We compute the relative orbifold Gromov–Witten invariants of $\left[{ℂ}^2∕{\mathbb{Z}}_{n+1}\right]\times {\mathbb{P}}^1$ with respect to vertical fibers. Via a vanishing property of the Hurwitz–Hodge bundle, 2-point rubber invariants are calculated explicitly using Pixton’s formula for the double ramification cycle, and the orbifold quantum Riemann–Roch. As a result parallel to its crepant resolution counterpart for ${\mathcal{𝒜}}_n$, the GW/DT/Hilb/Sym correspondence is established for $\left[{ℂ}^2∕{\mathbb{Z}}_{n+1}\right]$. The computation also implies the crepant resolution conjecture for the relative orbifold Gromov–Witten theory of $\left[{ℂ}^2∕{\mathbb{Z}}_{n+1}\right]\times {\mathbb{P}}^1$.

Shin and Templier studied families of automorphic representations with local restrictions: roughly, Archimedean components contained in a fixed $L$-packet of discrete series and non-Archimedean components ramified only up to a fixed level. They computed limiting statistics of local components as either the weight of the $L$-packet or level went to infinity. We extend their weight-aspect results to families where the Archimedean component is restricted to a single discrete-series representation instead of an entire $L$-packet.

We do this by using a so-called “hyperendoscopy” version of the stable trace formula of Ferarri. The main technical difficulties are first, defining a version of hyperendoscopy that works for groups without simply connected derived subgroup and second, bounding the values of transfers of unramified functions. We also present an extension to noncuspidal groups of Arthur’s simple trace formula since it does not seem to appear elsewhere in the literature.

We fill a gap in the literature by computing the Chow ring of the classifying space of the unitary group of a hermitian form on a finite-dimensional vector space over a separable quadratic field extension.

We show that the complete symmetric polynomials are dual generators of compressed artinian Gorenstein algebras satisfying the strong Lefschetz property. This is the first example of an explicit dual form with these properties.

For complete symmetric forms of any degree in any number of variables, we provide an upper bound for the Waring rank by establishing an explicit power sum decomposition.

Moreover, we determine the Waring rank, the cactus rank, the resolution and the strong Lefschetz property for any Gorenstein algebra defined by a symmetric cubic form. In particular, we show that the difference between the Waring rank and the cactus rank of a symmetric cubic form can be made arbitrarily large by increasing the number of variables.

We provide upper bounds for the Waring rank of generic symmetric forms of degrees four and five.

We initiate a systematic study of the cohomology of cluster varieties. We introduce the Louise property for cluster algebras that holds for all acyclic cluster algebras, and for most cluster algebras arising from marked surfaces. For cluster varieties satisfying the Louise property and of full rank, we show that the cohomology satisfies the curious Lefschetz property of Hausel and Rodriguez-Villegas, and that the mixed Hodge structure is split over $\mathbb{Q}$. We give a complete description of the highest weight part of the mixed Hodge structure of these cluster varieties, and develop the notion of a standard differential form on a cluster variety. We show that the point counts of these cluster varieties over finite fields can be expressed in terms of Dirichlet characters. Under an additional integrality hypothesis, the point counts are shown to be polynomials in the order of the finite field.