Applied Mathematics
Applied mathematics is the bridge between pure theory and working technology. Numerical analysis develops provably correct, efficient algorithms for scientific computing; optimization studies how to find the best solution under constraints; combinatorics probes discrete structure from graphs to codes; information theory quantifies communication; mathematical physics formalizes the theoretical physics frontier. All of these areas have grown explosively with modern computing.
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math.NA Numerical Analysis
Finite-element methods, spectral methods, multigrid, iterative solvers, high-performance computing.
math.OC Optimization and Control
Convex and non-convex optimization, stochastic optimization, optimal control.
math.CO Combinatorics
Enumerative, probabilistic, algebraic, additive, extremal combinatorics.
math.IT Information Theory
Shannon theory, coding theory, quantum information theory.
math-ph Mathematical Physics
Rigorous mathematical formulations of physical theories, integrable systems, quantum field theory.
Active research areas
The most active problem clusters right now.
Compressed sensing
Sparse recovery from undersampled linear measurements (Candès-Tao-Romberg, 2006).
Search arXiv →Additive combinatorics
Sum-product estimates, Freiman-Ruzsa, applications to number theory.
Search arXiv →Extremal graph theory
Szemerédi regularity, hypergraph regularity, the flag algebra method.
Search arXiv →Machine-learning theory
Gradient descent, neural tangent kernels, double descent, deep learning theory.
Search arXiv →Quantum information and codes
Stabilizer codes, topological codes, fault-tolerant quantum computation.
Search arXiv →Landmark results
Major results shaping the field.
- 1975
Szemerédi’s regularity lemma
A universal tool in modern extremal combinatorics.
- 2006
Candès–Tao–Romberg on compressed sensing
Sparse recovery revolutionized signal processing.
- 2022
Kahn–Narayanan–Park on thresholds
Proof of the Kahn-Kalai conjecture in probabilistic combinatorics.
- 1984
Karmarkar’s interior-point method
Polynomial-time linear programming, launching modern convex optimization.
Leading journals
Where current research in this area is published.