Schedule and Abstracts

Abstract: In this gentle introductory talk, I will explain how to cook up interesting abelian subcategories (or more precisely, t-structures) in your favourite triangulated category. These techniques are widely used with derived categories of coherent sheaves (especially in the context of Bridgeland stability), but most of the ideas go back much further to the work of Beilinson, Bernstein and Deligne on perverse sheaves in topology. I will illustrate everything with plenty of examples, drawn from both of these worlds.

Abstract: Perverse Schober is a categorification of a perverse sheaf proposed by Kapranov and Schechtman. I will explain the background on the theory of perverse schobers, together with my joint work with Genki Ouchi (Nagoya), where we constructed interesting examples of perverse schobers on the Riemann sphere coming from the mirror symmetry for Calabi-Yau hypersurfaces in the projective spaces.

Abstract: Kuznetsov conjectured the existence of equivalences between Kuznetsov categories of double quartic solids, and of Gushel-Mukai threefolds. I will explain an approach to this question via equivariant derived categories. This first disproves the existence of such equivalences (a statement first shown by Shizhuo Zhang). It also allows to prove that they are, instead, deformation equivalent. This is joint work with Alex Perry.

Abstract: I will start with introducing tilt stability/Bridgeland stability conditions and wall-crossing in dimension three. Then to see some applications to sheaf-theoretic moduli problems, I will introduce the notion of stable pairs and explain how wall-crossing can help understanding the moduli space of stable pairs for space curves of specific types. Time permitting, I will explain how combining this with derived algebraic geometry can extend the picture to handle the moduli space of stable pairs for general space curves.

Abstract: This will be an introductory talk about the interplay of representations, geometry, and the theory of stability conditions. I'll introduce a root system and present the representation theoretic structure it carries. Then I'll illustrate the general process of lifting this information to a categorical level through the stability manifold. I will mainly focus on the example of root systems arising from quotients of elliptic curves.

Abstract: This talk will survey the theory of compactified Jacobians associated with a singular curve and discuss how the recent development of derived algebraic geometry sheds new light on this old subject. The plan is as follows:

(1) For a smooth curve, there is an associated Abelian variety called Jacobian variety, which plays an essential role in studying the geometry of the curve; see Mumford’s book “Curves and their Jacobians” for a beautiful introduction.

(2) If the curve is singular, we associate a degenerate Abelian variety, called the compactified Jacobian. There is a growing literature on the study of compactified Jacobians since the 1980s, including the works of Altman, Kleiman, Hartshorne, etc; see Jesse Kass’s paper “Singular Curves and their Compactified Jacobian” for a nice survey. One central idea is to use Abel maps to describe the compactified Jacobians.

(3) Finally, we will mention how the framework of derived algebraic geometry sheds new light and provides us with further understanding of the Abel maps.