EPSRC Reference: 
EP/T012749/1 
Title: 
Bridgeland stability on Fukaya categories of CalabiYau 2folds 
Principal Investigator: 
Joyce, Professor D 
Other Investigators: 

Researcher CoInvestigators: 

Project Partners: 

Department: 
Mathematical Institute 
Organisation: 
University of Oxford 
Scheme: 
Standard Research 
Starts: 
01 July 2020 
Ends: 
30 June 2023 
Value (£): 
522,545

EPSRC Research Topic Classifications: 

EPSRC Industrial Sector Classifications: 
No relevance to Underpinning Sectors 


Related Grants: 

Panel History: 

Summary on Grant Application Form 
Two key ideas in mathematics are symmetry and classification.
Symmetry is ubiquitous in mathematics, and is the source of endless fascination and study. Many symmetries are wellknown, for example the symmetries of a cube or sphere, but others are far more mysterious and their study has led to great mathematical advances. Mirror symmetry of CalabiYau manifolds has excited much research in mathematics (for example, in Algebraic Geometry and Symplectic Topology), and also in theoretical physics through String Theory, but in general remains poorly understood. Mirror symmetry involves relating the geometry of two CalabiYau manifolds: one aspect of the symmetry is called the "Amodel" and the other is the "Bmodel". Whilst there have been advances in understanding the Bmodel, we seem to currently lack the tools to adequately tackle the Amodel. Our research proposal aims to give a complete understanding of the Amodel for CalabiYau 2folds, which would be a major achievement.
Classification results enable us to describe a large family of mathematical objects that are typically hard to understand in a simpler manner. A typical strategy for classification results in geometry, going back at least to Riemann's Uniformisation Theorem, is to find a special representative for a given class of geometric objects. The challenge then is to determine whether such a special representative exists and, when it does, whether it is unique. In our setting, the special representatives are called special Lagrangians and their uniqueness is known, but the problem of finding them in a given class has proven to be very difficult, despite many attempts to solve it. Our proposal aims to solve this problem for special Lagrangians completely in the setting of CalabiYau 2folds.
The proposed research will combine techniques from distinct areas of mathematics (Symplectic Topology and Geometric Analysis), and it is often the case that some of the most exciting breakthroughs in mathematics occur when different areas are brought together. The connections to further areas of mathematics and theoretical physics mean that the impact of the proposed research is likely to be farreaching and inspire many new research directions which will have a profound effect on the field.

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Organisation Website: 
http://www.ox.ac.uk 