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Since Jun Li's original definition, several other definitions of expanded pairs and expanded degenerations have appeared in the literature. We explain how these definitions are related and introduce several new variants and perspectives. Among these are the twisted expansions used by Abramovich and Fantechi as a basis for orbifold techniques in degeneation formulas., 41 pages

We define Donaldson-Thomas invariants of Calabi-Yau orbifolds and we develop a topological vertex formalism for computing them. The basic combinatorial object is the orbifold vertex, a generating function for the number of 3D partitions asymptotic to three given 2D partitions and colored by representations of a finite Abelian group G acting on C^3. In the case where G=Z_n acting on C^3 with transverse A_{n-1} quotient singularities, we give an explicit formula for the vertex in terms of Schur functions. We discuss applications of our formalism to the Donaldson-Thomas Crepant Resolution Conjecture and to the orbifold Donaldson-Thomas/Gromov-Witten correspondence. We also explicitly compute the Donaldson-Thomas partition function for some simple orbifold geometries: the local football and the local BZ_2 gerbe., 70 pages, lots of figures

We exhibit a set of recursive relations that completely determine all equivariant Gromov-Witten invariants of $[{\Bbb C}^3/{\Bbb Z}_3]$. We interpret such invariants as ${\Bbb Z}_3$-Hodge integrals, and produce relations among them via Atiyah-Bott localization on moduli spaces of twisted stable maps to gerbes over~${\Bbb P}^1$.

We use three different methods to count the number of lines in the plane whose intersection with a fixed general quintic has fixed cross-ratios. We compare and contrast these methods, shedding light on some classical ideas which underlie modern techniques.

The Gromov–Witten theory of Deligne–Mumford stacks is a recent development, and hardly any computations have been done beyond three-point genus 0 invariants. This paper provides explicit recursions which, together with some invariants computed by hand, determine all genus 0 invariants of the stack $\mathbb{P}^2_{D,2}$ . Here $D$ is a smooth plane curve and $\mathbb{P}^2_{D,2}$ is locally isomorphic to the stack quotient $[U/(\mathbb{Z}/(2))]$ , where $U\to V\subseteq \mathbb{P}^2$ is a double cover branched along $D\cap V$ . The introduction discusses an enumerative application of these invariants.

We use twisted stable maps to compute the number of rational degree d plane curves having prescribed contacts to a smooth plane cubic., Comment: 27 pages, v2: typos corrected and references added