrote about a helpful mental model for SEO (and marketers for that matter): the idea that your content, web pages, websites and other items published online exist on “surfaces”.

Mentioned in that post, you’ll often see the word “surface” common from AI/LLM platforms and related media.

While this can be thought of as content “surfacing” in different search moments or even the actual device surface that a user uses, looking closer at the internal mechanics of these platforms, the term “surfaces” can represent something a bit more technical – mathematical.

Tracing how information gets ingested into these spaces and eventually retrieved through pre-training, during inference or assisted with Retrieval Augmented Generation, when you abstract away the finer engineering details, you can see how that information gets embedded and exists on a surface – a manifold-like structure.

A manifold – loosely speaking – is a generalization of our usual 2D or 3D – and higher dimensional space (called “Euclidean”). It can be thought of as a collection of these spaces — when you zoom in on any manifold, they locally look exactly like a traditional Euclidean space.

Mentally, this might be hard to imagine (it’s hard for humans to imagine any space beyond our usual 3 dimension for that matter) so here’s an example:

When you look at the Earth from space, globally it looks like a sphere — for us locally, it looks flat. Every point on the 3D sphere (curved) looks 2D (flat). You can extend that to any dimension you’d like, but the point here is that the Earth is a loose collection of 2D spaces – flat everywhere at a point, but collectively form a curved sphere.

These manifold-like surfaces aren’t static in any way inside AI/LLM search spaces – they shift, bend and twist depending on the context of the retrieval and response during a prompt/search.

On top of context related shape-shifting, each model and related flavors has its own inherent geometry/shape based on how they were engineered and trained (and eventually fine-tuned in some cases). This shape-shifting contributes to the measurement and reliability challenges – very small changes in inputs (prompts/queries) and surrounding context can result in much different internal shapes and representations being formed when outputs are retrieved (responses) from those spaces.

We’ll set aside measurement for a bit (we’ll get to that soon), but before I push further I wanted to give a thoughtful rundown – a primer, of sorts – on the ambient spaces that are manifolds, which are part of the larger branch of mathematics called topology.

Topology is much too dense to explore in a single blog post (or even textbook), but what I hope to achieve here is a light introduction for SEOs and marketers (and any other readers that happen to be following along).

An Informal View Of Topology

“Rubber sheet geometry”.

This is likely the most common (perhaps cliche) alternative name for topology. It’s exactly how it sounds — you can take a piece of rubber and bend, twist and stretch it any way you’d like and the points on the sheet remain close to each other (continuous – more on that below).

Another cliche example here is “there is no difference between a donut and coffee mug in topology”. Like a ball of clay (loosely), you can slowly bend, twist and reshape a donut until it becomes a coffee mug (and vice-versa) – all without tearing or gluing pieces onto it.

The shape of these spaces matters less than how the shape changes shape – and how points in that space are related to each other, essentially.

A (Semi) Formal View Of Topology

A bit more formally, a topology is related to Set Theory — that is, take any set “S” (any collection of elements) and look at a family of subsets of that set, “τ” (a collection of sets of the set “S”).

If those subsets have the following properties, it is said to be a topology:

1. Both the empty set and “S” are elements of τ

2. A union of elements of τ is an element of τ

3. Any intersection of a finite amount of elements of τ is an element of τ.

Definitions of “empty set”, “union” and “intersection” can be explored elsewhere (and can be loosely inferred intuitively), but if τ is a topology on S, then (S, τ) forms a topological space.

What this is really saying is how things relate and are organized based on closeness or proximity.

If you combine two (or any amount of) elements of a set τ, the element that is formed by that union is also an element of that set τ.

If you look at how two (or any amount of) elements overlap (have common elements) that element formed by the intersection is also in that set.

This invariant nature – that is, those new elements formed by combining or overlapping elements, remain part of the same topology set hints at the invariant idea in the informal description above.

If you map one topological space to another topological space and the underlying topological properties are preserved, it’s what is known as a homeomorphism — this is the twisting, stretching or bending – deformation – mentioned above.

There is much, much more to the world of topology (continuity, compactness, connectedness, Hausdorff properties and metric spaces, to name a few important concepts), but these are the essential ingredients for what we need for a specific class of topological spaces we want to understand: manifolds.

Manifolds Are Topological Spaces

Coming all the way back around to manifolds “M”, as mentioned above these are topological spaces where the topology τ (a family of subsets, remember) is a set of Euclidean spaces ( like 2D and 3D spaces, for example ).

It might be difficult to imagine local points in space as spaces themselves, but that’s what makes this level of abstraction fun (to me anyway) – these are complex structures, but locally they’re quite simple in nature.

If a manifold is of “n” dimension, then locally it looks like an “n” dimensional Euclidean space, essentially.

A sphere, locally, is flat (2D), so it’s called a 2-manifold — it’s flat everywhere on its surface so the dimension is constant. One can map the Earth using a collection of flat 2D charts (which has a more technical definition) that collectively form an atlas (which, again, has a much more formal definition in topology).

Looking at the above formal definition of topology, you can see how sets of charts that form an atlas that covers the Earth is similar to the set properties mentioned – charts can overlap and be combined, but ultimately are part of the same topological space that is the manifold.

Connection To AI/LLM Search Spaces

So what does all of this have to do with AI/LLM search spaces?

Going back to last week’s post I mentioned how embedding spaces – the spaces formed by vectorizing sub-words, words, phrases, pages and other content online (the input manifold) and contextualized by the transformer and attention mechanisms in AI/LLM spaces for outputs (the output manifold) – collectively exhibit manifold-like properties.

The input manifolds are stretched, twisted, rotated, bent and other kinds of deformations as a prompt and related context are entered and pass through the transformer and attention mechanism to map to an output manifold (and eventually retrieve a response from that output manifold).

Ideally similar inputs and related input context matched with the input manifold would map to similar outputs in the output manifold, but the inherent, inconsistent contextuality that can warp, rotate or otherwise deform the space in the process can cause the outputs to change (sometimes dramatically).

Realistically, there is much more to these inconsistent responses, but this can be loosely imagined this way. One small change in word choice or context can autoregressively alter (or, recursively retrieve in a different direction) a response.

Final Takeaway And “Surviving Deformation”

No gimmicks, no funny business.

What you should takeaway from reading this is that mathematically – or topologically – the time for cheap or gimmicky tactics are coming to an end (mostly).

Search was likely much more forgiving in the time before transformers, but these days creating a high fidelity representation of you, your content, your website, business and other entities – and managing the context around that representation – is vital for surviving these new spaces and the deformations that occur during retrieval.

Any odd or strange gimmick, or mismatch in context you add to the digital space – and eventually bound to your digital representation – means more noise the decision mechanism in these spaces will have to fight through to understand if you’re relevant to particular contexts.

Keep it clean, keep it sharp – and choose quality over quantity.

Next week I’ll dig even further into some of the finer-grain mathematical concepts inside these spaces to help you understand them a bit further as I continue this series.

Stay tuned.