Abstract: This talk will showcase the emerging role played by convex programming in tackling challenging problems from physics that require the analysis and/or optimization of solutions to nonlinear partial differential equations (PDEs). In the first part of the talk, I will consider the problem of quantifying the average amount of heat transported by turbulent flows, which is a key question in fluid mechanics motivated by applications to heat exchangers, climate models, and planetary physics. I will show that ideas from ergodic optimization can be used to place rigorous bounds on the heat transport and that, crucially, such bounds can be computed efficiently by solving convex optimization problems in the form of semidefinite programs. Moreover, numerical bounds obtained in this way can often be turned into analytical scaling laws for the heat transport, thereby enabling scientists to make predictions even in flow conditions far beyond the reach of the most advanced  simulation tools. The second part of the talk, instead, will focus on the global minimization of nonconvex integral functionals, which arise for example in fields such as nonlinear elasticity, pattern formation, and optimal design. I will outline a "discretize then relax" approach that, under mild assumptions, produces convergent approximations to global optimizers through the solution of structured semidefinite programs. I will conclude the talk with an overview of promising future research directions.