The Poincaré conjecture is simple to state: every closed, simply connected three-dimensional manifold is a sphere, up to continuous deformation. For a hundred years nobody could prove it. In 2002 and 2003 Grigori Perelman posted three preprints to arXiv that settled the problem, then turned down both the Fields Medal and the million-dollar Clay prize and went on living in his flat in Kupchino, on the edge of St Petersburg.
Now for a few uncomfortable observations.
A coherent thought behaves like a closed manifold
When a person thinks something through in sequence, building an argument, reaching a conclusion, returning to the start, their neural activity leaves a trace through state space. That trace can be recorded, and it is: in fMRI, in EEG, in neural embeddings.
Over the past decade topological data analysis has been gaining ground in computational neuroscience. The idea is simple: instead of treating brain activity as a vector in Euclidean space, you look at its shape, at connectivity, holes and loops, and at which of them persist as the scale changes.
A strange regularity surfaces. Coherent thought resembles activity that closes into a loop; anxious, scattered thought resembles activity whose ends never meet. Some studies suggest that in meditation this activity has fewer "holes" than in ordinary waking.
The Poincaré conjecture itself plays no direct role here. But its language turns out to fit what we observe, and that is already something.
What it means
A few studies worth knowing.
— Giusti and colleagues (PNAS, 2015): topological analysis of correlations between hippocampal neurons reveals geometric structure that ordinary methods miss. In this picture geometry grows out of topology.
— Saggar and colleagues (Nature Communications, 2018): the Mapper algorithm turns fMRI data into a graph in which brain states during different tasks fall into distinguishable regions, and transitions between them can be tracked in a single person.
— Northoff and Huang (2017), the "temporo-spatial theory of consciousness": consciousness as a structure with characteristic dynamics in time and space.
These data are under no obligation to prove that consciousness is, in substance, a topological manifold. What they show is something else: topology is a fitting language for describing how consciousness works.
The distinction matters. Ontology answers "what is", methodology answers "how to describe", and so far we stand on the second.
Why look at this at all
Computational neuroscience today is roughly where physics was at the end of the nineteenth century: plenty of data, no general model. Everyone knows that neurons fire and that their activity is coordinated. But what is coordinated, and how, is described at a level Rutherford would have called stamp collecting.
Topology is an attempt to give the future model a mathematical skeleton. If coherent thoughts are closed loops in state space, if lucid awareness corresponds to particular persistent homologies, if déjà vu is current activity crossing a previously recorded trajectory, then thinking becomes something you can define formally.
"Mathematics has explained consciousness" claims more than we have. The careful version is that mathematics offers a tool with which we may at last begin to measure consciousness.
Where the evidence runs out
To prove the Poincaré conjecture, Perelman relied on the Ricci flow, which Richard Hamilton had been developing since the 1980s. The Ricci flow describes how the metric on a manifold changes under the influence of curvature.
Some papers try to apply the Ricci flow to graphs of neural connections. It is an elegant and fashionable move, but a premature one: a mathematical analogy becomes a theory only when it makes predictions that can be tested.
So far the topology of mind has produced language, metrics and hypotheses. It has not produced a single prohibition, not one "if this holds, that cannot be observed", and without prohibitions a hypothesis does not become science.
We are in the phase where language runs ahead of theory. That is how a field usually looks before a breakthrough, or before discovering that no breakthrough is coming. The second outcome is a useful result too.