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**School**

**Marco Robalo (Sorbonne Universite) - TBA**

**Adeel Khan (Academia Sinica) - TBA**

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**Jeongseok Oh (Imperial College)**

Title: An introduction to virtual cycles via classical algebraic geometry

Abstract: This is a classical preview of Hyeonjun Park's lecture.

We explain why we need a virtual fundamental cycle rather than fundamental cycle and a difficulty in its contsruction. Then we see how Li-Tian and Behrend-Fantechi overcame the difficulty in a reasonable circumstance.

Kiem-Li's seminal work, so called cosection localisation, has contributed to the theory significantly. We discuss what it is, how it has contributed, especially to counting stable sheaves on Calabi-Yau 4-folds.

**Hyeonjun Park (Korea Institute for Advanced Study)**

Title: An introduction to virtual cycles via derived algebraic geometry

Abstract: Modern enumerative geometry studies numerical invariants defined through virtual cycles on moduli spaces. Conceptually, these virtual fundamental cycles are the fundamental cycles of quasi-smooth derived enhancements. Although classical approximations called perfect obstruction theories are sufficient for constructing virtual cycles, their properties become more natural and conceptual from the perspective of derived algebraic geometry.

In the first talk, we discuss the original virtual fundamental cycles of Li-Tian and Behrend-Fantechi via derived algebraic geometry. In particular, we explain Kiem-Li’s cosection localization in terms of (-1)-shifted closed 1-forms and Graber-Pandharipande’s torus localization formula through derived mapping stacks.

In the second talk, we discuss the Donaldson-Thomas theory of Calabi-Yau 4-folds, where enumerative invariants are defined through a new type of virtual cycles introduced by Borisov-Joyce and Oh-Thomas. Heuristically, these new virtual cycles are the fundamental cycles of quasi-smooth Lagrangians on (-2)-shifted symplectic derived schemes. We present some basic properties of these new virtual cycles (e.g., virtual pullback formula, torus localization formula, cosection localization) and explain how to use them in practical situations.

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**Tasuki Kinjo (Kyoto University)**

(1) Title: An introduction to cohomological Donaldson-Thomas theory

Abstract: In this talk, I will provide a broad overview of the cohomological Donaldson-Thomas theory, which is a sheaf-theoretic categorification of the Donaldson-Thomas theory. First, I will explain the construction of the vanishing cycle functor. Then, I will discuss the work of Joyce and his collaborators (as well as independent work by Kiem-Li) on the construction of the Donaldson-Thomas perverse sheaves attached to (-1)-shifted symplectic derived Artin stacks (such as the derived moduli stack of coherent sheaves on Calabi-Yau threefolds). This perverse sheaf is a globalisation of the vanishing cycle complexes. If time permits, I will also discuss some recent progresses on cohomological Donaldson-Thomas theory.

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**Woonam Lim (ETH Zurich)**

Title: Moduli spaces of sheaves: an overview, curves, and surfaces

Abstract: This is an introduction to the moduli spaces of sheaves and enumerative problems therein. I will start with an overview of the theory focusing on generalities (moduli stack, construction of moduli spaces, tautological classes, deformation theory). The goal of the rest of the lectures is to convince that the moduli of sheaves on curves and surfaces are extremely rich sources of fascinating geometries and problems. In order to do so, I will touch upon the following topics; topology of the moduli spaces and Verlinde formulas for curves and Hilbert scheme of points, Vafa-Witten invariants, and one-dimensional sheaves for surfaces.

**Younghan Bae (ETH Zurich) **

Title: Sheaf counting theory for dimension three and four

Abstract: In this course, I will give an overview on sheaf counting problems for threefolds and Calabi-Yau fourfold. We will start from introducing the Donaldson-Thomas/Pandharipande-Thomas theory for smooth projective 3-folds. I will introduce various conjectures on the structure of invariants and conjectural relations to other curve counting theories such as the Gromov-Witten theory. In the second part, I will give an overview of recent developments on sheaf counting theories on Calabi-Yau 4-folds.