We compute the Hochschild cohomology and homology for arbitrary finite-dimensional quantum complete intersections. It turns out that their behavior varies widely, depending on the choice of commutation parameters, and we give precise criteria for when to expect what behavior.

We consider the one-parameter family of hypersurfaces in ${\mathbb{P}}^5$ over $\mathbb{Q}$ with projective equation $\left(X_1^5+X_2^5+X_3^5+X_4^5+X_5^5\right)=5t{X}_1{X}_2\dots X_5$, proving that the Galois representations attached to their cohomologies are potentially automorphic, and hence that the zeta function of the family has meromorphic continuation to the whole complex plane.

We give explicit equations for the Chow and Hilbert quotients of a projective scheme $X$ by the action of an algebraic torus $T$ in an auxiliary toric variety. As a consequence we provide geometric invariant theory descriptions of these canonical quotients, and obtain other GIT quotients of $X$ by variation of GIT quotient. We apply these results to find equations for the moduli space ${\overline{M}}_{0,n}$ of stable genus-zero $n$-pointed curves as a subvariety of a smooth toric variety defined via tropical methods.

In previous work we showed that two apparently unrelated formulas for the Hall–Littlewood polynomials of type $A$ are, in fact, closely related. The first is the tableau formula obtained by specializing $q=0$ in the Haglund–Haiman–Loehr formula for Macdonald polynomials. The second is the type $A$ instance of Schwer’s formula (rephrased and rederived by Ram) for Hall–Littlewood polynomials of arbitrary finite type; Schwer’s formula is in terms of so-called alcove walks, which originate in the work of Gaussent and Littelmann and of the author with Postnikov on discrete counterparts to the Littelmann path model. We showed that the tableau formula follows by “compressing” Ram’s version of Schwer’s formula. In this paper, we derive new tableau formulas for the Hall–Littlewood polynomials of type $B$ and $C$ by compressing the corresponding instances of Schwer’s formula.

After identification of the algebra of exponential generating series with the shuffle algebra of ordinary formal power series, the exponential map

$${exp}_{!}:X\mathbb{K}\left[\left[X\right]\right]\to 1+X\mathbb{K}\left[\left[X\right]\right]$$

for the associated Lie group with multiplication given by the shuffle product is well-defined over an arbitrary field $\mathbb{K}$ by a result going back to Hurwitz. The main result of this paper states that ${exp}_{!}$ and its reciprocal map ${log}_{!}$ induce a group isomorphism between the subgroup of rational, respectively algebraic series of the additive group $X\mathbb{K}\left[\left[X\right]\right]$ and the subgroup of rational, respectively algebraic series in the group $1+X\mathbb{K}\left[\left[X\right]\right]$ endowed with the shuffle product, if the field $\mathbb{K}$ is a subfield of the algebraically closed field ${\overline{\mathbb{F}}}_p$ of characteristic $p$.

Berger and Colmez (2008) formulated a theory of families of overconvergent étale $\left(\phi ,\Gamma \right)$-modules associated to families of $p$-adic Galois representations over $p$-adic Banach algebras. In contrast with the classical theory of $\left(\phi ,\Gamma \right)$-modules, the functor they obtain is not an equivalence of categories. In this paper, we prove that when the base is an affinoid space, every family of (overconvergent) étale $\left(\phi ,\Gamma \right)$-modules can locally be converted into a family of $p$-adic representations in a unique manner, providing the “local” equivalence. There is a global mod $p$ obstruction related to the moduli of residual representations.