Victoria Hoskins:  Moduli of representations of a quiver with multiplicities 

Abstract: There is a long and rich history connecting the representation theory of a quiver over a field and an associated symmetric Kac--Moody Lie algebra, where moduli spaces related to the quiver give rise to geometric representations of quantum groups associated to this Kac--Moody Lie algebra. Representations of a quiver with multiplicities are defined analogously by replacing the field with truncated polynomial rings, and they are related to a larger class of symmetrisable Kac-Moody Lie algebras. Although much of our motivation comes from geometric representation theory, this course will focus on geometric aspects of moduli of representations of a quiver with multiplicities. The main goal is to describe how to construct moduli spaces and quiver varieties for a quiver with multiplicities as quotients of a non-reductive group action, and describe the geometry of these moduli spaces. I will begin with reviewing the construction of moduli stacks and spaces in the classical case (without multiplicities) and then explain how to modify this in the case with multiplicities. This is all based on joint works with E. Hamilton, J. Jackson and T. Vernet. 

Zhiyu Liu 刘治宇 :  Bridgeland stability conditions on projective varieties

Abstract: In these two talks, I will begin by reviewing the classical theory of stable sheaves on varieties, and then introduce Bridgeland stability conditions on triangulated categories. Then I will explain some results in this direction, including Chunyi Li's recent construction of stability conditions on projective varieties and the generalization to the relative setting, as well as recent progress on the Bayer–Macrì–Toda Conjecture on threefolds.

Mauro Porta :  Derived tools in enumerative geometry

Abstract: In this 3h course I will go over the basics of derived geometry, discussing virtual fundamental classes. I will then explain how to use them to construct enumerative invariants and categorifications thereof, especially focusing on the (categorified) Hall product and the construction of quasi-BPS categories in various settings due to Padurariu and Toda.

Junliang Shen 沈俊亮:  Compact hyper-Kähler varieties: derived categories, cohomology, and algebraic cycles

Abstract: The derived Torelli theorem for K3 surfaces gives a complete cohomological description of when two K3 surfaces are derived equivalent. In higher dimensions, derived equivalent compact hyper-Kähler varieties are much more mysterious. On the one hand, a conjecture of Bondal–Orlov predicts that birational ones are derived equivalent; on the other hand, conjectures of Orlov and Fu–Vial predict that derived equivalent ones have isomorphic motives (the universal cohomology). I will explain some recent progress on these conjectures for varieties of K3[n]-type, including work of Buskin, Markman, Maulik–Shen–Yin–Zhang, and Maulik–Shen–Yin.

Yang Zhou 周杨:  Stable quasimaps and their wall-crossing formula

Abstract: The notation of quasimaps is a generalization of regular maps from curves into a large class of GIT quotients. For each positive number epsilon, the moduli of epsilon-stable quasimaps, introduced by Ciocan-Fontanine--Kim--Maulik, provides an alternative compactification of the space of regular maps with smooth domains. The space of stability conditions has a wall-and-chamber structure. As epsilon tends to infinity, one recovers stable maps and Gromov-Witten theory. The wall-crossing formula relating those different counting theories extracts the contribution of ``rational tails'' to Gromov-Witten invariants, which turns out to coincide with the mirror map. In the first lecture, I will explain the definition of epsilon-stable quasimaps and the wall-crossing formula. In the second lecture, I will explain a geometric proof of the wall-crossing formula via the so-called master space technique.