The research proposal is devoted to advancing several important aspects of the noncommutative analysis and its applications to partial differential equations (PDEs), fuelled by strong recent advances in the quantization theories and analysis on nilpotent Lie groups. On one hand, we will work on the development of the noncommutative Beals-Fefferman theory and its implications. This theory is one of the major breakthroughs of the classical theory of pseudo-differential operators, but very little is known so far about its noncommutative version. With a developed theory, as envisaged and outlined in this proposal, one will finally be able to attack several elusive problems, such as e.g. sharp Gärding inequalities on stratified and graded Lie groups. On the other hand, the project is devoted to advances in the global well-posedness theory of partial differential equations, based on the techniques of noncommutative analysis. One of the striking examples of such approach is the resolution by Rothschild and Stein of the question of finding sharp subelliptic orders and estimates for general Hörmander's sums of squares. The project aims at applying such philosophy for the determination of the sharp Fujita exponents for the global in time solvability/blow-up of solutions of semilinear heat equations for general Hörmander's sums of squares. Here the combination of global lifting techniques and analysis on stratified groups will be linked together to resolve this challenging problem.