**15 apr 2024 - 29 jun 2024**

CRM - Montréal

The last few years have seen spectacular progress on a variety of very difficult geometric problems. Probably the most familiar of these problems to the general mathematical audience is the solution of the Poincaré conjecture, but the general idea of taking a canonical variational flow to obtain geometric objects of interest of course has its origins much earlier, notably in the work of Yau, of Donaldson, and of several others in the 1970s and 1980s. The more recent Ricci flow solution of the Poincaré conjecture has stimulated a whole series of major breakthroughs in related areas of geometry, notably in complex and Kähler geometry, for example in the solution of the Kähler-Einstein existence problem for Fano varieties.

While there is a hard core of non-linear pde to the subject, to a certain degree the ideas are often reflections of finite dimensional ones; not only the quite classical idea of a gradient flow, but also the more technical notion of stability emerging from algebraic geometry, and a concomitant interest in moduli problems and in singularities.

This semester-long program in geometric analysis will focus mostly on complex geometry and Kähler geometry, but with the occasional excursion into real geometry. While the inspiration has deep geometrical roots, the tools are to a large degree those of partial differential equations.

A lot of the activity will centre on six workshops:

PDEs in Complex Geometry (April 15-19, 2024)

Special Riemannian Metrics in Dimensions 6,7,8 (April 22-26, 2024)

Analysis of Geometric Singularities (May 13-17, 2024)

Moduli Spaces and Singularities (May 20-24, 2024)

Current Trends in Kähler Metrics with Special Curvature Properties (June 17-21, 2024)

Current Trends in Geometric Flows (June 25-29, 2024)