For each integral homology sphere $Y$, we construct a function ${\Gamma }_Y$ on the set of integers. We establish that ${\Gamma }_Y$ depends only on the homology cobordism class of $Y$ and that it recovers the Frøyshov invariant. We state a relation between ${\Gamma }_Y$ and Fintushel and Stern’s $R$-invariant. We show that the value of ${\Gamma }_Y$ at each integer is related to the critical values of the Chern–Simons functional, and we give some topological applications of ${\Gamma }_Y$. In particular, we show that if ${\Gamma }_Y$ is trivial, then there is no simply connected homology cobordism from $Y$ to itself.

We establish the mean field convergence for systems of points evolving along the gradient flow of their interaction energy when the interaction is the Coulomb potential or a super-Coulombic Riesz potential, for the first time in arbitrary dimension. The proof is based on a modulated energy method using a Coulomb or Riesz distance, assumes that the solutions of the limiting equation are regular enough, and exploits a weak-strong stability property for them. The method can handle the addition of a regular interaction kernel and applies also to conservative and mixed flows. In the Appendix, it is also adapted to prove the mean field convergence of the solutions to Newton’s law with Coulomb or Riesz interaction in the monokinetic case to solutions of an Euler–Poisson type system.

We study the special fiber of the integral model for Shimura varieties of Hodge type with parahoric level structure recently constructed by Kisin and Pappas. We show that when the group at $p$ is residually split, the points in the mod $p$ isogeny classes have the form predicted by the Langlands–Rapoport conjecture. We also verify most of the He–Rapoport axioms for these integral models without the residually split assumption. This allows us to prove that all Newton strata are nonempty for these models. The verification of the axioms in full is reduced to a question on the connected components of affine Deligne–Lusztig varieties.