BIMSA > BIMSA AG Seminar BIMSA AG Seminar Donaldson-Uhlenbeck-Yau Correspondence for objects in the derived category
Donaldson-Uhlenbeck-Yau Correspondence for objects in the derived category
Organizers
Artan Sheshmani , Nanjun Yang , Beihui Yuan
Speaker
Artan Sheshmani
Time
Thursday, August 27, 2026 3:00 PM - 4:00 PM
Venue
A7-201
Online
Zoom 638 227 8222 (BIMSA)
Abstract
Donaldson-Uhlenbeck-Yau (DUY) theorem is one of the most profound established correspondences bridging Algebraic Geometry, Differential Geometry, theory of PDE and Mathematical Physics. While the original DUY correspondence provides a landmark equivalence between the existence of Hermitian-Einstein metrics on vector bundles and algebraic slope stability, extending this deep principle to objects of the derived category $D^b \mathrm{Coh}(X)$ has remained a decades-long open problem. I will discuss construction of a derived version of Donaldson- Uhlenbeck-Yau theorem. I introduce a derived-categorical framework for Hermitian metrics on objects of the derived category of coherent sheaves. I show that a Hermitian resolution on an object in derived category induces a nonlinear PDE whose satisfaction is in direct correspondence with Spencer Stability of the object in the derived category. I will discuss Spencer Stability and its connections to integrability of nonlinear PDE, based on my earlier joint work with Shing-Tung Yau and Jacob Kryczka.
Speaker Intro
Artan Sheshmani is a Professor of pure Mathematics, specialized in Algebraic geometry, Enumerative and Derived Geometry, and Mathematics of String Theory. He joined BIMSA as a Professor in September 2023. Prior to BIMSA he was a senior personnel at Simons Collaboration Program on Homological Mirror Symmetry at Harvard University Center for Mathematical Sciences and Applications (CMSA) for 7 years, during a portion of which he was jointly an Associate Professor of Mathematics at Institut for Mathematik (formerly the Center for Quantum Geometry of Moduli Spaces) at Aarhus University in Denmark (2016-2022). He is working on geometry of moduli spaces of sheaves and curves from enumerative geometry point of view as well as studying their structural properties from derived geometry and geometric representation theory point of view.