Workshop on Calabi-Yau Manifolds and Special Lagrangian Submanifolds
Description
Calabi-Yau metrics and special Lagrangians submanifolds are some of the most fundamental objects in complex geometry and geometric analysis, and they have been a very active subject in recent years. Insights from physics have led to deep conjectures regarding the existence and behaviour of Calabi-Yau manifolds and their special Lagrangian submanifolds. The Strominger–Yau–Zaslow (SYZ) conjecture, originating from string theory, predicts the existence of special Lagrangian torus fibrations within Calabi-Yau manifolds under “large complex structure limits” and offers insight into how these can be used to understand Mirror Symmetry of Calabi-Yau manifolds, while the Thomas–Yau conjecture predicts that existence of special Lagrangians submanifolds is governed by stability conditions. There have been many important developments in these directions in the past few years.
The SYZ conjecture has seen some spectacular progress in recent years due to the breakthroughs of Yang Li, which established a generic form of the conjecture for a large class of Calabi-Yau manifolds, while significant progress was also made towards the Thomas-Yau conjecture, including the work of Lotay–Schulze–Szekelyhidi on the singularities of Lagrangian mean curvature flow, and recent novel constructions of special Lagrangians due to Li and Chiu–Lin following a program of Donaldson and Scaduto. Other important recent developments include a refined understanding of Calabi-Yau metrics under “small complex structure degenerations” due to Sun–Zhang, new analytic developments leading to a regularity theory of singular Calabi-Yau metrics by the work of Guo–Phong–Song–Sturm and Szekelyhidi, and progress on constructions of complete Calabi-Yau metrics of Tian–Yau type by Collins–Li, Collins–Tong–Yau, and Collins–Tong. This is an exciting time for this subject. The tools developed in the past ten years have led to astonishing progress on many old questions, and a long list of conjectures still leaves open many avenues for future investigation.
The goal of this workshop is to bring together the experts in complex geometry, PDEs, and physics, alongside junior researchers in the field, to explore these developments, exchange ideas, and foster new research directions.
