A Library of Open Problems in Mathematics

This post collects links to open mathematical problems that I can find on the Internet.

1. General indexes and cross-field collections

• AIM Problem Lists — The American Institute of Mathematics hosts workshop-generated lists covering many areas of pure and applied mathematics.
• AMR Problem Lists — The Association of Mathematics Research provides a field-by-field directory of classical and specialist open-problem collections.
• UnsolvedMath — This searchable index aggregates more than two thousand questions from major notebooks and community collections.
• FrontierMath Open Problems — Epoch AI publishes expert-vetted research problems that can be filtered by field and status.
• Open Problem Garden — This community-maintained wiki organizes problems by subject, but individual status claims should be independently checked.
• MathWorld: Unsolved Problems — MathWorld offers short introductions to many famous unsolved problems with references for further reading.
• Clay Millennium Prize Problems — The Clay Mathematics Institute maintains the official pages for the Millennium problems and their status.
• Smale’s Problems — Stephen Smale’s 1998 list proposes eighteen mathematical problems for the twenty-first century.
• Arnold’s Problems — This book gathers 861 problems posed by Vladimir Arnold over more than forty years, with historical comments and status information.
• Open Problems in Mathematics — John Nash Jr. and Michael Th. Rassias edit a broad collection of expository chapters on major open directions.

2. Analysis

• Seventy Five (Thousand) Unsolved Problems in Analysis and Partial Differential Equations — Vladimir Maz’ya surveys seventy-five problem clusters in function theory, functional analysis, and linear and nonlinear PDE.
• Research Problems in Function Theory — Walter Hayman and Eleanor Lingham assemble more than five hundred problems in complex analysis together with progress notes.
• Some Problems in Harmonic Analysis — This article records sixteen problems discussed at the 2016 AIM workshop on harmonic analysis.
• Some Open Problems in Elasticity — John Ball lists foundational analytical questions in nonlinear elasticity and the calculus of variations.
• Some Open Problems and Research Directions in the Mathematical Study of Fluid Dynamics — Peter Constantin sketches major analytical challenges in incompressible fluid dynamics.
• Five Open Problems in Compressible Mathematical Fluid Dynamics — Denis Serre collects research questions about compressible flows, shocks, and conservation laws.
• Stony Brook Open Problems in Dynamical Systems — This portal gathers several conference and seminar problem lists in smooth and complex dynamics.
• Fifteen Problems in Mathematical Physics — Barry Simon proposes fifteen problems spanning spectral theory, Schrödinger operators, and mathematical physics.
• Michael Hochman’s Open Problems in Dynamics — This regularly updated page lists problems in ergodic theory, entropy, fractals, and related dynamics.
• Mike Boyle’s Open Problems in Symbolic Dynamics — This page collects problem lists and updates centered on symbolic and topological dynamics.
• Open Problems on Multiple Ergodic Averages — Nikos Frantzikinakis maintains questions on multiple recurrence, averages, and combinatorial applications.
• Jesse Peterson’s Open Problems — This maintained list focuses on operator algebras, measured group theory, and rigidity.
• Open Problems in One-Parameter Operator Semigroups — The survey collects research questions across semigroup theory, evolution equations, and operator theory.
• Some Open Problems in Low-Dimensional Dynamical Systems — Armengol Gasull presents thirty-three concrete problems about planar and low-dimensional dynamics.
• Some Open Problems in Random Matrix Theory and the Theory of Integrable Systems — Percy Deift connects open questions in random matrices, asymptotics, and integrable models.
• Open Problems in Mathematical Physics — Michael Aizenman’s page links a set of mathematically precise challenges motivated by physics.
• Open Problems in Integrable Systems — This multi-author collection surveys open questions across classical, quantum, and discrete integrable systems.

3. Geometry

• The Open Problems Project — This maintained database tracks dozens of problems in discrete and computational geometry, including status updates and references.
• Open Problems in Geometry — Shing-Tung Yau’s problem chapter ranges broadly across differential, complex, and algebraic geometry.
• Differential Geometry: Its Past and Its Future — Shiing-Shen Chern’s 1970 address is a historical source of influential questions and research directions.
• Problems from the Manifolds and Their Differential Geometry Conference — Frank Morgan and Pierre Pansu record concise problems contributed at a 2018 differential-geometry meeting.
• 101 Questions, Problems and Conjectures around Scalar Curvature — Mikhail Gromov organizes a large research agenda around scalar curvature and metric geometry.
• Open Problems in Geometry of Curves and Surfaces — Mohammad Ghomi collects approachable but research-level questions about curves, surfaces, convexity, and knots.
• Questions in Differential Geometry — Ken Richardson curates links to problem lists in Riemannian, contact, symplectic, and related geometry.
• Unsolved Problems in Geometry — Croft, Falconer, and Guy present classical intuitive problems in convex, discrete, and combinatorial geometry.
• Research Problems in Discrete Geometry — Brass, Moser, and Pach assemble an Erdős-style collection spanning incidence, packing, covering, and combinatorial geometry.
• Some Open Problems in Sub-Riemannian Geometry — Andrei Agrachev lists foundational questions about geodesics, regularity, and metric structure in sub-Riemannian spaces.
• Some Open Problems in Holomorphic Symplectic Geometry — Arnaud Beauville surveys conjectures and classification questions for irreducible holomorphic symplectic manifolds.
• Open Problems in Symplectic Topology — McDuff and Salamon’s book chapter provides a structured list of problems in symplectic geometry and topology.
• Some Open Problems in Symplectic Topology — This compact conference handout records concrete questions about Hamiltonian actions, Hofer geometry, and Gromov–Witten theory.
• Open Problems on Billiards and Quantitative Symplectic Geometry — This recent survey links billiard dynamics with symplectic capacities and quantitative questions.
• Geometry of Continued Fractions: Open Problems and Conjectures — Oleg Karpenkov collects questions on multidimensional continued fractions and their geometric structures.
• A Gallery of Open Problems in Geometry — This illustrated collection presents a varied set of current geometric questions in a compact format.

4. Topology

• Mark Hovey’s Algebraic Topology Problem List — This restored problem list organizes questions across homotopy theory, spectra, classifying spaces, and generalized cohomology.
• Open Problems and Questions in Algebraic Geometry, Topology, and Differential Geometry — This 2026 AMS volume centered on K3 surfaces contains nearly four hundred problems and many further subproblems.
• Kirby’s Problems in Low-Dimensional Topology — Robion Kirby’s classic list is a historical hub for problems about 3- and 4-manifolds, knots, and geometric topology.
• Problems on Mapping Class Groups and Related Topics — Benson Farb’s edited volume contains twenty-three chapters of problems linking topology, geometry, and dynamics.
• Problems, Questions, and Conjectures about Mapping Class Groups — Dan Margalit surveys a large set of questions about mapping class groups and their relatives.
• Problems on Invariants of Knots and 3-Manifolds — Ohtsuki and collaborators compile questions on quantum, finite-type, and classical invariants.
• Virtual Knot Theory: Unsolved Problems — This survey collects problems in virtual, combinatorial, and diagrammatic knot theory.
• Problems in Foliations and Laminations of 3-Manifolds — Danny Calegari lists research questions connecting foliations, laminations, contact structures, and group actions.
• Problems in Groups, Geometry, and Three-Manifolds — This workshop list gathers problems at the interface of geometric group theory and 3-manifold topology.
• John Etnyre’s Problems in Contact Topology — This page collects papers and lists devoted to open questions in contact topology.
• Some Open Problems in Symplectic Topology — This 2025 list emphasizes Floer homotopy theory and adjacent symplectic-topological questions.
• Open Problems in Algebraic Topology and Homotopy Theory — This MathOverflow thread is a community index of concrete questions and references in modern homotopy theory.
• Wild Topology Problem List — This specialist list focuses on wild embeddings, decomposition spaces, and pathological phenomena in topology.
• Open Problems in Non-Hausdorff Topology and Domain Theory — This maintained site records numbered questions in non-Hausdorff topology, order theory, and theoretical computer science.
• Open Problems in Arithmetic Topology — This survey formulates analogies and questions linking knots, 3-manifolds, number fields, and arithmetic invariants.
• Voevodsky’s Open Problems in Motivic Stable Homotopy Theory — Vladimir Voevodsky’s page states problems about the motivic stable homotopy category and its invariants.
• Problems in the Topology of Manifolds — This MATRIX workshop report compiles questions about manifold groups, embeddings, foliations, and geometric structures.
• Open Problems in Topological Graph Theory — Dan Archdeacon surveys questions about graph embeddings, genus, colorings, crossing numbers, and geometric representations.
• Open Problems in Topology II: A Status Report — This cumulative report tracks solutions and progress for the roughly eleven hundred problems in the classic topology volumes.

5. Algebra

• The Kourovka Notebook — The long-running notebook’s twenty-first edition collects new questions in group theory and updates many earlier entries.
• The World of Groups: Open Problems — This directory links more than two hundred problems in combinatorial and computational group theory.
• Questions in Geometric Group Theory — Mladen Bestvina’s influential list records problems on hyperbolic groups, boundaries, automorphisms, and group actions.
• The Lyapin Problem Notebook — This 2026 collection presents unsolved problems across several major directions in semigroup theory.
• Open Problems in Commutative Ring Theory — Sarah Glaz and collaborators assemble questions across homological, coherent, and non-Noetherian commutative algebra.
• Further Problems in Commutative Algebra — This appendix offers a concise set of research-level problems complementing a graduate text.
• Open Problems on Central Simple Algebras — A. S. Merkurjev’s survey lists structural questions about division algebras, Brauer groups, and algebraic groups.
• Open Problems in Hopf Algebras — This BIRS workshop report records problems on classification, tensor categories, Nichols algebras, and quantum groups.
• Open Problems in the Representation Theory of Artin Algebras — Auslander, Reiten, and Smalø conclude their monograph with a focused list of representation-theoretic problems.
• Open Problems in the Theory of Simple Modular Lie Algebras — Georgia Benkart, Thomas Gregory, and Alexander Premet survey classification and representation questions in positive characteristic.
• nLab: Open Problems in Category Theory — The nLab community maintains a cross-linked collection of conjectures and questions in category and higher-category theory.
• Open Problems in Operad Theory — This chapter gathers questions about operads, homotopy algebras, deformation theory, and applications.
• Open Problems on Open Algebraic Varieties — Shulim Kaliman and Mikhail Zaidenberg survey questions concerning affine and open complex algebraic varieties.
• Algebraic Stories from One and from the Other Pockets — Vladimir Shpilrain’s essay presents accessible open problems across combinatorial and computational algebra.
• The Finite Lattice Representation Problem — This site explains the history, current status, partial results, and related questions surrounding a central problem in universal algebra.
• Open Problems in Representation Theory and Around — This MathOverflow thread gathers community-contributed questions and references across representation theory.
• Open Problems in Hopf Algebras on MathOverflow — This community thread complements formal surveys with specific Hopf-algebra questions and literature pointers.
• Open Problems on Pro-Lie Groups — Hofmann and Morris list questions about infinite-dimensional Lie theory and projective limits of Lie groups.
• Selected Open Problems in Topological Algebra — Dikran Dikranjan and Michael Megrelishvili survey problems involving topological groups, compactifications, and dynamics.

Keeping the directory current

If you know a substantial maintained list that is missing, or if a linked problem collection has moved or received a major status update, please send me the canonical source so that both language versions can be updated together.