We show that the Virasoro conjecture in Gromov–Witten theory holds for the the total space of a toric bundle E → B if and only if it holds for the base B. The main steps are: (i) we establish a localization formula that expresses Gromov–Witten invariants of E, equivariant with respect to the fiberwise torus action, in terms of genus-zero invariants of the toric fiber and all-genus invariants of B; and (ii) we pass to the non- equivariant limit in this formula, using Brown’s mirror theorem for toric bundles.

For a gerbe Y over a smooth proper Deligne-Mumford stack B banded by a finite group G, we prove a structure result on the Gromov-Witten theory of Y, expressing Gromov-Witten invariants of Y in terms of Gromov-Witten invariants of B twisted by various flat U(1)-gerbes on B. This is interpreted as a Leray-Hirsch type of result for Gromov-Witten theory of gerbes.

We prove a Givental-style mirror theorem for toric Deligne--Mumford stacks X. This determines the genus-zero Gromov--Witten invariants of X in terms of an explicit hypergeometric function, called the I-function, that takes values in the Chen--Ruan orbifold cohomology of X.

In this article, we survey recent developments in defining the quasi-local mass in general relativity. We discuss various approaches and the properties and applications of the different definitions. Among the expected properties, we focus on the rigidity property: for a surface in the Minkowski spacetime, one expects that the mass should vanish. We describe the Wang-Yau quasi-local mass whose definition is motivated by this rigidity property and by the Hamilton-Jacobi analysis of the Einstein-Hilbert action. In addition, we survey recent results on the minimizing property the Wang-Yau quasi-local mass.

Pop-up books are a fascinating form of paper art with intriguing geometric properties. In this paper, we present a systematic study of a simple but common class of pop-ups consisting of patches falling into four parallel groups, which we call v-style pop-ups. We give sufficient conditions for a v-style paper structure to be pop-uppable. That is, it can be closed flat while maintaining the rigidity of the patches, the closing and opening do not need extra force besides holding two patches and are free of intersections, and the closed paper is contained within the page border. These conditions allow us to identify novel mechanisms for making pop-ups. Based on the theory and mechanisms, we developed an interactive tool for designing v-style pop-ups and an automated construction algorithm from a given geometry, both of which guaranteeing the popuppability of the results.