Seminar on Complex Analysis and Complex Geometry

                                              Fall 2026

Please watch for changing meeting times due to speakers' travel plans. For more information, please send an email to Gordon Heier.

Wednesday, October 14, 2026, 11-12, room PGH 648

Title: Moduli and derived categories of sheaves on algebraic surfaces

Speaker: Yinbang Lin, University of Houston

Abstract: Moduli theory is concerned with the classification of objects, a basic problem throughout mathematics. The moduli space of vector bundles (or sheaves) on an algebraic surface is expected to reflect the geometry of the surface in a highly non-trivial way. Instead of a single vector bundle, one can consider a complex of vector bundles, which leads to the notion of a derived category. One can then study moduli of complexes, which offers much more flexibility for a variety of problems. To obtain moduli spaces of finite type, one must impose a stability condition. In this talk, I will introduce some basic notions from moduli theory and derived categories, including stability conditions and Quot schemes. This is an introductory talk for graduate students for my talk next week.

Wednesday, October 21, 2026, 11-12, room PGH 648

Title: Extensions and Segre stratifications on algebraic surfaces

Speaker: Yinbang Lin, University of Houston

Abstract: Motivated by the Verlinde/Segre correspondence and the strange duality conjecture, we study two closely related problems on algebraic surfaces: the stability of extensions of stable sheaves by stable sheaves, and the maximal subsheaves of a given stable sheaf. For the former, we construct complete families of sheaves via extensions and prove stability in certain cases. The latter leads to a refinement of the Segre stratification of the moduli space of sheaves. We obtain the expected dimension of certain refined Segre strata over rational surfaces and K3 surfaces. Over the projective plane, we prove that all refined Segre strata corresponding to line bundles are irreducible of the expected dimension and are nested under taking closures. These results suggest that refined Segre strata are much better behaved than Brill–Noether strata. Our study relies crucially on Bridgeland stability conditions and on the work of Li and Zhao. This is joint work with Thomas Goller (arXiv:2609.13684).