An Introduction to Derived Algebraic Geometry.- An Introduction to Shifted Symplectic Structures.- An Introduction to Virtual Cycles via Classical Algebraic Geometry.- An Introduction to Virtual Cycles via Derived Algebraic Geometry.- An Introduction to Cohomological Donaldson Thomas Theory.- Moduli Spaces of Sheaves: An Overview, Curves and Surfaces.- Sheaf Counting Theory in Dimension Three and Four.

Enumerative geometry is a core area of algebraic geometry that dates back to Apollonius in the second century BCE. It asks for the number of geometric figures with desired properties and has many applications from classical geometry to modern physics. Typically, an enumerative geometry problem is solved by first constructing the space of all geometric figures of fixed type, called the moduli space, and then finding the subspace of objects satisfying the desired properties. Unfortunately, many moduli spaces from nature are highly singular, and an intersection theory is difficult to make sense of. However, they come with deeper structures, such as perfect obstruction theories, which enable us to define nice subsets, called virtual fundamental classes. Now, enumerative numbers, called virtual invariants, are defined as integrals against the virtual fundamental classes.

Derived algebraic geometry is a relatively new area of algebraic geometry that is a natural generalization of Serre’s intersection theory in the 1950s and Grothendieck’s scheme theory in the 1960s. Many moduli spaces in enumerative geometry admit natural derived structures as well as shifted symplectic structures.

The book covers foundations on derived algebraic and symplectic geometry. Then, it covers foundations on virtual fundamental classes and moduli spaces from a classical algebraic geometry point of view. Finally, it fuses derived algebraic geometry with enumerative geometry and covers the cutting-edge research topics about Donaldson–Thomas invariants in dimensions three and four.