Let $Y\to B$ be a relative smooth projective curve over an affine integral base scheme $B$ of positive characteristic. We provide for all prime characteristics example classes of vector bundles $\mathcal{S}$ over $Y$ such that $\mathcal{S}$ is generically strongly semistable and semistable but not strongly semistable for some special fibre.

In this paper, we return to the study of the Watson kernel for the Abel summability of Jacobi polynomial series. These estimates have been studied for over more than 40 years. The main innovations are in the techniques used to get the estimates that allow us to handle the cases $0<\alpha$ as well as $-1<\alpha<0$, with essentially the same methods. To that effect, we use an integral superposition of Natanson kernels, and the A. P. Calderón-Kurtz, B. Muckenhoupt $A_{p}$-weight theory. We consider also a generalization of a theorem due to Zygmund in the context of Borel measures. The proofs are different from the ones given in (Sobre la conjugación y sumabilidad de series de Jacobi (1971) Universidad de Buenos Aires, Studia Math. 49 (1974) 217–224, Colloq. Math. 30 (1974) 277–288 and Illinois J. Math. 41 (1997) 237–265). We will discuss in detail the Calderón–Zygmund decomposition for nonatomic Borel measures in $\mathbb{R}$. We prove that the Jacobi measure is doubling and following (Studia Math. 57 (1976) 297–306), we study the $A_{p}$ weight theory in the context of Abel summability of Jacobi expansions. We consider power weights of the form $(1-x)^{\overline{\alpha}}$, $(1+x)^{\overline{\beta}}$, $-1<{\overline{\alpha}}<0$, $-1<{\overline{\beta}}<0$. Finally, as an application of the weight theory we obtain $L^{p}$ estimates for the maximal operator of Abel summability of Jacobi function expansions for suitable values of $p$.

The coeffective differential complex on a symplectic manifold is extended both in length and in scope, unifying the constructions of various other authors.

We construct a family of canonical connections and surrounding basic theory for almost complex manifolds that are equipped with an affine connection. This framework provides a uniform approach to treating a range of geometries. In particular, we are able to construct an invariant and efficient calculus for conformal almost Hermitian geometries, and also for almost complex structures that are equipped with a projective structure. In the latter case, we find a projectively invariant tensor the vanishing of which is necessary and sufficient for the existence of an almost complex connection compatible with the path structure. In both the conformal and projective setting, we give torsion characterisations of the canonical connections and introduce certain interesting higher order invariants.

We establish conditions under which the universal and reduced norms coincide for a Fell bundle over a groupoid. Specifically, we prove that the full and reduced $C^{*}$-algebras of any Fell bundle over a measurewise amenable groupoid coincide, and also that for a groupoid $G$ whose orbit space is $T_{0}$, the full and reduced algebras of a Fell bundle over $G$ coincide if the full and reduced algebras of the restriction of the bundle to each isotropy group coincide.

In this paper, we define and discuss Eichler integrals for Maass cusp forms of half-integral weight on the full modular group. We discuss nearly periodic functions associated to the Eichler integrals, introduce period functions for such Maass cusp forms, and show that the nearly periodic functions and the period functions are closely related. Those functions are extensions of the periodic functions and period functions for Maass cusp forms of weight $0$ on the full modular group introduced by Lewis and Zagier.

Motivated by a geometric meaning of Mahler’s measure, we introduce two operator analogues of Mahler’s measure. This leads to some interesting equalities and inequalities between the two operator-theoretic Mahler measures and the classical Mahler measure. In order to apply these results to the operator version of Lehmer’s problem, we introduce and study an important class of operators, the so-called subharmonic operators. It is shown that the operator version of Lehmer’s problem fails under some mild condition.

We prove that even Coxeter groups, whose Coxeter diagrams contain no $(4,4,2)$ triangles, are conjugacy separable. In particular, this applies to all right-angled Coxeter groups or word hyperbolic even Coxeter groups. For an arbitrary Coxeter group $W$, we also study the relationship between Coxeter generating sets that give rise to the same collection of parabolic subgroups. As an application, we show that if an automorphism of $W$ preserves the conjugacy class of every sufficiently short element then it is inner. We then derive consequences for the outer automorphism groups of Coxeter groups.

We study the loop clusters induced by Poissonian ensembles of Markov loops on a finite or countable graph (Markov loops can be viewed as excursions of Markov chains with a random starting point, up to re-rooting). Poissonian ensembles are seen as a Poisson point process of loops indexed by ‘time’. The evolution in time of the loop clusters defines a coalescent process on the vertices of the graph. After a description of some general properties of the coalescent process, we address several aspects of the loop clusters defined by a simple random walk killed at a constant rate on three different graphs: the integer number line $\mathbb{Z}$, the integer lattice $\mathbb{Z}^{d}$ with $d\geq2$ and the complete graph. These examples show the relations between Poissonian ensembles of Markov loops and other models: renewal process, percolation and random graphs.

We discuss methods of producing random walks on a compact homogeneous space $X$ and examine how they lead to approximate evaluation of integrals of elements of various function spaces, including $L^{p}$ spaces, $L^{p}$-Sobolev spaces, and Hölder spaces.

We study bounded quasiregular mappings and mappings of finite distortion $f:\mathbb{H}^{n}\to\mathbb{R}^{n}$, $n\geq3$. We show that almost every $k$-dimensional cone-like cusp with vertex in $\partial\mathbb{H}^{n}$ maps to a set of finite $k$-dimensional, $k\in\{2,\ldots,n-1\}$, measure under these mappings.

For a pair of fibre preserving continuous functions $f,g:E_{1}\rightarrow E_{2}$ between two compact smooth fibre bundles over $B$, we construct a transfer map $T(f,g):H^{*}(B)\rightarrow H^{*}(B)$ that generalizes Lefschetz number $\lambda_{f,g}$ of the pair of maps. If the pair $(f,g)$ is smooth satisfying a transversality condition and $T(f,g)$ is non-zero, then there is a surjective submersion from any connected component of $\{x\mid f(x)=g(x)\}$ to $B$. This yields a necessary and sufficient condition for a principal $G$-bundle over a simply connected compact manifold to be trivial and we also get a necessary condition for every smooth map from $S^{2n+1}$ to $S^{1}$ for all $n\geq1$.

Let $(R,\mathfrak{m})$ be a local ring of prime characteristic $p$ and $q$ a varying power of $p$. We study the asymptotic behavior of the socle of $R/I^{[q]}$ where $I$ is an $\mathfrak{m}$-primary ideal of $R$. In the graded case, we define the notion of diagonal $F$-threshold of $R$ as the limit of the top socle degree of $R/\mathfrak{m}^{[q]}$ over $q$ when $q\to\infty$. Diagonal $F$-threshold exists as a positive number (rational number in the latter case) when: (1) $R$ is either a complete intersection or $R$ is $F$-pure on the punctured spectrum; (2) $R$ is a two dimensional normal domain. In the latter case, we also discuss its geometric interpretation and apply it to determine the strong semistability of the syzygy bundle of $(x^{d},y^{d},z^{d})$ over the smooth projective curve in $\mathbb{P}^{2}$ defined by $x^{n}+y^{n}+z^{n}=0$. The rest of this paper concerns a different question about how the length of the socle of $R/I^{[q]}$ vary as $q$ varies. We give explicit calculations of the length of the socle of $R/\mathfrak{m}^{[q]}$ for a class of hypersurface rings which attain the minimal Hilbert–Kunz function. We finally show, under mild conditions, the growth of such length function and the growth of the second Betti numbers of $R/\mathfrak{m}^{[q]}$ differ by at most a constant, as $q\to\infty$.