American Institute of Mathematics, San Jose, California
Gabor Szekelyhidi, Valentino Tosatti, and Ben Weinkove
The main topics for the workshop are:
Constant scalar curvature Kaehler metrics. The Yau-Tian-Donaldson conjecture relates the existence of a constant scalar curvature Kaehler metric to an algebro-geometric condition, called K-stability, of the underlying complex manifold. Works of Yau, Tian and Chen-Donaldson-Sun culminated in a solution of this conjecture in the Kaehler-Einstein case, but the general constant scalar curvature problem remains largely open and will require substantial new ideas in the geometric analysis of higher order elliptic PDE. The case of toric manifolds is of particular interest, partly because it has been solved in the 2-dimensional case by Donaldson, but also because of its close connections to other fourth order equations such as those in affine geometry, studied by Trudinger-Wang and Li-Jia.
Intrinsic geometric flows. The Ricci flow has enjoyed tremendous success, with Perelman's solution of the Poincare conjecture and works of Bohm-Wilking and Brendle-Schoen on manifolds satisfying positive curvature conditions, among others. In other directions there is still significant scope for progress, such as in the work of Munteanu, Wang and others on classifying Ricci solitons in four dimensions, and the program of Song-Tian relating the singularity formation in the Kaehler case along the Ricci flow to the minimal model program. Beyond this there are other parabolic flows tailored to different problems, such as the fourth order Calabi flow which is the parabolic analog of constant scalar curvature metrics, and also flows on non-Kaehler complex manifolds such as the Chern-Ricci flow.
Geometry of compact complex manifolds. In recent years, much progress has been achieved in the study of positive cones in the cohomology of compact complex manifolds, including the Demailly-Paun's numerical characterization of the Kaehler cone, and a description of the dual cone of the pseudoeffective cone of a projective manifold (Boucksom-Demailly-Paun-Peternell and Witt Nystrom). All of these results use fundamentally nonlinear PDEs, mostly of complex Monge-Ampere type or involving more general symmetric functions of the complex Hessian, to attack basic questions in algebraic geometry and their Kaehler generalizations. New nonlinear PDEs have also been recently introduced on general (possibly non-Kaehler) compact complex manifolds, including the Chern-Ricci flow, a Monge-Ampere equation for $(n-1, n-1)$ forms, and the Fu-Yau version of the Strominger system. This is the right time to further develop these analytic tools, and create new ones, in order to make progress on the many outstanding problems in this area, and to outline future conjectures for the field.
The workshop schedule.
A report on the workshop activities.