Gromov-Witten invariants involve the enumerative geometry of curves in varieties. I will introduce the moduli spaces of stable maps, virtual classes, and the CohFT axioms. The basic techniques of computation (localization, degeneration, and classification) will be discussed in several examples. Open questions about Gromov-Wittengenerating functions and Gromov-Witten cycle classes will be presented.

Symmetric obstruction theories and weighted Euler characteristics: Definition and basic properties of counting invariants of Donaldson-Thomas type. Lecture 2: Beyond numbers: motivic enhancements or categorification, to be determined

We will give an introduction to various versions of the crepant resolution conjecture. This is a whole collection of theorems and conjectures which relate the geometry of a Calabi-Yau orbifold to the geometry of a crepant resolution. It includes the classical and derived McKay correspondences, Ruan's cohomological crepant resolution conjecture, and various versions of the Gromov-Witten and Donaldson-Thomas crepant resolution conjectures.

I will survey wall-crossing phenomena in derived categories and their applications to enumerative geometry, developed in the past 10 years.

Quasimap theory is concerned with curve counting on certain GIT quotients. In fact, one has a family of curve counting theories, with Gromov-Witten included, depending on the linearization in the GIT problem. I will present a wall-crossing formula, in all genera and at the level of virtual classes, as the size of the linearization changes. Some numerical consequences will be discussed as well. The talk will focus primarily on the recently established case of complete intersections in projective space (in this case, stable quasimaps coincide with stable quotients). This is joint work with Bumsig Kim

We decribe the relation between knot contact homology of a knot in the three sphere and the Gromov-Witten theory of its Lagrangian conormal in the resolved conifold.

In my lectures, I will try to give an overview of some of the recent activity regarding enumerative geometry of quiver varieties, both in cohomology and in K-theory, and discuss connections with representation theory and DT theory (time permitting).

I will describe a pair of recent applications of enumerative geometry: one is to classic problems coming from integrable lattice models, and the other to the geometric Langlands correspondence. This is based on joint works with Andrei Okounkov and Edward Frenkel.

In my lectures, I will try to give an overview of some of the recent activity regarding enumerative geometry of quiver varieties, both in cohomology and in K-theory, and discuss connections with representation theory and DT theory (time permitting).