Mean Field Limit for Coulomb-Type Flows

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.Comment: Final version with expanded introduction, to appear in Duke Math Journal. 35 page

Online Appendix to Efficient Timing of Retirement

Post-retirement, the model in the main text (published in the Review of Economic Dynamics) reduces to the Merton (1969) problem, which has of course an exact solution. Pre-retirement, however, the agent holds an American option, namely, retire now or keep working. Problems involving American options are generally difficult to solve exactly. This appendix describes an approximate solution to the agent's pre-retirement problem.retirement, life cycle model, optimal stopping problem

Torsion order of smooth projective surfaces

To a smooth projective variety $X$ whose Chow group of $0$-cycles is $\mathbf Q$-universally trivial one can associate its torsion index $\mathrm{Tor}(X)$, the smallest multiple of the diagonal appearing in a cycle-theoretic decomposition \`a la Bloch-Srinivas. We show that $\mathrm{Tor}(X)$ is the exponent of the torsion in the N\'eron-Severi-group of $X$ when $X$ is a surface over an algebraically closed field $k$, up to a power of the exponential characteristic of $k$.Comment: A few more minor changes in Colliot-Th\'el\`ene's appendi