Gergely Bérczi 

I am an associate professor at Aarhus University and head of the Aarhus DFF Algebraic Geometry Group. 

Before this, I spent 2 years at ETH Zürich in the group of Rahul Pandharipande and 9 years in Oxford working with Frances Kirwan. I obtained my PhD in 2008 under the supervision of András Szenes.  

My research lies at the intersection of algebraic geometry, algebraic topology, global singularity theory, and, more recently, the emerging use of machine learning and AI in mathematical problem solving and reasoning. 

Join our growing Math&AI Zulip community. 

News and activities:

• ProofCouncil: https://arxiv.org/abs/2607.09474
  ProofCouncil  is an agentic harness we developed for a submission to the second batch of FirstProof, a challenge consisting of 10 real-world mathematical problems that agents must solve autonomously. Its submissions for 6 of the 10 problems were judged by the referees to be correct up to at most minor revisions, showing the best performance among participating teams.

• A beautiful and still emerging story in AI-assisted algebraic geometry https://arxiv.org/abs/2605.29151. 

• Two Google DeepMind papers on AI and mathematics.
  I am a co-author of DeepMind's new AlphaProof Nexus paper:
  I was also among early mathematicians who tested and gave feedback on DeepMind's AI Co-Mathematician system. 

• I organize (with Geordie Williamson and François Charton) the Aarhus Math&AI Workshop at AIAS, January 27-30 2026

• IMProofBench - An AI Benchmark for Mathematical Reasoning.  

• Invited speaker at the Matrix-MFO Mathematics and Machine Learning Workshop, Oberwolfach, 2025 September

• Invited speaker at the Harvard Mathematics and Machine Learning Reunion Workshop, Harvard CMSA, 2025 September

• Since February 2025 I am a contributor of Project Numina.

• I also work on EpochAI's FrontierMath benchmarks (currently Tier 4.5). Here’s a somewhat biased and overly enthusiastic article covering our Frontier Math weekend in Berkeley.

• Invited speaker at the KIAS Machine Learning and Mathematics workshop, KIAS Seoul, 2025 July

• I was an invited participant of the 2024 Mathematics and Machine Learning program at Harvard CMSA, Boston, and an invited speaker at the AMS Joint Math Meetings 2025 in Seattle, whose theme is Math&AI. 

• Since January 2025 I am the AI Chair for Aarhus University for the CircleU AI Knowledge Hub.

• From January - June 2024, I co-organised a semester at the Isaac Newton Institute Cambridge on New equivariant methods in algebraic and differential geometry. This included the workshop  Singularity theory and hyperbolicity and the workshop Moduli stack and enumerative geometry. 

• I co-organised the Algebraic Geometry Session of the 29th Nordic Congress of Mathematicians 3-7 July 2023 in Aalborg, Denmark.

• I co-organised a Swissmap workshop Enumerative geometry of the Hilbert scheme of points at the SwissMAP Research Station in Les Diablerets in January 2024.

Preprints:

• (with J Schmitt, T Gehrunger, J. Dekoninck, U. Kreitner, L. Price, D. Holmes)  ProofCouncil: An LLM Agent for Solving Open Mathematical Problems. https://arxiv.org/abs/2607.09474

• (with Y-H Kiem) Real-rootedness of the Poincaré polynomial of moduli of curves: an AI-assisted proof. https://arxiv.org/abs/2605.29151

• (with L. M. Fehér) Positivity in classical enumerative geometry: a case study in synchronized AI-assisted mathematics. https://arxiv.org/abs/2605.25271
  We develop a workflow combining AI tools with human insight. 

• The AlphaEvolve series
  A series of papers reporting on my experiments with AlphaEvolve, an evolutionary search model by DeepMind.
  Part 1: Evolving Ranking Functions for Canonical Blow-Ups in Positive Characteristic, arxiv.2602.06553
  Part 2: Evolving Local Corrections for Global Constructions in Combinatorics, arXiv:2603.06692 

• (with Bin Dong, Haocheng Ju, Tianyi Xu) Connected Theorems: A Graph-Based Approach to Evaluating Mathematical Results, arXiv:2508.17596

• (with Jonas Klüver and Baran Hashemi) Flow-based Extremal Mathematical Structure Discovery, arXiv.2601.18005
  We introduce FlowBoost, a flow-based evolutionary search model. 

• (with Felix Minddal) Virtual invariants from the non-associative Hilbert scheme, arxiv.2512.11538
  We introduce a non-associate Hilbert scheme model as a smooth ambient space for the Hilbert scheme of points on affine spaces, and develop virtual classes and intersection theory. 

• (with Jonas Klüver) Reinforcement Learning the Chromatic Symmetric Function, arXiv:2410.19189
  Motivated by the Stanley-Stembridge positivity conjecture, we propose a conjectural counting formula for the coefficients of the chromatic symmetric function of unit interval graphs using reinforcement learning.

• (with Jonas Svendsen) Fixed point distribution on Hilbert scheme of points, arXiv:2306.11521
  We answer an old question by showing that all monomial ideals (i.e torus fixed points) on the punctual Hilbert scheme of points of the affine space A_k^n sit in the curvilinear component. This implies that the punctual Hilbert scheme is connected, which was only known for the full Hilbert scheme due to Hartshorne. We prove analogous results for the Quot scheme of points in higher rank. 

• Tautological integrals on Hilbert scheme of points I. arXiv:2303.14807
  Tautological integrals on Hilbert scheme of points II: Geometric subsets, arXiv:2303.14812
  Tautological integrals on Hilbert scheme of points III: Applications, in preparation
  We aim to develop a new method to study intersection theory of the Hilbert scheme of points on complex manifolds. The main result is an iterated residue formula for tautological integrals over the main component and certain geometric subsets with applications in classical enumerative geometry We also formulate a Chern-Segre-type positivity conjecture.

• Multiple-point residue formulas for holomorphic maps (with A. Szenes), arXiv:2112.15502
  We develop a formula for the multipoint loci of holomorphic maps between complex manifolds, and prove a conjecture of Kazarian and Rimányi.

• Non-reductive geometric invariant theory and Thom polynomials, arXiv:2012.06425
  We build a non-reductive GIT model for the Morin singularity locus of holomorphic maps, and develop a toric localisation formula for their Thom polynomials. 

Recent talks

• MathematicsxAI: Challenges and Opportunities, KIAS, Seoul, June 29 – July 3, 2026
  Invited speaker

• Algebraic Geometry and Machine Learning, ICARM, Carnegie Mellon University, June 22 – 26, 2026
  Invited speaker

• Using AI to find examples: CRM, Montreal,  April 27 - May 1, 2026
  Invited speaker

• Geometric Structures and Stability , ICTS Bangalore, 2026 February
  Mini-course on non-reductive group actions

• Mathematics and Machine Learning Reunion Workshop, Harvard CMSA, 2025 September
  Invited speaker

•  KIAS Machine Learning and Mathematics workshop, KIAS Seoul, 2025 July
  Invited speaker

•  Matrix-MFO Mathematics and Machine Learning Workshop, Oberwolfach, 2025 September
  Invited speaker

• Helvetic Algebraic Geometry Workshop, SRS Les Diablerets, 2025
  Invited speaker: The non-associative Hilbert scheme and virtual classes on Hilbert scheme of points

• Géométrie Algébrique et Géométrie Complexe 2024, CIRM Luminy
  A survey on non-reductive group actions and applications, Talk 1  , Talk 2 , Talk 3 

• Jean-Pierre Demailly Memorial Conference, Grenoble, 2024, Talk  

• GHiSP 2024, Levico Terme
  Tautological intersection theory of the Hilbert scheme of points, Talk  

• Mathematics and Machine Learning, Harvard CMSA, 2024, Short talk  
  A very informal summary of my recent Math&ML work, MLSummary