We define and study a series indexed by rooted trees and with coefficients in $\mathbb{Q}\left(q\right)$. We show that it is related to a family of Lie idempotents. We prove that this series is a $q$-deformation of a more classical series and that some of its coefficients are Carlitz $q$-Bernoulli numbers.

Monodromy arguments and deformation-and-specialization are used to prove existence of a pencil of Enriques surfaces with no section and index 1. The same technique “completes” the strategy from Graber et al. (2005) proving that the family of witness curves for dimension $d$ depends on the integer $d$.

We present a method for compactifying stacks of PGL${}_n$-torsors (Azumaya algebras) on algebraic spaces. In particular, when the ambient space is a smooth projective surface we use our methods to show that various moduli spaces are irreducible and carry natural virtual fundamental classes. We also prove a version of the Skolem–Noether theorem for certain algebra objects in the derived category, which allows us to give an explicit description of the boundary points in our compactified moduli problem.

We prove the effectiveness of the log Iitaka fibration in Kodaira codimension two for varieties of dimension $\le 4$. In particular, we finish the proof of effective log Iitaka fibration in dimension two. Also, we show that for the log Iitaka fibration, if the fiber is of dimension two, the denominator of the moduli part is bounded.

In previous work we have developed a general formalism for Schubert calculus. Here we show how this theory can be adapted to give a formalism for equivariant Schubert calculus consisting of a basis theorem, a Pieri formula and a Giambelli formula. Our theory specializes to a formalism for equivariant cohomology of grassmannians. We interpret the results in a ring that can be considered as the formal generalized analog of localized equivariant cohomology of infinite grassmannians.