Let $G$ be a locally compact group and let $\omega$ be a continuous weight on $G$. We show that for each of the Banach algebras ${{L}^{1}}\left( G,\,\omega \right),\,M\left( G,\,\omega \right),\,LUC{{\left( G,\,{{\omega }^{-1}} \right)}^{*}}$, and ${{L}^{1}}{{\left( G,\,\omega \right)}^{**}}$, the order structure combined with the algebra structure determines the weighted group.

The dual immaculate functions are a basis of the ring QSym of quasisymmetric functions and form one of the most natural analogues of the Schur functions. The dual immaculate function corresponding to a composition is a weighted generating function for immaculate tableaux in the same way as a Schur function is for semistandard Young tableaux; an immaculate tableau is defined similarly to a semistandard Young tableau, but the shape is a composition rather than a partition, and only the first column is required to strictly increase (whereas the other columns can be arbitrary, but each row has to weakly increase). Dual immaculate functions were introduced by Berg, Bergeron, Saliola, Serrano, and Zabrocki in arXiv:1208.5191, and have since been found to possess numerous nontrivial properties.

In this note, we prove a conjecture of M. Zabrocki that provides an alternative construction for the dual immaculate functions in terms of certain “vertex operators”. The proof uses a dendriform structure on the ring QSym; we discuss the relation of this structure to known dendriform structures on the combinatorial Hopf algebras FQSym and WQSym.

We continue the investigation of the isomorphism problem for multiplier algebras of reproducing kernel Hilbert spaces with the complete Nevanlinna-Pick property. In contrast to previous work in this area, we do not study these spaces by identifying them with the restrictions of a universal space, namely the Drury-Arveson space. Instead, we work directly with the Hilbert spaces and their reproducing kernels. In particular, we show that two multiplier algebras of Nevanlinna-Pick spaces on the same set are equal if and only if the Hilbert spaces are equal. Most of the article is devoted to the study of a special class of complete Nevanlinna-Pick spaces on homogeneous varieties. We provide a complete answer to the question of when two multiplier algebras of spaces of this type are algebraically or isometrically isomorphic.This generalizes results of Davidson, Ramsey,Shalit, and the author.

Under the local Langlands correspondence, the conductor of an irreducible representation of $\text{G}{{\text{l}}_{n}}\left( F \right)$ is greater than the Swan conductor of the corresponding Galois representation. In this paper, we establish the geometric analogue of this statement by showing that the conductor of a categorical representation of the loop group is greater than the irregularity of the corresponding meromorphic connection.

Let $\mathbf{k}$ be an algebraically closed field complete with respect to a non-Archimedean absolute value of arbitrary characteristic. Let ${{D}_{1}},...,\,{{D}_{n}}$ be effective nef divisors intersecting transversally in an $n$-dimensional nonsingular projective variety $X$. We study the degeneracy of non-Archimedean analytic maps from $\mathbf{k}$ into $X\,\backslash \,\cup _{i=1}^{n}\,{{D}_{i}}$ under various geometric conditions. When $X$ is a rational ruled surface and ${{D}_{1}}$ and ${{D}_{2}}$ are ample, we obtain a necessary and sufficient condition such that there is no non-Archimedean analytic map from $\mathbf{k}$ into $X\,\backslash \,{{D}_{1}}\,\cup \,{{D}_{2}}$. Using the dictionary between non-Archimedean Nevanlinna theory and Diophantine approximation that originated in earlier work with T. T. H. An, we also study arithmetic analogues of these problems, establishing results on integral points on these varieties over $\mathbb{Z}$ or the ring of integers of an imaginary quadratic field.

We consider Schubert problems with respect to flags osculating the rational normal curve. These problems are of special interest when the osculation points are all real. In this case, for zero-dimensional Schubert problems, the solutions are “ as real as possible”. Recent work by Speyer has extended the theory to the moduli space $\overline{{{\mathcal{M}}_{0,\,r}}}$ allowing the points to collide. This gives rise to smooth covers $\overline{{{\mathcal{M}}_{0,\,r}}}\left( \mathbb{R} \right)$, with structure and monodromy described by Young tableaux and jeu de taquin.

In this paper, we give analogous results on one-dimensional Schubert problems over $\overline{{{\mathcal{M}}_{0,\,r}}}$.Their(real) geometry turns out to be described by orbits of Schützenberger promotion and a related operation involving tableau evacuation. Over ${{\mathcal{M}}_{0,\,r}}$, our results show that the real points of the solution curves are smooth.

We also find a new identity involving “first-order” $\text{K}$-theoretic Littlewood-Richardson coefficients, for which there does not appear to be a known combinatorial proof.

The preservation principle of local theta correspondences of reductive dual pairs over a $p$-adic field predicts the existence of a sequence of irreducible supercuspidal representations of classical groups. Adams and Harris-Kudla-Sweet have a conjecture about the Langlands parameters for the sequence of supercuspidal representations. In this paper we prove modified versions of their conjectures for the case of supercuspidal representations with unipotent reduction.

We study the Chern-Ricci flow, an evolution equation of Hermitian metrics, on a family of Oeljeklaus–Toma $\left( \text{OT-} \right)$ manifolds that are non-Kähler compact complex manifolds with negative Kodaira dimension. We prove that after an initial conformal change, the flow converges in the Gromov–Hausdorff sense to a torus with a flat Riemannian metric determined by the $\text{OT}$-manifolds themselves.