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.

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Starting with a commutative ring $R$ and an ideal $I$ , it is possible to define a family of rings $R(I)_{a,b}$ , with $a,b \in R$ , as quotients of the Rees algebra $\oplus_{n \geq0} I^{n}t^{n}$ ; among the rings appearing in this family we find Nagata’s idealization and amalgamated duplication. Many properties of these rings depend only on $R$ and $I$ and not on $a$ , $b$ ; in this paper we show that the Gorenstein and the almost Gorenstein properties are independent of $a$ , $b$ . More precisely, we characterize when the rings in the family are Gorenstein, complete intersection, or almost Gorenstein and we find a formula for the type.

The Lp dual curvature measure was introduced by Lutwak, Yang & Zhang in an attempt to unify the Lp Brunn– Minkowski theory and the dual Brunn–Minkowski theory. The characterization problem for Lp dual curvature measure, called the Lp dual Minkowski problem, is a fundamental problem in this unifying theory. The Lp dual Minkowski problem contains the Lp Minkowski problem and the dual Minkowski problem, two major problems in modern convex geometry that remain open in general. In this paper, existence results on the Lp dual Minkowski problem in the weak sense will be provided. Moreover, existence and uniqueness of the solution in the smooth category will also be demonstrated.