DeepMind's Co-Mathematician AI helps prove 15-year-old algebraic geometry conjecture

In a paper published on arXiv, mathematicians Gergely Bérczi and Young-Hoon Kiem have proven a long-standing conjecture about the Poincaré polynomials of the Deligne-Mumford moduli space \(\overline{\mathcal M}_{0,n}\), which parameterizes stable rational curves with n marked points. The conjecture, originally posed by Aluffi, Chen, and Marcolli, asserted that the polynomial \(P_n(t) = \sum_{i=0}^{n-3} \dim H^{2i}(\overline{\mathcal M}_{0,n};\mathbb{Q}) t^i\) has only real roots—a property that implies the Betti numbers form an ultra-log-concave sequence. The proof was achieved with substantial assistance from Co-Mathematician, an agentic frontier-model AI system developed by Google DeepMind. The AI helped explore bivariate deformations of the polynomial, leading to the discovery of a hidden interlacing structure that a standard one-variable recurrence had obscured. The human mathematicians guided the process, evaluated AI-generated attempts, and assembled the final human-verifiable argument.

The key technical innovation is a bivariate deformation \(F_m(y,t)\) of the Poincaré polynomial. For fixed \(t<0\), the zeros in the \(y\)-direction are controlled by a Sturm-Rolle argument on the interval \(0<y<1-t\). The original polynomial is recovered on the slice \(y=1\), and the ordered crossings of moving roots through this slice yield both real-rootedness and strict interlacing. The team also extended the same deformation strategy to prove analogous results for the Fulton-MacPherson space \(\mathbb{P}^1[n]\) of n ordered points in degenerations of the complex projective line. This work demonstrates how AI can uncover non-obvious algebraic structures and accelerate progress in pure mathematics, while still requiring human insight to frame problems and compile rigorous proofs.