  In this work, we study the convergence of the empirical measure of moderately
interacting particle systems with singular interaction kernels. First, we prove
quantitative convergence of the time marginals of the empirical measure of
particle positions towards the solution of the limiting nonlinear Fokker-Planck
equation. Second, we prove the well-posedness for the McKean-Vlasov SDE
involving such singular kernels and the convergence of the empirical measure
towards it (propagation of chaos).
  Our results only require very weak regularity on the interaction kernel,
including the Biot-Savart kernel, and attractive kernels such as Riesz and
Keller-Segel kernels in arbitrary dimension. For some of these important
examples, this is the first time that a quantitative approximation of the PDE
is obtained by means of a stochastic particle system. In particular, this
convergence still holds (locally in time) for PDEs exhibiting a blow-up in
finite time.
  The proofs are based on a semigroup approach combined with a fine analysis of
the regularity of infinite-dimensional stochastic convolution integrals.
