Antoine Song will receive his PhD in 2019 from Princeton University, where he has been working under the guidance of Fernando Codá Marques.
Song has already established himself as an expert in geometric analysis, solving longstanding problems of fundamental importance concerning the nature of minimal hypersurfaces in compact Riemannian manifolds. First he proved that in dimensions 3 to 7 the closed minimal hypersurface of least area in such a manifold is always embedded. Then, in joint work with Codá Marques and Neves, he showed that for generic metrics on closed manifolds in these dimensions, one can always find a sequence of minimal embedded hypersurfaces that become equidistributed in the sense that the average of the induced measures on the first n hypersurfaces in the sequence converges to the normalised volume measure on the ambient manifold as n tends to infinity. This was a dramatic improvement in the state of the art concerning a circle of problems inspired by Yau's 1982 conjecture that every closed 3-dimensional Riemannian manifold contains infinitely many closed minimal surfaces. Building on work of Codá Marques and Neves, in 2018 Song proved Yau's conjecture in complete generality.
Antoine has been appointed as a Clay Research Fellow for a term of five years beginning 1 July 2019. He will be based at the University of California at Berkeley.