Many scientific, engineering and financial applications require solving high-dimensional PDEs. However, traditional tensor product based algorithms suffer from the so called "curse of dimensionality". We shall construct a new sparse spectral method for high-dimensional problems, and present, in particular, rigorous error estimates as well as efficient numerical algorithms for elliptic equations. We shall also propose a new weighted weak formulation for the Fokker-Planck equation of FENE dumbbell model, and prove its well-posedness in weighted Sobolev spaces. Based on the new formulation, we are able to design simple, efficient, and unconditionally stable semi-implicit Fourier-Jacobi schemes for the Fokker-Planck equation of FENE dumbbell model. It is hoped that the combination of the two new approaches would make it possible to directly simulate the five or six dimensional Navier-Stokes Fokker-Planck system.