In real algebraic geometry, I describe how to use SDP hierarchies to approximate as closely as desired exact projections of semialgebraic sets. In nonlinear optimization, SDP hierarchies allow to compute Pareto curves (associated with multicriteria problems) as well as solutions of transcendental problems.

These hierarchies can also be easily interleaved with computer assisted proofs. An appealing motivation is to solve efficiently thousands of nonlinear inequalities occurring in the formal proof of Kepler Conjecture by Hales. Finally, SDP can provide precise information to automatically tune reconfigurable hardware (e.g. FPGA) to algorithm specifications.