Two theorems found by machine pattern-spotting in knot theory and representation theory
Networks trained to predict one mathematical invariant from another, then probed for which inputs mattered, pointed mathematicians to a proved theorem linking a knot's signature to a new geometric quantity, and to structure behind the combinatorial invariance conjecture.
- Model
- Supervised networks with gradient-based attribution
- Field
- Mathematics
- Date
- 2021-12-01
- Human collaborators
- Alex Davies, Marc Lackenby, Geordie Williamson
Sources
Original work
Announcement
Challenge
What was found
The recipe was to train a network to predict one invariant from another, then use attribution to see which inputs carried the signal, and hand that to a mathematician. In knot theory it led Marc Lackenby to define a new quantity, the natural slope of a knot, and to prove a theorem bounding the signature in terms of it. In representation theory it led Geordie Williamson to structure in Bruhat interval graphs bearing on the combinatorial invariance conjecture for Kazhdan–Lusztig polynomials. Published in Nature.
Novelty check
Both theorems are new and neither had a prior form in the literature; both were proved conventionally by the human mathematicians after the models indicated where to look. The general idea of using computation to generate conjectures is old; what was new was the attribution step turning a black-box predictor into a usable hint.
Caveats and known objections
The mathematics was done by humans; the models indicated which relationships were worth studying, which is why autonomy is ai-assisted. Ernest Davis's review argues the "guiding human intuition" framing overstates the contribution and that the knot-theory signal was within reach of simpler statistical methods. The theorems themselves are not disputed.
Independent checks
Ernest Davis (critical review of the framing): theorems accepted; disputes how much the deep learning contributed · link ↗
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Entry history (1 event)
- AddedEntered the registry graded Peer reviewed and AI-assisted.
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Graded peer reviewed for verification and ai-assisted for autonomy. What these mean.
Cite this entry
whataifound.org. (2021). Two theorems found by machine pattern-spotting in knot theory and representation theory. whataifound.org: A Registry of AI Scientific and Mathematical Discoveries. https://whataifound.org/finding/2021-12-01-knot-theory-intuitionBibTeX
@misc{whataifound-googledeepmind-2021-intuition,
title = {Two theorems found by machine pattern-spotting in knot theory and representation theory},
author = {{whataifound.org}},
year = {2021},
howpublished = {whataifound.org: A Registry of AI Scientific and Mathematical Discoveries},
note = {Result by Google DeepMind (with Oxford and Sydney). Verification: Peer reviewed. Autonomy: AI-assisted.},
url = {https://whataifound.org/finding/2021-12-01-knot-theory-intuition}
}