ICMConjectures: A Collection of Open Conjectures from the ICM Proceedings

Abstract

We introduce ICMConjectures, a curated collection of mathematical conjectures extracted from the Proceedings of the International Congress of Mathematicians. The collection aims to provide genuine mathematical conjectures spanning multiple fields for the evaluation of AI systems. Unlike collections centered on a single mathematician or a narrow range of subjects, the ICM Proceedings provide a historically authoritative source representing a broad spectrum of mathematical research. ICMConjectures currently contains 2,898 conjectures across 18 fields. In this report, we describe the pipeline used to construct the collection. The collection is available at https://icmconjectures.com.

Introduction

Large language models (LLMs) have recently begun to demonstrate substantially stronger capabilities in research-level mathematics. Beyond solving competition-style problems or reproducing standard arguments, frontier models have in several cases contributed to the resolution of open problems. In May 2026, for example, an internal general-purpose OpenAI model constructed an infinite family of point configurations that disproved a longstanding conjecture about the planar unit-distance problem; the argument was subsequently checked by external mathematicians OpenAI’s report. More recently, Levent Alpöge, working with Anthropic’s Claude Fable 5 model, announced an explicit three-dimensional counterexample to the Jacobian Conjecture Alpöge’s announcement, Tao’s discussion. The example refutes the conjecture in dimensions $n\geq 3$, although the two-dimensional case remains open. Such results suggest that general-purpose AI systems are beginning to make original contributions at the research frontier.

These developments create a need for collections built from genuinely open, research-level mathematical problems. Such collections should offer broad disciplinary coverage, since evaluating AI systems on a narrow class of questions cannot reveal their capabilities across different areas of mathematics. The Erdős Problems collection is a valuable and influential resource Erdős Problems, and has already supported systematic experiments with mathematical agents Feng et al.. By its origin and emphasis, however, it is concentrated largely in combinatorics and number theory. A complementary collection should span as many mathematical fields as possible in order to assess the breadth of an agent’s research capabilities.

The problems should also be mathematically meaningful: they should be connected to substantial bodies of theory, regarded as important by researchers, and embedded in active research programs. The Proceedings of the International Congress of Mathematicians (ICM) provide a particularly strong source with these properties IMU Proceedings. Written by leading mathematicians from many fields, ICM articles are high-level expository and research documents that survey central developments, formulate major questions, and describe directions for future work. A conjecture appearing in such a venue is therefore likely to be mathematically substantive, well connected to its surrounding theory, and representative of a direction valued by the research community. The disciplinary scope and historical depth of the Proceedings make them especially well suited to constructing a broad, source-grounded evaluation of AI research in mathematics.

In this report, we introduce ICMConjectures, a collection of open mathematical problems drawn from the ICM Proceedings. We used a Codex-based agent to process the Proceedings from 1893 through 2022, extract conjectures and open questions, expand their statements with necessary context, and rewrite them as self-contained problems. The agent assigned each problem labels from a two-level subject classification and surveyed the subsequent literature to trace its progress; only problems found to remain unresolved were retained. For problems that have been partially resolved, we further record the remaining open content and identify selected open subproblems proposed in the literature. These finer-grained targets may provide more accessible starting points for both AI agents and human researchers. ICMConjectures currently contains 2,898 conjectures across 18 fields. We will continuously revise the status of these problems using emerging frontier models to improve the quality of the collection. Because a collection extracted and curated by AI systems cannot be fully reliable without expert validation, we have manually reviewed a subset of the problems in algebraic geometry and geometry. We now invite mathematicians across these fields to help review the remaining problems and improve the accuracy and completeness of the collection.

Collection Construction

In this section, we describe the construction of the collection, including its source corpus, processing procedure, metadata schema, and expert review protocol.

Source Corpus

The source corpus consists of the ICM Proceedings published between 1893 and 2022. We downloaded the proceedings in PDF format from the official archive maintained by the International Mathematical Union IMU Proceedings. We then applied Mistral OCR to the PDF files and converted their contents into Markdown. Finally, we segmented the resulting Markdown by individual proceedings article, producing one Markdown file for each paper. The resulting article-level corpus serves as the input to the subsequent conjecture-extraction pipeline.

Conjecture Processing

We first applied an article-level extraction pass to identify conjectures and open-problem statements in each proceedings paper. Each extracted statement, together with its surrounding source context, was then processed by a Codex-based agent through the following pipeline.

Step 1: Statement normalization and translation. The agent first determined whether the original statement was self-contained. If definitions, hypotheses, notation, or other information needed to understand the problem appeared only in the surrounding text, the agent expanded the statement using that context. It then checked whether the expanded statement was a well-formulated mathematical problem rather than a narrative description of the problem’s history, status, or motivation. Narrative or underspecified passages were rewritten as clear mathematical problems while preserving their intended content. Finally, any statement not written in English was translated into English.
Step 2: Literature-based status assessment. The agent surveyed the literature to determine whether the problem had been resolved. We used two status labels: resolved for statements that have been proved or disproved, and unresolved for statements that have not been completely settled. Thus, a problem with substantial partial progress was still labeled unresolved unless the full statement had been resolved; in such cases, the literature documenting the partial progress was recorded. Only unresolved problems were retained in the final collection.
Step 3: Difficulty and research-activity assessment. The agent assigned two ordinal scores to each problem. Estimated mathematical difficulty was scored from 1 to 3, with 1 denoting the lowest and 3 the highest difficulty. Research activity---how actively the exact problem or conjecture has been studied by the research community---was scored on the same scale, from 1 (low activity) to 3 (high activity).
Step 4: Identification of literature-grounded subproblems. The agent searched the literature for narrower problems that mathematicians had explicitly proposed, or clearly identified, as open parts, special cases, or intermediate questions related to the original problem. When such a subproblem remained open, we recorded both a self-contained statement of the subproblem and its source. This step was restricted to subproblems grounded in the literature rather than new decompositions invented by the agent.

As an example, consider the problem:

Determine whether every algebraic variety over an arbitrary ground field admits a resolution of singularities.

Its some_remaining_open_problems field includes the following literature-grounded subproblems:
1. Statement: Let $K/k$ be an algebraic function field of positive characteristic and transcendence degree greater than $3$, and let $\nu$ be a valuation of $K$ having a center on an integral $k$-variety $X$ with function field $K$. Must there be a proper birational model $X’\to X$ on which the center of $\nu$ is a regular point (local uniformization)?
  
  Reference: Michael Temkin, “Inseparable Local Uniformization,” Journal of Algebra 373 (2013), 65–119, open problem stated in the abstract and §1.1, DOI: 10.1016/j.jalgebra.2012.09.023, https://doi.org/10.1016/j.jalgebra.2012.09.023.
2. Statement: Does embedded resolution hold for threefold singularities in positive characteristic: given a reduced three-dimensional variety embedded in a smooth ambient variety over a field of positive characteristic, can one obtain a smooth strict transform having normal crossings with the exceptional divisor by a finite sequence of blowups in permissible smooth centers?
  
  Reference: Herwig Hauser and Stefan Perlega, “Resolving Surface Singularities in Positive Characteristic,” Publications of the Research Institute for Mathematical Sciences 60, no. 4 (2024), 767–813, Introduction (explicitly states that embedded resolution for threefolds is open), DOI: 10.4171/PRIMS/60-4-5, https://doi.org/10.4171/PRIMS/60-4-5.
3. Statement: Let $V$ be a smooth scheme over a perfect field $k$ of characteristic $p \gt 0$, and let $X\subset V$ be a hypersurface of maximum multiplicity $n$. Is there always a finite sequence of monoidal transformations whose successive centers lie in the $n$-fold loci of the successive strict transforms of $X$, whose exceptional hypersurfaces have normal crossings, and whose final strict transform has no point of multiplicity $n$?
  
  Reference: Angélica Benito and Orlando E. Villamayor U., “Monoidal Transforms and Invariants of Singularities in Positive Characteristic,” Compositio Mathematica 149, no. 8 (2013), 1267–1311, open simplification problem in §1.1, DOI: 10.1112/S0010437X1200084X, https://doi.org/10.1112/S0010437X1200084X.
4. Statement: In the positive-characteristic hypersurface setting of Benito–Villamayor, can one construct a sequence of permissible monoidal transformations that takes a hypersurface whose highest-multiplicity locus is in their monomial case (or weak monomial case) to one in their strong monomial case, so that the remaining highest-multiplicity points can then be eliminated combinatorially?
  
  Reference: Angélica Benito and Orlando E. Villamayor U., “Monoidal Transforms and Invariants of Singularities in Positive Characteristic,” Compositio Mathematica 149, no. 8 (2013), 1267–1311, §1.1 (the explicitly identified gap between the monomial/weak monomial case and the strong monomial case), DOI: 10.1112/S0010437X1200084X, https://doi.org/10.1112/S0010437X1200084X.
Step 5: Two-level subject classification. Each problem was assigned both a broad field and a more specific second-level category. For the first level, we adopted the 20-field taxonomy introduced in Connected Theorems; 18 of these fields are represented in ICMConjectures. The second-level categories were generated by GPT-5.5 (xhigh) based on the corresponding MSC codes within each field. The resulting categories are summarized in Table subject taxonomy.

The two-level subject taxonomy used in the current release.
Primary field	Second-level categories
Algebra	Associative and Noncommutative Ring Theory Categories in Geometry and Topology Computational and Combinatorial Algebra Differential and Difference Algebra Division Rings, Orders, and Semisimple Algebras Field Extensions and Galois Theory General Category Theory and Categorical Algebra General Commutative Ring Theory Higher Categories and Homotopical Algebra Homological and Derived Algebra Hopf Algebras and Quantum Groups Linear and Multilinear Algebra Local and Noetherian Algebra Logic and Model Theory in Algebra Matrix Theory Module Theory and Morita Theory Modules and Ideals over Commutative Rings Monoidal Categories and Operads Polynomial and General Field Theory Representation Theory of Associative Algebras and Quivers Special Classes of Commutative Rings Universal Algebra and General Algebraic Systems Valuation Theory and Topological Fields
Analysis	Abstract Harmonic Analysis Approximations and Expansions Functional Analysis Functions of a Complex Variable Harmonic Analysis on Euclidean Spaces Measure and Integration Operator Theory Potential Theory Real Functions Sequences, Series, and Summability Several Complex Variables and Analytic Spaces Special Functions
Biology & Other Natural Sciences	Artificial Life, Biological Neural Networks, and Biological Rhythms Biophysics and Biomechanics Genetics, Genomics, and Evolution Mathematical Chemistry Population Dynamics, Epidemiology, and Ecology
Combinatorics & Discrete Mathematics	Algebraic Combinatorics Boolean Algebras Designs and Configurations Enumerative Combinatorics Extremal Combinatorics Graph Theory
Economics & Social Sciences / Game Theory	Actuarial Science and Mathematical Finance Combinatorial, Algorithmic and Applied Game Theory Cooperative Game Theory Decision, Utility, Risk and Preference Theory Dynamic, Stochastic and Differential Games Evolutionary Games, Learning and Congestion Games General Equilibrium and Economic Dynamics Microeconomics, Firms, Labor, Contracts and Information
Lie & Rep	Combinatorial and Geometric Group Theory Finite Group Theory General Topological Groups and Topological Algebraic Systems Homological and Categorical Group Theory Lie Algebras and Lie Superalgebras Lie Groups and Their Representations Linear Algebraic Groups and Groups of Lie Type Locally Compact Abelian and Compact Groups Locally Compact Groups and Their Representations Matrix Groups and Arithmetic Groups Probabilistic Group Theory Representation Theory of Groups Structure and Classification of Groups Transformation Groups and Homogeneous Spaces
Logic	Computability and Recursion Theory General Logic Model Theory Proof Theory and Constructive Mathematics Set Theory
Mathematical Physics & Mechanics	Astronomy and Astrophysics Fluid Mechanics Geophysics Mechanics of Deformable Solids Mechanics of Particles and Systems Optics and Electromagnetic Theory Quantum Theory Relativity and Gravitational Theory Statistical Mechanics and Structure of Matter
Numerical Analysis & Computer Science	Algorithms in Computer Science Artificial Intelligence and Machine Learning Computer Aspects of Numerical Algorithms Computer Systems, Networks, and Security Error Analysis and Interval Analysis Numerical Analysis in Abstract Spaces and Inverse Problems Numerical Approximation and Computational Geometry Numerical Methods for Dynamical Systems Numerical Methods for Partial Differential Equation Boundary-Value Problems Numerical Methods for Time-Dependent Partial Differential Equations Numerical Optimization and Variational Methods Numerical Solution of Nonlinear Equations Probabilistic and Stochastic Numerical Methods Theory of Computation and Computational Complexity Theory of Data and Information Management
Optimization & Operations Research	Combinatorial and Network Optimization Convex and Conic Optimization Dynamic Programming and Markov Decision Processes Location and Assignment Nonlinear and Global Optimization Optimization Theory, Algorithms, and Complexity
PDE & Dyn	Bifurcation and Random Dynamical Systems Complex and Arithmetic Dynamics Computational and Applied Dynamics Difference and Functional Equations Differential Equations in Abstract Spaces and with Randomness Differential Operators and Complex Differential Equations Elliptic, Hypoelliptic, and Subelliptic Equations Ergodic Theory Existence, Optimality, and Variational Analysis General Theory and Solution Methods for PDEs Geometric Variational Problems and Optimal Transport Hamiltonian and Integrable Systems Hyperbolic and Low-Dimensional Dynamics Hyperbolic, Conservation-Law, and Mixed-Type Equations Infinite-Dimensional Dynamical Systems Integral Transforms and Integral Equations Inverse, Free-Boundary, Nonlocal, and Stochastic PDEs Optimal Control and Hamilton—Jacobi Theory PDEs of Mathematical Physics and Applications Parabolic Equations and Systems Pseudodifferential and Fourier Integral Operators Qualitative, Stability, and Asymptotic Theory of ODEs Smooth Dynamical Systems Spectral and Scattering Theory for PDEs Topological and Symbolic Dynamics
Statistics	Design of Statistical Experiments Foundational Topics in Statistics Linear Inference and Regression Multivariate Analysis Nonparametric Inference Statistical Decision Theory Statistical Distribution Theory Statistics on Algebraic and Topological Structures
Systems & Control / Information Theory	Control System Models and Methods Controllability, Observability, and System Structure Cryptography and Information Security Error-Correcting Codes and Decoding Information Theory and Entropy Signal Processing, Sampling, and Detection Stability and Stabilization Stochastic, Adaptive, and Learning Control
Topology	Algebraic, Arithmetic, and Hermitian K-Theory Classical Algebraic Topology and Fixed Point Theory Cohomology Operations, Obstruction Theory, and Spectral Sequences Compactness, Separation, and Covering Properties Continuum and Dimension Theory Differential Topology, Surgery, and Cobordism Fiber Bundles, Classifying Spaces, and Characteristic Classes General and Set-Theoretic Topology Geometric, Topological, and Operator-Algebraic K-Theory Homology and Cohomology Theories Homotopy Theory and Homotopy Groups Knot and Link Theory Low-Dimensional Manifold Topology Maps, Function Spaces, and Shape Theory Metric and Uniform Topology Topological Algebra and Topological Groups Topological, PL, and Generalized Manifolds Transformation Groups and Equivariant Topology
Algebraic geometry	Foundations of algebraic geometry Local theory in algebraic geometry Cycles and subschemes Families and fibrations in algebraic geometry Birational geometry Cohomology theory in algebraic geometry Arithmetic problems in algebraic geometry Diophantine geometry Curves in algebraic geometry Surfaces and higher-dimensional varieties Abelian varieties and schemes Algebraic groups Special varieties Projective and enumerative algebraic geometry Real algebraic and real-analytic geometry Computational aspects in algebraic geometry Affine geometry Tropical geometry
Geometry	Incidence, axiomatic, and classical geometry Metric and topological geometry Convex geometry Discrete and polyhedral geometry Classical and local differential geometry Global/Riemannian geometry Symplectic and contact geometry Geometric flows and geometric PDE Manifolds and global analysis Singularity theory Applications of geometry/global analysis
Number theory	Elementary number theory Sequences and sets Polynomials and matrices Diophantine equations Forms and linear algebraic groups Discontinuous groups and automorphic forms Arithmetic algebraic geometry (Diophantine geometry) Geometry of numbers Diophantine approximation, transcendental number theory Probabilistic theory: distribution modulo 1 metric theory of algorithms Exponential sums and character sums Zeta and $L$-functions: analytic theory Multiplicative number theory Additive number theory partitions Algebraic number theory: global fields Algebraic number theory: local fields Finite fields and commutative rings (number-theoretic aspects) Connections of number theory and logic Computational number theory Miscellaneous applications of number theory
Probability	Foundations of probability theory Probability theory on algebraic and topological structures Combinatorial probability Geometric probability and stochastic geometry Distribution theory Limit theorems in probability theory Stochastic processes Stochastic analysis Markov processes Special processes Rough analysis

After all statements had been processed individually, we performed a lightweight semantic deduplication pass. A Codex-based agent read the normalized statements, identified entries that in fact asked the same mathematical question despite differences in wording or source context, and selected one representative entry from each detected duplicate group. Because this is a heuristic semantic comparison rather than an exhaustive mathematical equivalence check, some duplicate questions may remain in the collection.

Metadata Schema

Each retained problem is represented as a structured record. The schema preserves source provenance, the normalized problem statement, subject labels, literature-based progress information, AI-generated assessments, and the status of human review. Table metadata schema summarizes the fields included in the current release.

Metadata fields in the current ICMConjectures release.
Field	Description
`id`	A unique identifier for the collection entry.
`conjecture_name`	The conventional name of the conjecture or problem, when one exists. At present, this field is completed by the authors only for human-reviewed entries; it is left empty for all other entries.
`statement`	The normalized, self-contained English statement of the conjecture or open problem.
`source`	The ICM Proceedings article from which the original statement was extracted.
`url`	The URL of the corresponding item in the ICM Proceedings archive.
`category`	The primary, first-level subject classification.
`subfield`	The more specific, second-level subject classification.
`progress`	A summary of progress on the problem, including relevant partial results identified in the literature.
`reviewed`	An indicator of whether the entry has been reviewed by a human expert.
`difficulty`	The AI-assigned difficulty score on the 1—3 scale. This estimate is a model annotation and may differ from the assessment of human experts.
`activity`	The AI-assigned research-activity score on the 1—3 scale. This estimate is a model annotation and may differ from the assessment of human experts.
`some_remaining_open_problems`	Narrower related problems that were explicitly proposed or identified in the literature and remain open, together with their sources.

Expert Review of Selected Problems

Because the conjectures in ICMConjectures were extracted, normalized, and annotated by AI agents, the resulting entries cannot be assumed to be completely reliable. Human expert review is therefore an essential part of the curation process. Reviewers may need to verify the mathematical accuracy of a problem statement, determine whether the problem is genuinely still open, identify important omissions or errors in the recorded progress, correct its subject classification, detect duplicate entries, and check the statements, open status, and references of the subproblems listed under some_remaining_open_problems. The initial phase of this process asked experts to review a selected set of problems in algebraic geometry and geometry.

Reviewer instructions.

For each entry, reviewers were asked to examine three components: Problem statement, Progress, and Some remaining open problems. After reviewing these components, if the conjecture or problem had a well-accepted name, the reviewer was asked to add it to the name field displayed at the top left of the problem entry. The detailed instructions were as follows.

Problem statement. Reviewers should check that the statement is mathematically accurate, complete, unambiguous, and clearly formulated, and should edit any inaccurate, incomplete, ambiguous, or misleading formulation. If a reviewer knows that the problem has been fully resolved, they should leave a comment explaining this so that an administrator can remove the entry. If the entry duplicates another problem on the website---that is, if both entries ask essentially the same mathematical question---the reviewer should leave a comment and, when possible, provide the duplicate problem’s ID so that an administrator can remove the duplicate.
Progress. Reviewers are not required to check the long AI-generated narrative in this section line by line. Instead, they should inspect the listed references, verify that they are relevant, and identify any major missing references. When a good survey of the problem is available, it may be listed in place of an exhaustive bibliography. Reviewers may also remove the AI-generated narrative and retain only the references.
Some remaining open problems. If the original problem remains fully open, this section may be left empty. For a partially resolved problem, it should contain selected open parts, special cases, or related unresolved questions. Each such entry consists of a statement and a reference. Reviewers should verify that the statement is mathematically accurate, that the cited reference is the source of the statement, and that the subproblem remains open. They are also encouraged to add further relevant open subproblems, provided that both a statement and a source reference are supplied.

Limitations

ICMConjectures has several limitations arising from the scope and digitization of its source corpus, the use of AI-assisted processing, and the continually evolving mathematical literature.

Although the ICM Proceedings are a rich and authoritative source of important research questions, they do not provide an exhaustive record of open problems in mathematics.

The construction of ICMConjectures depends on OCR of the ICM Proceedings. OCR errors may corrupt mathematical notation, omit parts of formulas or surrounding prose, and disrupt the structure of a document. Such errors can propagate through the extraction pipeline, causing conjectures to be missed or source passages to be interpreted incorrectly.

The problem statements in the collection are not necessarily verbatim statements from the Proceedings. They were extracted, normalized, and often rewritten to be self-contained by AI agents. This process may omit necessary hypotheses or context, alter quantifiers or notation, conflate related questions, or otherwise change the intended mathematical meaning. A clear and plausible formulation in the collection should therefore not be taken as a guarantee that the statement is mathematically accurate or equivalent to its source.

The literature-based annotations are subject to further uncertainty. The recorded progress may be incomplete or inaccurate because relevant results can be missed, misinterpreted, or superseded by later work. Consequently, a problem labeled as open may already have been resolved, either before the collection was assembled or after its latest review. The statements listed under some_remaining_open_problems may likewise be formulated incorrectly, may not follow from the cited sources, or may no longer be open. These lists are selective rather than exhaustive. Users should verify each statement, citation, and open-status claim against the original sources and the current literature. Continued expert review is needed to correct these issues and keep the collection up to date.

Conclusion

ICMConjectures is a collection of open conjectures drawn from the Proceedings of the International Congress of Mathematicians, a broad and historically significant source of research problems across mathematics. It currently contains 2,898 conjectures spanning 18 fields.

For each conjecture, we document subsequent progress in the literature and, when available, identify narrower open cases, intermediate problems, and their sources. We hope that ICMConjectures will support the evaluation of AI systems’ research-level mathematical reasoning abilities and provide a useful resource for studying AI-assisted mathematical research. At the same time, the collection requires continued expert validation. We therefore call on the mathematical community to help review, correct, and refine its entries as the collection develops.

Acknowledgments

This work is supported in part by the Fundamental and Interdisciplinary Disciplines Breakthrough Plan of the Ministry of Education of China (JYB2025XDXM113), the National Key R&D Program of China grant 2024YFA1014000, and the New Cornerstone Investigator Program.