The equation that keeps airplanes flying, predicts hurricanes, explains ocean currents, and powers modern engineering is the Navier Stokes equation.
It describes how fluids such as air, water, and gases move by combining the effects of acceleration, pressure, viscosity, and external forces. From weather forecasting to aircraft design, it is one of the most important equations in physics and engineering.
Navier Stokes equation:
ρ(∂u/∂t + (u · ∇)u) = −∇p + μ∇²u + f
Where
ρ = fluid density
u = fluid velocity
p = pressure
μ = dynamic viscosity
f = external forces such as gravity
Despite its compact form, this equation remains one of the greatest unsolved challenges in mathematics. Proving whether smooth solutions always exist in three dimensions is a Clay Millennium Prize Problem worth $1 million.
Mikhail Leonidovich Gromov, commonly known as Mikhail Gromov, was born on December 23, 1943, in Boksitogorsk, Russian SFSR, Soviet Union.
In Riemannian geometry and geometric group theory, Gromov introduced the Gromov-Hausdorff distance between metric spaces and established associated compactness theorems for sequences of Riemannian manifolds with bounded Ricci curvature and diameter. He proved that every finitely generated group of polynomial growth is virtually nilpotent, thereby resolving the Milnor-Wolf conjecture.
In symplectic geometry, Gromov founded a major new direction in 1985 by introducing the theory of pseudoholomorphic curves, also called J-holomorphic curves, on symplectic manifolds. This framework led directly to the definition of Gromov-Witten invariants and the creation of modern symplectic topology, with applications extending to string theory and quantum field theory.
Gromov has received many major awards recognizing his mathematical achievements. These include the Oswald Veblen Prize in Geometry in 1981, the Wolf Prize in Mathematics in 1993, the Balzan Prize in 1999, the Kyoto Prize in 2002, and the Nemmers Prize in 2004. In 2009 he was awarded the Abel Prize by the Norwegian Academy of Science and Letters for his revolutionary contributions to geometry.