Perverse schobers and the McKay correspondence

Project Overview

This project focuses on the study of wall-crossing problems in algebraic geometry and theoretical physics, particularly examining how moduli spaces change as stability parameters vary. The student will explore perverse schobers, a recent mathematical framework packaging wall-crossing correspondences into a perverse sheaf of triangulated categories, unlocking insights into hidden higher categorical wall-crossings. The work centers on the classical 2-dimensional McKay correspondence for ADE groups, a well-studied and computable case with significant UK expertise.

What You Will Do

The student will analyze perverse schobers within the McKay correspondence context, studying transformations and autoequivalences of moduli spaces and the associated spherical functors. Engagement with research groups at Bath, Warwick, Glasgow, and the Institute for Physics and Mathematics of the Universe (IPMU) at the University of Tokyo is expected, fostering collaboration and interdisciplinarity.

Expected Outcomes

The student will develop new understanding of wall-crossing phenomena and categorical equivalences encoded by perverse schobers, potentially revealing new mathematical structures and physical interpretations. The research will contribute to both mathematical theory and the physics of string theory compactifications.

Why This Matters

Wall-crossing mathematics connects deep algebraic geometry with string theory, impacting mathematical physics and providing structural insights into moduli spaces and stability phenomena. This research has foundational importance in geometry and theoretical physics, offering tools for further advances in both fields.