Gergely Bérczi

Gergely Bérczi, Honglu Fan, Mingcong Zeng

The solution set of a system of polynomial equations typically contains ill-behaved, singular points. Resolution is a fundamental process in geometry in which we replace singular points with smooth points, while keeping the rest of the solution set unchanged. Resolutions are not unique: the usual way to describe them involves repeatedly performing a fundamental operation known as "blowing-up", and the complexity of the resolution highly depends on certain choices. The process can be translated into various versions of a 2-player game, the so-called Hironaka game, and a winning strategy for the first player provides a solution to the resolution problem. In this paper we introduce a new approach to the Hironaka game that uses reinforcement learning agents to find optimal resolutions of singularities. In certain domains, the trained model outperforms state-of-the-art selection heuristics in total number of polynomial additions performed, which provides a proof-of-concept that recent developments in machine learning have the potential to improve performance of algorithms in symbolic computation.

Gergely Bérczi, Young-Hoon Kiem

We prove real-rootedness for the Poincaré polynomial \[ P_n(t)=\sum_{i=0}^{n-3} \dim H^{2i}(\overline{\mathcal M}_{0,n};\mathbb{Q})t^i \] of the Deligne--Mumford moduli space $\overline{\mathcal M}_{0,n}$ of stable $n$-pointed rational curves, proving a conjecture of Aluffi--Chen--Marcolli. The proof starts from the Keel--Manin--Getzler recurrence, but its main new idea is a bivariate deformation $F_m(y,t)$ of the Poincaré polynomial. This deformation reveals a hidden interlacing structure not visible in the one-variable recurrence. For fixed $t<0$, the zero set of $F_m$ in the $y$-direction is 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 the moving roots through this slice give both real-rootedness and strict interlacing. Consequently, the Betti numbers of $\overline{\mathcal M}_{0,n}$ form an ultra-log-concave sequence. We further prove real-rootedness and ultra-log-concavity for the Poincaré polynomial of the Fulton--MacPherson space $\mathbb{P}^1[n]$ of $n$ ordered points in degenerations of the complex projective line. The proof for $\overline{\mathcal M}_{0,n}$ was obtained through an iterative AI-assisted workflow with Co-Mathematician, an agentic frontier-model system developed by Google DeepMind. The human role was to pose the problem, evaluate successive attempts, request repairs of gaps, compare the evolving argument with the literature, and assemble the final human-verifiable proof. Our additional human contribution was to observe that a similar residual deformation strategy applies to the Fulton--MacPherson spaces $\mathbb P^1[n]$, yielding the corresponding real-rootedness theorem.

Johannes Schmitt, Gergely Bérczi, Jasper Dekoninck et al.

As the mathematical capabilities of large language models (LLMs) improve, it becomes increasingly important to evaluate their performance on research-level tasks at the frontier of mathematical knowledge. However, existing benchmarks are limited, as they focus solely on final-answer questions or high-school competition problems. To address this gap, we introduce IMProofBench, a private benchmark consisting of 39 peer-reviewed problems developed by expert mathematicians. Each problem requires a detailed proof and is paired with subproblems that have final answers, supporting both an evaluation of mathematical reasoning capabilities by human experts and a large-scale quantitative analysis through automated grading. Furthermore, unlike prior benchmarks, the evaluation setup simulates a realistic research environment: models operate in an agentic framework with tools like web search for literature review and mathematical software such as SageMath. Our results show that current LLMs can succeed at the more accessible research-level questions, but still encounter significant difficulties on more challenging problems. Quantitatively, Grok-4 achieves the highest accuracy of 52% on final-answer subproblems, while GPT-5 obtains the best performance for proof generation, achieving a fully correct solution for 22% of problems. IMProofBench will continue to evolve as a dynamic benchmark in collaboration with the mathematical community, ensuring its relevance for evaluating the next generation of LLMs.

Gergely Bérczi, Adam Zsolt Wagner

We apply a generative AI pattern-recognition technique called PatternBoost to study bootstrap percolation on hypercubes. With this, we slightly improve the best existing upper bound for the size of percolating subsets of the hypercube.