In this paper we give an explicit formula for the flag curvature of homogeneous Randers spaces of Douglas type and apply this formula to obtain some interesting results. We first deduce an explicit formula for the flag curvature of an arbitrary left invariant Randers metric on a two-step nilpotent Lie group. Then we obtain a classification of negatively curved homogeneous Randers spaces of Douglas type. This results, in particular, in many examples of homogeneous non-Riemannian Finsler spaces with negative flag curvature. Finally, we prove a rigidity result that a homogeneous Randers space of Berwald type whose flag curvature is everywhere nonzero must be Riemannian.

We establish a formula for the variation of integrals of differential forms on cubic chains in the context of diffeological spaces. Then we establish the diffeological version of Stokes’ theorem, and we apply that to get the diffeological variant of the Cartan–Lie formula. Still in the context of Cartan–De Rham calculus in diffeology, we construct a chain-homotopy operator $K$, and we apply it here to get the homotopic invariance of De Rham cohomology for diffeological spaces. This is the chain-homotopy operator that is used in symplectic diffeology to construct the moment map.

Let $S$ be the semigroup $S=\sum\nolimits_{i=1}^{\oplus k}{{{S}_{i}}}$, where for each $i\in I,{{S}_{i}}$ is a countable subsemigroup of the additive semigroup ${{\mathbb{R}}_{+}}$ containing 0. We consider representations of $S$ as contractions ${{\left\{ {{T}_{s}} \right\}}_{s\in S}}$ on a Hilbert space with the Nica-covariance property: $T_{s}^{*}{{T}_{t}}={{T}_{t}}T_{s}^{*}$ whenever $t\wedge s=0$. We show that all such representations have a unique minimal isometric Nica-covariant dilation.

This result is used to help analyse the nonself-adjoint semicrossed product algebras formed from Nica-covariant representations of the action of $S$ on an operator algebra $\mathcal{A}$ by completely contractive endomorphisms. We conclude by calculating the ${{C}^{*}}$-envelope of the isometric nonself-adjoint semicrossed product algebra (in the sense of Kakariadis and Katsoulis).

In this paper we prove a certain ${{L}^{2}}$-estimate for multilinear Fourier multiplier operators with multipliers of limited smoothness. As a consequence, we extend the result of Calderón and Torchinsky in the linear theory to the multilinear case. The sharpness of our results and some related estimates in Hardy spaces are also discussed.

We study locally compact quantum groups $\mathbb{G}$ through the convolution algebras ${{L}_{1}}\left( \mathbb{G} \right)$ and $\left( T\left( {{L}_{2}}\left( \mathbb{G} \right) \right),\triangleright \right)$. We prove that the reduced quantum group ${{C}^{*}}$ -algebra ${{C}_{0}}\left( \mathbb{G} \right)$ can be recovered from the convolution $\triangleright $ by showing that the right $T\left( {{L}_{2}}\left( \mathbb{G} \right) \right)$-module $\left\langle K\left( {{L}_{2}}\left( \mathbb{G} \right) \right)\,\triangleright \,T\left( {{L}_{2}}\left( \mathbb{G} \right) \right) \right\rangle $ is equal to ${{C}_{0}}\left( \mathbb{G} \right)$. On the other hand, we show that the left $T\left( {{L}_{2}}\left( \mathbb{G} \right) \right)$-module $\left\langle T\left( {{L}_{2}}\left( \mathbb{G} \right) \right)\,\triangleright \,K\left( {{L}_{2}}\left( \mathbb{G} \right) \right) \right\rangle $ is isomorphic to the reduced crossed product ${{C}_{0}}\left( \widehat{\mathbb{G}} \right){{\,}_{r}}\,\ltimes \,{{C}_{0}}\left( \mathbb{G} \right)$, and hence is a much larger ${{C}^{*}}$ -subalgebra of $B\left( {{L}_{2}}\left( \mathbb{G} \right) \right)$.

We establish a natural isomorphism between the completely bounded right multiplier algebras of ${{L}_{1}}\left( \mathbb{G} \right)$ and $\left( T\left( {{L}_{2}}\left( \mathbb{G} \right) \right),\,\triangleright \right)$, and settle two invariance problems associated with the representation theorem of Junge–Neufang–Ruan (2009). We characterize regularity and discreteness of the quantum group $\mathbb{G}$ in terms of continuity properties of the convolution $\triangleright $ on $T\left( {{L}_{2}}\left( \mathbb{G} \right) \right)$. We prove that if $\mathbb{G}$ is semiregular, then the space $\left\langle T\left( {{L}_{2}}\left( \mathbb{G} \right) \right)\,\triangleright \,B\left( {{L}_{2}}\left( \mathbb{G} \right) \right) \right\rangle $ of right $\mathbb{G}$-continuous operators on ${{L}_{2}}\left( \mathbb{G} \right)$, which was introduced by Bekka (1990) for ${{L}_{\infty }}\left( G \right)$, is a unital ${{C}^{*}}$-subalgebra of $B\left( {{L}_{2}}\left( \mathbb{G} \right) \right)$. In the representation framework formulated by Neufang–Ruan–Spronk (2008) and Junge–Neufang–Ruan, we show that the dual properties of compactness and discreteness can be characterized simultaneously via automatic normality of quantum group bimodule maps on $B\left( {{L}_{2}}\left( \mathbb{G} \right) \right)$. We also characterize some commutation relations of completely bounded multipliers of $\left( T\left( {{L}_{2}}\left( \mathbb{G} \right) \right),\,\triangleright \right)$ over $B\left( {{L}_{2}}\left( \mathbb{G} \right) \right)$.

We provide a framework for studying randomly coloured point sets in a locally compact second-countable space on which a metrizable unimodular group acts continuously and properly. We first construct and describe an appropriate dynamical system for uniformly discrete uncoloured point sets. For point sets of finite local complexity, we characterize ergodicity geometrically in terms of pattern frequencies. The general framework allows us to incorporate a random colouring of the point sets. We derive an ergodic theorem for randomly coloured point sets with finite-range dependencies. Special attention is paid to the exclusion of exceptional instances for uniquely ergodic systems. The setup allows for a straightforward application to randomly coloured graphs

We introduce generalized triple homomorphisms between Jordan–Banach triple systems as a concept that extends the notion of generalized homomorphisms between Banach algebras given by K. Jarosz and B. E. Johnson in 1985 and 1987, respectively. We prove that every generalized triple homomorphism between $\text{J}{{\text{B}}^{*}}$-triples is automatically continuous. When particularized to ${{C}^{*}}$-algebras, we rediscover one of the main theorems established by Johnson. We will also consider generalized triple derivations from a Jordan–Banach triple $E$ into a Jordan–Banach triple $E$-module, proving that every generalized triple derivation from a $\text{J}{{\text{B}}^{*}}$-triple $E$ into itself or into ${{E}^{*}}$is automatically continuous.

Let $X$ be an $n$-dimensional, finite, simply connected $\text{CW}$ complex and set

$${{\alpha }_{X}}\,=\,\underset{i}{\mathop{\lim \,\sup }}\,\frac{\log \,\text{rank}\,{{\pi }_{i}}\left( X \right)}{i}$$

When $0<{{\alpha }_{X}}<\infty $, we give upper and lower bounds for $\sum\nolimits_{i=k+2}^{k+n}{\,\text{rank}}\text{ }{{\pi }_{i}}\left( X \right)$ for $k$ sufficiently large. We also show for any $r$ that $\alpha x$ can be estimated from the integers $\text{rk }{{\pi }_{i}}\left( X \right),\,i\,\le \,nr$ with an error bound depending explicitly on $r$.

This paper studies the $K$-theoretic invariants of the crossed product ${{C}^{*}}$-algebras associated with an important family of homeomorphisms of the tori ${{\mathbb{T}}^{n}}$ called Furstenberg transformations. Using the Pimsner–Voiculescu theorem, we prove that given $n$, the $K$-groups of those crossed products whose corresponding $n\,\times \,n$ integer matrices are unipotent of maximal degree always have the same rank ${{a}_{n}}$. We show using the theory developed here that a claim made in the literature about the torsion subgroups of these $K$-groups is false. Using the representation theory of the simple Lie algebra $\mathfrak{s}\mathfrak{l}\left( 2,\,\mathbb{C} \right)$, we show that, remarkably, ${{a}_{n}}$ has a combinatorial significance. For example, every ${{a}_{2n+1}}$ is just the number of ways that 0 can be represented as a sum of integers between –$n$ and $n$ (with no repetitions). By adapting an argument of van Lint (in which he answered a question of Erdős), a simple explicit formula for the asymptotic behavior of the sequence $\{{{a}_{n}}\}$ is given. Finally, we describe the order structure of the ${{K}_{0}}$-groups of an important class of Furstenberg crossed products, obtaining their complete Elliott invariant using classification results of H. Lin and N. C. Phillips.

We introduce the notion of families of $n$-marked, smooth, rational tropical curves over smooth tropical varieties and establish a one-to-one correspondence between (equivalence classes of) these families and morphisms from smooth tropical varieties into the moduli space of $n$-marked, abstract, rational, tropical curves ${{\mathcal{M}}_{n}}$.

We define and study the so-called extreme version of the notion of a projective normed module. The relevant definition takes into account the exact value of the norm of the module in question, in contrast with the standard known definition that is formulated in terms of norm topology.

After the discussion of the case where our normed algebra $A$ is just $\mathbb{C}$, we concentrate on the case of the next degree of complication, where $A$ is a sequence algebra satisfying some natural conditions. The main results give a full characterization of extremely projective objects within the subcategory of the category of non-degenerate normed $A$-modules, consisting of the so-called homogeneous modules. We consider two cases, ‘non-complete’ and ‘complete’, and the respective answers turn out to be essentially different.

In particular, all Banach non-degenerate homogeneous modules consisting of sequences are extremely projective within the category of Banach non-degenerate homogeneous modules. However, neither of them, provided it is infinite-dimensional, is extremely projective within the category of all normed non-degenerate homogeneous modules. On the other hand, submodules of these modules consisting of finite sequences are extremely projective within the latter category.

We introduce the notion of orbital $L$-functions for the space of binary cubic forms and investigate their analytic properties. We study their functional equations and residue formulas in some detail. Aside from their intrinsic interest, the results from this paper are used to prove the existence of secondary terms in counting functions for cubic fields. This is worked out in a companion paper.

We construct a bipartite Euler system in the sense of Howard for Hilbert modular eigenforms of parallel weight two over totally real fields, generalizing works of Bertolini–Darmon, Longo, Nekovar, Pollack–Weston, and others. The construction has direct applications to Iwasawa's main conjectures. For instance, it implies in many cases one divisibility of the associated dihedral or anticyclotomic main conjecture, at the same time reducing the other divisibility to a certain nonvanishing criterion for the associated $p$-adic $L$-functions. It also has applications to cyclotomic main conjectures for Hilbert modular forms over $\text{CM}$ fields via the technique of Skinner and Urban.

In this paper we use the recent developments in the representation theory of locally compact quantum groups, to assign to each locally compact quantum group $\mathbb{G}$ a locally compact group $\tilde{\mathbb{G}}$ that is the quantum version of point-masses and is an invariant for the latter. We show that “quantum point-masses” can be identified with several other locally compact groups that can be naturally assigned to the quantum group $\mathbb{G}$. This assignment preserves compactness as well as discreteness (hence also finiteness), and for large classes of quantum groups, amenability. We calculate this invariant for some of the most well-known examples of non-classical quantum groups. Also, we show that several structural properties of $\mathbb{G}$ are encoded by $\tilde{\mathbb{G}}$ the latter, despite being a simpler object, can carry very important information about $\mathbb{G}$.

Given a non-oscillating gradient trajectory $\left| \text{ }\!\!\gamma\!\!\text{ } \right|$ of a real analytic function $f$, we show that the limit $v$ of the secants at the limit point $0$ of $\left| \text{ }\!\!\gamma\!\!\text{ } \right|$ along the trajectory $\left| \text{ }\!\!\gamma\!\!\text{ } \right|$ is an eigenvector of the limit of the direction of the Hessian matrix Hess$\left( f \right)$ at $0$ along $\left| \text{ }\!\!\gamma\!\!\text{ } \right|$. The same holds true at infinity if the function is globally sub-analytic. We also deduce some interesting estimates along the trajectory. Away from the ends of the ambient space, this property is of metric nature and still holds in a general Riemannian analytic setting.

We show that quasimap Floer cohomology for varying symplectic quotients resolves several puzzles regarding displaceability of toric moment fibers. For example, we present a compact Hamiltonian torus action containing an open subset of non-displaceable orbits and a codimension four singular set, partly answering a question of McDuff, and we determine displaceability for most of the moment fibers of a symplectic ellipsoid.

Nous considérons les perturbations $H\,:=\,{{H}_{0}}\,+\,V$ et $D\,:=\,{{D}_{0}}\,+\,V$ des Hamiltoniens libres ${{H}_{0}}$ de Pauli et ${{D}_{0}}$ de Dirac en dimension 3 avec champ magnétique non constant, $V$ étant un potentiel électrique qui décroıt super-exponentiellement dans la direction du champ magnétique. Nous montrons que dans des espaces de Banach appropriés, les résolvantes de $H$ et $D$ définies sur le demi-plan supérieur admettent des prolongements méromorphes. Nous définissons les résonances de $H$ et $D$ comme étant les pôles de ces extensions méromorphes. D’une part, nous étudions la répartition des résonances de $H$ prés de l’origine 0 et d’autre part, celle des résonances de $D$ près de $\pm m$ où m est la masse d’une particule. Dans les deux cas, nous obtenons d’abord des majorations du nombre de résonances dans de petits domaines au voisinage de 0 et $\pm m$. Sous des hypothèses supplémentaires, nous obtenons des développements asymptotiques du nombre de résonances qui entraınent leur accumulation près des seuils 0 et $\pm m$. En particulier, pour une perturbation $V$ de signe défini, nous obtenons des informations sur la répartition des valeurs propres de $H$ et $D$ près de 0 et $\pm m$ respectivement.

This article concerns a class of open billiards consisting of a finite number of strictly convex, non-eclipsing obstacles $K$. The non-wandering set ${{M}_{0}}$ of the billiard ball map is a topological Cantor set, and its Hausdorff dimension has been previously estimated for billiards in ${{\mathbb{R}}^{2}}$ using well-known techniques. We extend these estimates to billiards in ${{\mathbb{R}}^{n}}$ and make various refinements to the estimates. These refinements also allow improvements to other results. We also show that in many cases, the non-wandering set is confined to a particular subset of ${{\mathbb{R}}^{n}}$ formed by the convex hull of points determined by period 2 orbits. This allows more accurate bounds on the constants used in estimating Hausdorff dimension.

Permutation products and their various “fat diagonal” subspaces are studied from the topological and geometric points of view. We describe in detail the stabilizer and orbit stratifications related to the permutation action, producing a sharp upper bound for its depth and then paying particular attention to the geometry of the diagonal stratum. We exhibit an expression for the fundamental group of any permutation product of a connected space $X$ having the homotopy type of a CW complex in terms of ${{\pi }_{1}}(X)$ and ${{H}_{1}}(X;\,\mathbb{Z})$. We then prove that the fundamental group of the configuration space of $n$-points on $X$, of which multiplicities do not exceed $n/2$, coincides with ${{H}_{1}}(X;\,\mathbb{Z})$. Further results consist in giving conditions for when fat diagonal subspaces of manifolds can be manifolds again. Various examples and homological calculations are included.