Riemannian Convergence Theory: November 9 – 13, 2015

Organized by Simon Donaldson, Kenji Fukaya, Hans-Joachim Hein, Song Sun

Dates: November 9 – 13, 2015

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The subject of the workshop is the Gromov-Hausdorff convergence of sequences of Riemannian manifolds under a variety of curvature and other natural geometric bounds. Such Gromov-Hausdorff limits are typically highly singu- lar metric spaces. By studying their regularity properties, one obtains uniform structural results for the class of all Riemannian manifolds satisfying a given col- lection of geometric bounds. The Gromov-Hausdorff convergence and regularity theory provides much of the technical underpinning of the more differential- geometric side of the 3 month SCGP program Moduli spaces and singularities in algebraic and Riemannian geometry which this workshop is a part of. It has a long and rich history (work of Cheeger, Fukaya and Gromov on sectional curva- ture bounds and of Cheeger and Colding on Ricci curvature bounds), with spec- tacular recent developments such as the proof of the codimension-4 conjecture for Ricci limit spaces by Cheeger and Naber. On the other hand, applications to algebraic geometry such as the ones that the SCGP program will focus on (in particular those related to the “collapsing” of Calabi-Yau manifolds) will require significant further development of the Riemannian theory. While the importance of this Riemannian convergence theory, and some of its main results, are now widely appreciated the more technical aspects of the theory are much less well-known. One purpose of the workshop is to gather together some of the main experts on Riemannian convergence theory to review recent progress in the area, learn about new challenges from the algebraic geom- etry side, and interact with the other participants of the 3-month programme coming from a variety of different backgrounds. The topics discussed will in- clude both the more traditional convergence theory (Riemannian manifolds of bounded sectional or Ricci curvature) and on more recent, less-developed, aspects of the subject such as Kahler metrics of constant scalar curvature and limits of geometric flows.