We prove that the number of right ideals of codimension $n$ in the algebra of noncommutative Laurent polynomials in two variables over the finite field ${{\mathbb{F}}_{q}}$ is equal to

$${{\left( q-1 \right)}^{n+1}}_{{}}{{q}^{\frac{\left( n+1 \right)\left( n-2 \right)}{2}}}\sum\limits_{\theta }{{{q}^{inv\left( \theta \right)}}}$$

where the sum is over all indecomposable permutations in ${{S}_{n+1}}$ and where inv $\left( \theta \right)$ stands for the number of inversions of $\theta $ .

We prove a Torelli theorem for the moduli space of semistable parabolic Higgs bundles over a smooth complex projective algebraic curve under the assumption that the parabolic weight systemis generic. When the genus is at least two, using this result we also prove a Torelli theoremfor the moduli space of semistable parabolic bundles of rank at least two with generic parabolic weights. The key input in the proofs is a method of J.C. Hurtubise.

In this paper we introduce two rearrangement optimization problems, one being a maximization and the other a minimization problem, related to a nonlocal boundary value problem of Kirchhoff type. Using the theory of rearrangements as developed by G. R. Burton, we are able to show that both problems are solvable and derive the corresponding optimality conditions. These conditions in turn provide information concerning the locations of the optimal solutions.The strict convexity of the energy functional plays a crucial role in both problems. The popular case in which the rearrangement class (i.e., the admissible set) is generated by a characteristic function is also considered. We show that in this case, the maximization problem gives rise to a free boundary problem of obstacle type, which turns out to be unstable. On the other hand, the minimization problem leads to another free boundary problem of obstacle type that is stable. Some numerical results are included to conûrm the theory.

Let $G$ be a finite group. A faithful $G$ -variety $X$ is called strongly incompressible if every dominant $G$ -equivariant rationalmap of $X$ onto another faithful $G$ -variety $Y$ is birational. We settle the problem of existence of strongly incompressible $G$ -curves for any finite group $G$ and any base field $k$ of characteristic zero.

Let $K/\mathbb{Q}$ be Galois and let $\eta \,\in \,{{K}^{\times }}$ be such that $\text{Re}{{\text{g}}_{\infty }}\left( \eta \right)\,\ne \,0$. We define the local $\theta $-regulator $\Delta _{p}^{\theta }\left( \eta \right)\,\in \,{{\mathbb{F}}_{p}}$ for the ${{\mathbb{Q}}_{p}}$-irreducible characters $\theta $ of $G=\,\text{Gal}\left( K/\mathbb{Q} \right)$. Let ${{V}_{\theta }}$ be the $\theta $-irreducible representation. A linear representation ${{\mathfrak{L}}^{\theta }}\,\simeq \,\delta \,{{V}_{\theta }}$ is associated with $\Delta _{p}^{\theta }\left( \eta \right)$ whose nullity is equivalent to $\delta \,\ge \,1$. Each $\Delta _{p}^{\theta }\left( \eta \right)$ yields $\text{R}eg_{p}^{\theta }\left( \eta \right)$ modulo $p$ in the factorization ${{\Pi }_{\theta }}{{\left( \text{Reg}_{p}^{\theta }\left( \eta \right) \right)}^{\phi \left( 1 \right)}}$ of $\text{Reg}_{p}^{G}\,\left( \eta \right)\,:=\frac{\text{Re}{{\text{g}}_{p}}\left( \eta \right)}{_{p}[K\,:\,\mathbb{Q}]}$ (normalized $p$-adic regulator). From Prob $\left( \Delta _{p}^{\theta }\left( \eta \right)=0\,\text{and}\,{{\mathfrak{L}}^{\theta }}\simeq \delta {{V}_{\theta }} \right)\,\le {{p}^{-f{{\delta }^{2}}}}$ ($f\,\ge \,1$ is a residue degree) and the Borel-Cantelli heuristic, we conjecture that for $p$ large enough, $\text{Reg}_{p}^{G}\left( \eta \right)$ is a $p$-adic unit or ${{p}^{\phi \left( 1 \right)}}\,||\,\text{Reg}_{p}^{G}\left( \eta \right)$ (a single $\theta $ with $f\,=\,\delta \,=\,1$); this obstruction may be led assuming the existence of a binomial probability law confirmed through numerical studies (groups ${{C}_{3,}}\,{{C}_{5}},\,{{D}_{6}}$) is conjecture would imply that for all $p$ large enough, Fermat quotients, normalized $p$-adic regulators are $p$-adic units and that number fields are $p$-rational.We recall some deep cohomological results that may strengthen such conjectures.

The morphism $f\,:\,{{\mathbb{P}}^{N}}\,\to \,{{\mathbb{P}}^{N}}$ is called post-critically finite $\left( \text{PCF} \right)$ if the forward image of the critical locus, under iteration of $f$ , has algebraic support. In the case $N\,=\,1$ , a result of Thurston implies that there are no algebraic families of PCF morphisms, other than a well-understood exceptional class known as the flexible Lattés maps. A related arithmetic result states that the set of PCF morphisms corresponds to a set of bounded height in the moduli space of univariate rational functions. We prove corresponding results for a certain subclass of the regular polynomial endomorphisms of ${{\mathbb{P}}^{N}}$ for any $N$ .

We study a natural discrete Bochner-type inequality on graphs, and explore its merit as a notion of “curvature” in discrete spaces. An appealing feature of this discrete version of the so-called ${{\Gamma }_{2}}$-calculus (of Bakry-Émery) seems to be that it is fairly straightforward to compute this notion of curvature parameter for several specific graphs of interest, particularly, abelian groups, slices of the hypercube, and the symmetric group under various sets of generators. We further develop this notion by deriving Buser-type inequalities (à la Ledoux), relating functional and isoperimetric constants associated with a graph. Our derivations provide a tight bound on the Cheeger constant (i.e., the edge-isoperimetric constant) in terms of the spectral gap, for graphs with nonnegative curvature, particularly, the class of abelian Cayley graphs, a result of independent interest.

Continua $X$ and $Y$ are monotone equivalent if there exist monotone onto maps $f\,:\,X\,\to \,Y$ and $g:\,Y\to \,X.\,\text{A}$ . A continuum $X$ is isolated with respect to monotone maps if every continuumthat is monotone equivalent to $X$ must also be homeomorphic to $X$ . In this paper we show that a dendrite $X$ is isolated with respect to monotone maps if and only if the set of ramification points of $X$ is finite. In this way we fully characterize the classes of dendrites that are monotone isolated.

The notion of families of quantum invertible maps (${{C}^{*}}$-algebra homomorphisms satisfying Podleś condition) is employed to strengthen and reinterpret several results concerning universal quantum groups acting on finite quantum spaces. In particular, Wang's quantum automorphism groups are shown to be universal with respect to quantum families of invertible maps. Further, the construction of the Hopf image of Banica and Bichon is phrased in purely analytic language and employed to define the quantum subgroup generated by a family of quantum subgroups or, more generally, a family of quantum invertible maps.