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Recently, Fei-Fei Li and Justin Johnson, co-founders of the newly established AI company World Labs, which has successfully raised $230 million in funding, featured in an insightful interview on the a16z podcast, hosted by partner Martin Casado. During the discussion, they revealed World Labs to be a deep-tech enterprise focused on developing spatial general artificial intelligence (AGI), with the ambitious aim of enabling machines to reason and operate within the physical world. The founders emphasised the profound significance of visual-spatial intelligence, a domain of AI that extends far beyond the capabilities of language comprehension alone. At its core, this level of intelligence encompasses a machine's ability to perceive, reason, and act within four dimensions—three spatial and one temporal.
To us, as mathematicians familiar with some historical interplay between mathematics and physics, this moment resonates with Einstein's key realisation that, in order to formulate his theory of general relativity, he needed to transcend the confines of Euclidean geometry. In his 1922 Kyoto address, Einstein reflected on this profound insight, stating:
"If all systems are equivalent, then Euclidean geometry cannot hold in all of them. To throw out geometry and keep laws is equivalent to describing thoughts without words. We must search for words before we can express thoughts. What must we search for at this point? This problem remained insoluble to me until 1912, when I suddenly realized that Gauss's theory of surfaces holds the key for unlocking this mystery. I realized that Gauss's surface coordinates had a profound significance. However, I did not know at that time that Riemann had studied the foundations of geometry in an even more profound way. I suddenly remembered that Gauss's theory was contained in the geometry course given by Geiser when I was a student...
I realized that the foundations of geometry have physical significance. My dear friend the mathematician Grossmann was there when I returned from Prague to Zurich. From him I learned for the first time about Ricci and later about Riemann. So I asked my friend whether my problem could be solved by Riemann's theory, namely, whether the invariants of the line element could completely determine the quantities I had been looking for".
It was Einstein's close friend, Marcel Grossman, who introduced him to Bernhard Riemann's pioneering work in differential geometry, which ultimately provided the mathematical framework for expressing how mass warps spacetime. Just as differential geometry revolutionised our understanding of the universe, we believe it holds immense potential for breakthroughs in spatial AGI, as described by Fei-Fei Li and Justin Johnson. AI researchers aiming to unlock such intelligence could greatly benefit from the matured structures of differential geometry, differential topology, and algebraic topology. Interestingly, category theory offers a powerful mechanism for connecting these mathematical fields.
Just as Einstein recognised the need to explore new mathematical tools, today’s AI researchers must similarly adopt advanced mathematics to enhance the development of systems that could potentially bring us closer to AGI—capable of reasoning and acting in 3D and 4D space-time. While deep learning has transformed fields such as computer vision and natural language processing, the next frontier could indeed be spatial intelligence as described by Fei-Fei and Justin. However, only by harnessing sophisticated mathematical frameworks can machines truly navigate and comprehend the complexities of the real world in its full dimensionality.
Finally, as spatial intelligence becomes more central to the development efforts of AGI, the application of advanced branches of mathematics will be vital in guiding this research. The algorithms that will power such systems must draw on ideas that are not only computationally feasible but also geometrically rich, capable of interpreting and interacting with the fabric of reality itself. Just as differential geometry once bridged gaps in physics, it now holds the key to advancing AI toward spatial reasoning in both virtual and real-world contexts.
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Best wishes,
The QF Team
Differential Geometry Course
If your AI research team needs an introductory crash course (2-4 weeks) in Differential Geometry with emphasis on Smooth Manifolds and their potential applications in AI, feel free to contact us at hello@mathematicaltalent.com or visit https://mathematicaltalent.com/mathematical-upskilling to learn more about our upskilling program. All courses are taught by PhD-level mathematicians, combining mathematical rigour with practical applications. For researchers working on open-source projects involving Measure Theory and Probability, we offer free credits as a way to support the accessibility of their work to the broader community.
Free Mathematical Proof Writing Course
We’re excited to announce a special two-lecture crash course on proof writing. It will begin this coming Tuesday, September 24, at 5 PM BST.
