Progress on the Langlands Program
New connections discovered between number theory and harmonic analysis, advancing our understanding of this grand unifying theory of mathematics.
Mathematical Research Hub
Discover the latest breakthroughs and ongoing research in mathematics
Research Areas
🔢 Number Theory
Prime numbers, Riemann hypothesis, and modern cryptography applications
- Twin Prime Conjecture
- Goldbach's Conjecture
- Elliptic Curves
📐 Geometry & Topology
Shape of space, knot theory, and geometric structures
- Poincaré Conjecture (Solved)
- Hodge Conjecture
- Manifold Theory
🧮 Applied Mathematics
Real-world applications in physics, engineering, and data science
- Machine Learning Theory
- Quantum Computing
- Climate Modeling
∞ Analysis
Calculus, differential equations, and functional analysis
- Navier-Stokes Equations
- Harmonic Analysis
- Partial Differential Equations
Recent Breakthroughs
Breakthrough in Ramsey Theory
Mathematicians found new bounds for Ramsey numbers, solving a problem that stood for decades.
Aperiodic Monotile Discovery
The "einstein" tile - a single shape that tiles the plane aperiodically - was discovered, revolutionizing tiling theory.
Famous Unsolved Problems
P vs NP Problem
Can every problem whose solution can be verified quickly also be solved quickly?
$1,000,000 PrizeRiemann Hypothesis
All non-trivial zeros of the Riemann zeta function have real part 1/2.
$1,000,000 PrizeYang-Mills Existence
Prove quantum Yang-Mills theory exists and has a mass gap.
$1,000,000 PrizeCollatz Conjecture
Does the 3n+1 sequence always reach 1 for any positive integer?
Open since 1937