Algebra
Algebra is one of the largest and most active research areas in mathematics. It encompasses classical structure theory (groups, rings, fields, modules), the geometric interpretation of algebraic equations (algebraic geometry, schemes, motives, derived categories), the arithmetic of number fields and elliptic curves (number theory, L-functions, the Langlands program), and the representation-theoretic study of symmetry (Lie groups, Lie algebras, quantum groups). Modern algebra is profoundly interconnected — results in one subarea often have immediate consequences in another.
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math.AG Algebraic Geometry
Schemes, sheaves, cohomology, moduli spaces, derived algebraic geometry, motives.
math.NT Number Theory
Algebraic, analytic, and Diophantine number theory; L-functions, automorphic forms, arithmetic geometry.
math.GR Group Theory
Finite, infinite, and geometric group theory; representation growth.
math.RT Representation Theory
Representations of groups, Lie algebras, quantum groups, Hecke algebras.
math.AC Commutative Algebra
Rings, ideals, modules; Noetherian and Cohen-Macaulay theory.
math.KT K-Theory and Homology
Algebraic K-theory, cyclic homology, motivic cohomology.
math.QA Quantum Algebra
Hopf algebras, quantum groups, non-commutative geometry.
math.RA Rings and Algebras
Non-commutative algebras, associative and non-associative structures.
Active research areas
The most active problem clusters right now.
The Langlands program
Conjectural correspondences between automorphic representations and Galois representations; a unifying vision of much of modern number theory.
Search arXiv →Derived algebraic geometry
Gluing schemes along higher homotopical information; Lurie’s ∞-categorical framework.
Search arXiv →Arithmetic geometry
Heights, rational points, Mordell–Weil, Birch–Swinnerton-Dyer conjecture.
Search arXiv →Geometric representation theory
Perverse sheaves, categorification, Koszul duality.
Search arXiv →Geometric group theory
Group actions on spaces, CAT(0) geometry, mapping class groups.
Search arXiv →Landmark results
Major results shaping the field.
- 1980–2004
Classification of finite simple groups
Complete list of finite simple groups in 18 infinite families plus 26 sporadic groups; tens of thousands of pages of proof.
- 1994
Wiles’s proof of Fermat’s Last Theorem
Via the modularity theorem for semistable elliptic curves — a 358-year-old problem resolved.
- 2012
Scholze’s perfectoid spaces
New foundational framework for p-adic geometry (Fields Medal 2018).
- ongoing
Geometric Langlands
Gaitsgory, Arinkin, and co-authors announced a proof of the geometric Langlands conjecture in 2024.
Leading journals
Where current research in this area is published.