Mathematics

Formal foundations, proof systems and the theory everything else is built on.

Sub-fields

Number TheoryPrimes, Diophantine equations and the arithmetic structures behind cryptography and much of modern mathematics.3 entries · 12 open problems
Analysis & PDELimits, function spaces and the partial differential equations that model fluids, fields and diffusion.5 entries · 4 open problems
CombinatoricsCounting, graphs and extremal structures, where small certificates often settle big questions.3 entries · 13 open problems
Algebra & GeometryGroups, rings, varieties and the shapes they describe, from representation theory to algebraic geometry.2 entries · 2 open problems
Formal Proof (Lean & Mathlib)Machine-checked mathematics: proof assistants, Mathlib and the libraries that let a computer verify a proof.9 entries · 3 open problems

Open problems, most leverage first

All open problems →

#VotesProblemWork mixDVLStatus

010Riemann Hypothesis: all non-trivial zeros of ζ(s) lie on Re s = 1/2Lean proofThe Riemann zeta function and prime distribution3 links3 bottlenecks
963095OpenD 96

Research

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Analysis & PDETheoryActive

Harmonic analysis

The study of decomposing functions into waves (Fourier analysis) and of the operators that act on them. Its modern core is geometric: restriction, decoupling and Kakeya-type incidence problems.

2 problems5 dependents0 maintainers

CombinatoricsTheoryActive

Extremal and probabilistic combinatorics

How large can a discrete structure be without containing a forbidden pattern? Ramsey theory, Turán problems and set systems, attacked mostly by random constructions, entropy and iterative algorithms.

12 problems1 dependent maintainers

Mathematics

Sub-fields

Open problems, most leverage first

Research

Harmonic analysis

Extremal and probabilistic combinatorics

Mathematics

Sub-fields

Open problems, most leverage first

Research

Harmonic analysis

Extremal and probabilistic combinatorics

Lattices and sphere packing

Mathematical quantum field theory

Algebraic geometry and Hodge theory

Elliptic curves

The Navier–Stokes equations

The Riemann zeta function and prime distribution

Geometric measure theory and incidence geometry

Graph minors and structural graph theory

The probabilistic method and concentration of measure

Sobolev spaces and energy methods for PDE

Analytic number theory: Dirichlet series and the explicit formula

Machine-assisted theorem proving

Lean 4 & Mathlib

Formal Conjectures: open problems stated in Lean

Lean proof-search agents and their benchmarks (miniF2F, PutnamBench, ProofNet, FATE)

Large formalization projects: blueprints, Liquid Tensor, PFR, equational theories, FLT

Autoformalization

Lean tactic frameworks and automation (simp, aesop, omega, grind)

Mathlib's algebraic and order hierarchy

Lean 4 kernel and dependent type theory