Noncommutative smooth spaces
  We make an attempt to develop "noncommutative algebraic geometry" in which noncommutative affine schemes are in one-to-one correspondence with associative algebras.   In the first part we discuss various aspects of smoothness in affine noncommutative algebraic geometry. The main example for us is the noncommutative affine space. The algebra of functions on this space is free finitely generated algebra. We propose a general correspondence principle which allows us to give definitions of various differential-geometric structures in the noncommutative setting.   In the second part we propose a definition of noncommutative spaces based on a version of flat topology. Our category of noncommutative spaces contains as full subcategories both the category of quasi-compact separated schemes and the opposite category to the category of associative algebras.   In the third part we describe the noncommutative projective space and the derived category of coherent sheaves on it. 
Maxim Kontsevich, Alexander Rosenberg