Analysis
Analysis studies continuous phenomena: limits, derivatives, integrals, and the functions and function spaces on which they act. Research in modern analysis ranges from the abstract (functional analysis, operator algebras, ergodic theory) to the concrete (PDE theory, harmonic analysis, complex and real variables), with extensive links to mathematical physics, probability, and geometry. Some of the hardest open problems in mathematics (Navier-Stokes, mass gap) sit at the interface of analysis and physics.
Browse live results
Jump directly to the public APIs filtered on this domain.
math.AP Analysis of PDEs
Existence, uniqueness, regularity, and qualitative behavior of partial differential equations.
math.CA Classical Analysis and ODEs
Real analysis, integration, special functions, ordinary differential equations.
math.CV Complex Variables
Complex analysis in one and several variables, several complex variables.
math.FA Functional Analysis
Banach spaces, Hilbert spaces, operator theory, distributions.
math.OA Operator Algebras
C*-algebras, von Neumann algebras, non-commutative geometry.
math.SP Spectral Theory
Spectra of operators, pseudodifferential operators, semiclassical analysis.
math.DS Dynamical Systems
Ergodic theory, hyperbolic dynamics, chaos, smooth dynamics.
Active research areas
The most active problem clusters right now.
PDE regularity
Nonlinear PDEs, free boundary problems, De Giorgi-Nash-Moser theory, and its modern extensions.
Search arXiv →Stochastic analysis
SPDEs, regularity structures (Hairer), paracontrolled distributions (Gubinelli-Imkeller-Perkowski).
Search arXiv →Operator algebras
Classification of C*-algebras, free probability, subfactors.
Search arXiv →Harmonic analysis
Restriction theorems, Kakeya problem, decoupling (Bourgain-Demeter-Guth).
Search arXiv →Ergodic theory
Szemerédi’s theorem via Furstenberg, polynomial ergodic theorems, measure rigidity.
Search arXiv →Landmark results
Major results shaping the field.
- 2014
Hairer’s regularity structures
Solution theory for singular SPDEs (Fields Medal 2014).
- 2018
Figalli on Monge-Ampère
Regularity of solutions to Monge-Ampère-type equations (Fields Medal 2018).
- 2016
Bourgain–Demeter–Guth decoupling
Breakthrough in Fourier-analytic methods with applications to number theory.
- 2019
Tao on averaged Navier-Stokes
Finite-time blow-up for an averaged variant of 3D Navier-Stokes.
Leading journals
Where current research in this area is published.