This volume covers some of the most recent and important developments in geometry and theoretical physics today. Topics include Monge-Ampère equations, Kähler-Ricci flows, and other fully non-linear elliptic and parabolic equations; canonical metrics in Kähler geometry; notions of quasi-local mass in general relativity and geometric properties of gauge theories; and new algebro-geometric and symplectic methods. The topics are all at the interface of several major branches of mathematics (geometry, analysis, and mathematical physics), and they are written by many of the most respected authorities in their fields worldwide.

Topics of particular interest within this volume are:



Localised estimates at the boundary for sections of solutions to the real Monge-Ampère equation, pluripotential aspects of complex Monge-Ampère equations on compact Hermitian manifolds, geometric modelling techniques for certain types of Monge-Ampère equations arising in affine hypersurface theory, a new gradient estimate for complex Monge-Ampère equations and properties of singular solutions of nonlinear elliptic equations.
The Kähler-Ricci flow on singular Calabi-Yau varieties, construction of Calabi-Yau metrics on Kummer surfaces, and small deformations of constant scalar curvature Kähler manifolds.
New methods for solutions of Einstein’s field equations with singularities, numerical properties of the new quasilocal mass in general relativity, framework and history of the Chern conjecture for isoparametric hypersurfaces in spheres, geometry of minimal energy Yang-Mills connections on bundles over manifolds of special holonomy.
The space of cyclic covers of a fixed topological type between complex projective curves and its irreducibility in the case of smooth curves and stable curves, categorical approach to the theory of manifolds with corners with applications to symplectic geometry, parametric singularities and their symplectic invariants, the global braid monodromy factorisation of the branch curves of a surface.