Asymmetric nuclear matter presents an ideal testing ground for nuclear forces and at the same time allows to constrain important astrophysical quantities, such as the symmetry energy, without using empirical inputs. We present several applications of an improved normal-ordering method for including three-nucleon (3N) forces up to next-to-next-to-next-to leading order (N3LO) in many-body calculations. Based on explicit calculations for different proton fractions, we discuss saturation properties, the symmetry energy, and its slope parameter for commonly-used Hamiltonians with N3LO NN plus N2LO 3N forces. For neutron matter we assess the convergence of many-body perturbation theory (MBPT) by comparing order-by-order results for the energy to the nonperturbative method of self-consistent Green’s functions. In addition, we present results for the BCS pairing gaps in the 1S0 and 3P2-3F2 channels based on recent local and semilocal NN potentials up to N2LO and N4LO, respectively, including 3N forces and novel uncertainty estimates. Finally, we discuss recent advances towards a new and efficient approach to evaluate energy diagrams using Monte Carlo techniques. We show preliminary results up to fourth order in MBPT. As one application, this enables constraining fits of next-generation chiral interactions in terms of their saturation and other nuclear matter quantities.
