When it comes to computational complexity theory, the simple understanding is "Can problems that can be verified quickly be solved quickly?"

hodge conjecture

Relates to algebraic topology and cohomology theory.

Poincaré conjecture

When it comes to algebraic topology, a simple understanding is whether balloons made of three-dimensional films can be pulled at will. ...

Riemann hypothesis

The zero point of Riemann zeta function is very important for the study of prime number distribution.

Yang-Mills Existence and Quality Gap

It involves quantum field theory and standard model in theoretical physics. (This Yang is known as Grandpa Yang ...)

The existence and smoothness of Naville-Stokes

It involves fluid mechanics and nonlinear analysis, and is very important for the study of turbulence.

Birch and Swinerton-Dale conjecture

I know nothing about number theory.