In this talk, we establish refined Strichartz estimates for the Schrödinger equation on tori within the framework of partial regularity. As a result, we reveal that the solution of the free Schrödinger equation has better regularity in mixed Lebesgue spaces. As an application, we obtain local well-posedness for non-gauge-invariant nonlinearities with partially regular initial data. On the other hand, we extend refined Strichartz estimates for infinite systems of orthonormal functions. As an application, we establish well-posedness for the Hartree equation for infinitely many fermions in some Schatten spaces.