Fields
Explore the full MSC2020 hierarchy, then follow a subject to its problems, journals and reviewed articles. Community fields below provide a broader route across mathematics and the sciences.
Browse by mathematical subject
63 classes, 534 sections and 6,006 detailed subjects from the official downloadable classification. Counts include mapped published entries of every status. Broad community mappings are labelled as inferred; an empty category means no assigned entries yet.
Classification © Mathematical Reviews (AMS) and zbMATH. CC BY-NC-SA 4.0. Official PDF · Journals & article coverage →
Research communities
Mathematics
11349 entriesNumber theory
Primes, Diophantine equations, L-functions — the field with the highest ratio of easily stated to hopelessly hard.
Combinatorics
Extremal and additive questions about finite structures, where a single clever construction can end a decades-old problem.
Graph theory
Colourings, minors, expansion and the conjectures that organise them.
Analysis
Harmonic analysis, operator theory and the fine structure of functions and spectra.
Partial differential equations
Regularity, blow-up and long-time behaviour of the equations of physics.
Geometry and topology
Manifolds, knots, packings and the stubborn gap between the smooth and the topological.
Algebra
Groups, rings and the representation theory that keeps them honest.
Probability
Random walks, percolation and scaling limits — where physics intuition outruns proof by decades.
Dynamical systems
Iteration, ergodic theory and the geometry of long-run behaviour.
Control and information theory
Control systems, stability, communication, and information.
Logic and foundations
Set theory, model theory and the limits of what axioms decide.
Mathematics · general
Mathematical questions awaiting a more specific subject classification.
Computer science
285 entriesComputational complexity
What can be computed with bounded resources. Home of the discipline's largest embarrassments.
Algorithms and data structures
Upper bounds: the running times we can achieve, and the ones we cannot rule out.
Cryptography
Security reductions and the hardness assumptions the modern internet is standing on.
Computational geometry
Points, distances and arrangements, counted and computed.
Machine learning
70 entriesLearning theory
Sample complexity, regret and the formal limits of learning from data. Source of the COLT open-problem track.
Optimization
Convergence rates, landscapes and the complexity of finding a good point.
Deep learning theory
Why the methods that work in practice work at all — the field's largest open questions are still qualitative.
Physics
339 entriesQuantum information
Entanglement, channels and the structure of quantum correlations. Unusually rich in crisply stated problems.
Mathematical physics
Rigorous foundations for the models physicists already trust.
Condensed matter
Phases of matter, and the many-body models nobody can solve.