Let $F$ denote a binary form of order $d$ over the complex numbers. If $r$ is a divisor of $d$, then the Hilbert covariant ${{H}_{r,\,d}}\,\left( F \right)$ vanishes exactly when $F$ is the perfect power of an order $r$ form. In geometric terms, the coefficients of $H$ give defining equations for the image variety $X$ of an embedding ${{\text{P}}^{r}}\,\to \,{{\text{P}}^{d}}$. In this paper we describe a new construction of the Hilbert covariant and simultaneously situate it into a wider class of covariants called the Göttingen covariants, all of which vanish on $X$. We prove that the ideal generated by the coefficients of $H$ defines $X$ as a scheme. Finally, we exhibit a generalisation of the Göttingen covariants to $n$-ary forms using the classical Clebsch transfer principle.

Soit $G$ le groupe $\text{GL}\,\left( N,\,F \right)$, où $F$ est un corps localement compact et non archimédien. En utilisant la théorie des types simples de Bushnell et Kutzko, ainsi qu'une idée originale d'Henniart, nous construisons des pseudo-coefficients explicites pour les représentations de la série discrète de $G$. Comme application, nous en déduisons des formules inédites pour la valeur du charactère d'Harish- Chandra de certaines telles représentations en certainséléments elliptiques réguliers.

Combinatorial structures that compose and decompose give rise to Hopf monoids in Joyal's category of species. The Hadamard product of two Hopf monoids is another Hopf monoid. We prove two main results regarding freeness of Hadamard products. The first one states that if one factor is connected and the other is free as a monoid, their Hadamard product is free (and connected). The second provides an explicit basis for the Hadamard product when both factors are free.

The first main result is obtained by showing the existence of a one-parameter deformation of the comonoid structure and appealing to a rigidity result of Loday and Ronco that applies when the parameter is set to zero. To obtain the second result, we introduce an operation on species that is intertwined by the free monoid functor with the Hadamard product. As an application of the first result, we deduce that the Boolean transform of the dimension sequence of a connected Hopf monoid is nonnegative.

Moduli spaces of real bundles over a real curve arise naturally as Lagrangian submanifolds of the moduli space of semi-stable bundles over a complex curve. In this paper, we adapt the methods of Atiyah–Bott's “Yang–Mills over a Riemann Surface” to compute $\mathbb{Z}/2$–Betti numbers of these spaces.

We study approximately differentiable functions on metric measure spaces admitting a Cheeger differentiable structure. The main result is a Whitney-type characterization of approximately differentiable functions in this setting. As an application, we prove a Stepanov-type theorem and consider approximate differentiability of Sobolev, $BV$, and maximal functions.

Suppose that $\widetilde{G}$ is a connected reductive group defined over a field $k$, and $\Gamma$ is a finite group acting via $k$-automorphisms of $\widetilde{G}$ satisfying a certain quasi-semisimplicity condition. Then the identity component of the group of $\Gamma$-fixed points in $\widetilde{G}$ is reductive. We axiomatize the main features of the relationship between this fixed-point group and the pair $\left( \tilde{G},\Gamma \right)$, and consider any group $G$ satisfying the axioms. If both $\widetilde{G}$ and $G$ are $k$-quasisplit, then we can consider their duals $\widetilde{{{G}^{*}}}$ and ${{G}^{*}}$. We show the existence of and give an explicit formula for a natural map from the set of semisimple stable conjugacy classes in ${{G}^{*}}\,(k)$ to the analogous set for $\widetilde{{{G}^{*}}}\,(k)$. If $k$ is finite, then our groups are automatically quasisplit, and our result specializes to give a map of semisimple conjugacy classes. Since such classes parametrize packets of irreducible representations of $G(k)$ and $\widetilde{G}\,(k)$, one obtains a mapping of such packets.

We answer the natural question: when is a transversely holomorphic symplectic foliation induced by a generalized complex structure? The leafwise symplectic form and transverse complex structure determine an obstruction class in a certain cohomology, which vanishes if and only if our question has an affirmative answer. We first study a component of this obstruction, which gives the condition that the leafwise cohomology class of the symplectic form must be transversely pluriharmonic. As a consequence, under certain topological hypotheses, we infer that we actually have a symplectic fibre bundle over a complex base. We then show how to compute the full obstruction via a spectral sequence. We give various concrete necessary and sufficient conditions for the vanishing of the obstruction. Throughout, we give examples to test the sharpness of these conditions, including a symplectic fibre bundle over a complex base that does not come from a generalized complex structure, and a regular generalized complex structure that is very unlike a symplectic fibre bundle, i.e., for which nearby leaves are not symplectomorphic.

We study separable metric spaces with few types of countable dense sets. We present a structure theorem for locally compact spaces having precisely $n$ types of countable dense sets: such a space contains a subset $S$ of size at most $n-1$ such that $S$ is invariant under all homeomorphisms of $X$ and $X\,\backslash \,S$ is countable dense homogeneous. We prove that every Borel space having fewer than $\mathfrak{c}$ types of countable dense sets is Polish. The natural question of whether every Polish space has either countably many or $\mathfrak{c}$ many types of countable dense sets is shown to be closely related to Topological Vaught's Conjecture.

Let $E/F$ be a quadratic extension of $p$-adic fields and let $d,\,m$ be nonnegative integers of distinct parities. Fix admissible irreducible tempered representations $\pi $ and $\sigma $ of $\text{G}{{\text{L}}_{d}}\left( E \right)$ and $\text{G}{{\text{L}}_{m}}\left( E \right)$ respectively. We assume that $\pi $ and $\sigma $ are conjugate-dual. That is to say $\pi \,\simeq \,{{\pi }^{\vee \text{,}\,c}}$ and $\sigma \,\simeq \,{{\sigma }^{\vee ,c}}$ where $c$ is the nontrivial $F$-automorphism of $E$. This implies that we can extend $\pi $ to an unitary representation $\tilde{\pi }$ of a nonconnected group $\text{G}{{\text{L}}_{d}}\left( E \right)\,\rtimes \,\left\{ 1,\,0 \right\}$. Define $\tilde{\sigma }$ the same way. We state and prove an integral formula for $\in \left( 1/2,\,\pi \,\times \,\sigma ,\,{{\psi }_{E}} \right)$ involving the characters of $\tilde{\pi }$ and $\tilde{\sigma }$. This formula is related to the local Gan–Gross–Prasad conjecture for unitary groups.

We exhibit a set of minimal generators of the defining ideal of the Rees Algebra associated with the ideal of three bivariate homogeneous polynomials parametrizing a proper rational curve in projective plane, having a minimal syzygy of degree 2.

Let $h_{\vee }^{\infty }\,\left( \text{B} \right)$ and $h_{\vee }^{\infty }\,\left( \text{B} \right)$ be the spaces of harmonic functions in the unit disk and multidimensional unit ball admitting a two-sided radial majorant $v\left( r \right)$. We consider functions $v$ that fulfill a doubling condition. In the two-dimensional case let

$$u\left( r{{e}^{i\theta }},\xi \right)\,=\,\sum\limits_{j=0}^{\infty }{\left( {{a}_{j0}}{{\xi }_{j0}}{{r}^{j}}\,\cos \,j\theta \,+\,{{a}_{j1}}{{\xi }_{j1}}{{r}^{j}}\,\sin \,j\theta \right)}$$

where $\xi \,=\,\left\{ {{\xi }_{ji}} \right\}$ is a sequence of random subnormal variables and ${{a}_{ji}}$ are real. In higher dimensions we consider series of spherical harmonics. We will obtain conditions on the coefficients ${{a}_{ji}}$ that imply that $u$ is in $h_{\vee }^{\infty }\,\left( \text{B} \right)$ almost surely. Our estimate improves previous results by Bennett, Stegenga, and Timoney, and we prove that the estimate is sharp. The results for growth spaces can easily be applied to Bloch-type spaces, and we obtain a similar characterization for these spaces that generalizes results by Anderson, Clunie, and Pommerenke and by Guo and Liu.

Suppose that $Y$ is a cyclic cover of projective space branched over a hyperplane arrangement $D$ and that $U$ is the complement of the ramification locus in $Y$. The first theorem in this paper implies that the Beilinson–Hodge conjecture holds for $U$ if certain multiplicities of $D$ are coprime to the degree of the cover. For instance, this applies when $D$ is reduced with normal crossings. The second theorem shows that when $D$ has normal crossings and the degree of the cover is a prime number, the generalized Hodge conjecture holds for any toroidal resolution of $Y$. The last section contains some partial extensions to more general nonabelian covers.

We study the asymptotic behaviour of random walks in i.i.d. random environments on ${{\mathbb{Z}}^{d}}$. The environments need not be elliptic, so some steps may not be available to the random walker. We prove a monotonicity result for the velocity (when it exists) for any 2-valued environment, and show that this does not hold for 3-valued environments without additional assumptions. We give a proof of directional transience and the existence of positive speeds under strong but non-trivial conditions on the distribution of the environment. Our results include generalisations (to the non-elliptic setting) of 0-1 laws for directional transience and, in 2-dimensions, the existence of a deterministic limiting velocity.

We introduce a new basis of the algebra of non-commutative symmetric functions whose images under the forgetful map are Schur functions when indexed by a partition. Dually, we build a basis of the quasi-symmetric functions that expand positively in the fundamental quasi-symmetric functions. We then use the basis to construct a non-commutative lift of the Hall–Littlewood symmetric functions with similar properties to their commutative counterparts.

A simple finite dimensional module ${{V}_{\lambda }}$ of a simple complex algebraic group $G$ is naturally endowed with a filtration induced by the PBW-filtration of $U\,(\text{Lie}\,G)$. The associated graded space $\text{V}_{\text{ }\!\!\lambda\!\!\text{ }}^{a}$ is a module for the group ${{G}^{a}}$, which can be roughly described as a semi-direct product of a Borel subgroup of $G$ and a large commutative unipotent group $\mathbb{G}_{a}^{M}$. In analogy to the flag variety ${{\mathcal{F}}_{\lambda }}\,=\,G.[{{v}_{\lambda }}]\,\,\subset \,\,\mathbb{P}({{V}_{\lambda }})$, we call the closure $\overline{{{G}^{a}}\,.\,[{{v}_{\text{ }\!\!\lambda\!\!\text{ }}}]}\,\,\subset \,\,\mathbb{P}\,(V_{\text{ }\!\!\lambda\!\!\text{ }}^{a})$ of the ${{G}^{a}}$-orbit through the highest weight line the degenerate flag variety $\mathcal{F}_{\text{ }\!\!\lambda\!\!\text{ }}^{a}$. In general this is a singular variety, but we conjecture that it has many nice properties similar to that of Schubert varieties. In this paper we consider the case of $G$ being the symplectic group. The symplectic case is important for the conjecture because it is the first known case where, even for fundamental weights $\omega$, the varieties $\mathcal{F}_{\text{ }\!\!\omega\!\!\text{ }}^{a}$ differ from ${{\mathcal{F}}_{\text{ }\!\!\omega\!\!\text{ }}}$. We give an explicit construction of the varieties $\text{Sp}\,\mathcal{F}_{\text{ }\!\!\lambda\!\!\text{ }}^{a}$ and construct desingularizations, similar to the Bott–Samelson resolutions in the classical case. We prove that $\text{Sp}\,\mathcal{F}_{\text{ }\!\!\lambda\!\!\text{ }}^{a}$ are normal locally complete intersections with terminal and rational singularities. We also show that these varieties are Frobenius split. Using the above mentioned results, we prove an analogue of the Borel–Weil theorem and obtain a $q$-character formula for the characters of irreducible $\text{S}{{\text{p}}_{2\pi }}$-modules via the Atiyah–Bott–Lefschetz fixed points formula.

Given a Bratteli diagram $B$, we study the set ${{\mathcal{O}}_{B}}$ of all possible orderings on $B$ and its subset ${{\mathcal{P}}_{B}}$ consisting of perfect orderings that produce Bratteli–Vershik topological dynamical systems (Vershik maps). We give necessary and sufficient conditions for the ordering $\omega $ to be perfect. On the other hand, a wide class of non-simple Bratteli diagrams that do not admit Vershik maps is explicitly described. In the case of finite rank Bratteli diagrams, we show that the existence of perfect orderings with a prescribed number of extreme paths constrains significantly the values of the entries of the incidence matrices and the structure of the diagram $B$. Our proofs are based on the new notions of skeletons and associated graphs, defined and studied in the paper. For a Bratteli diagram $B$ of rank $k$, we endow the set ${{\mathcal{O}}_{B}}$ with product measure $\mu $ and prove that there is some $1\,\le \,j\,\le \,k$ such that $\mu $-almost all orderings on $B$ have $j$ maximal and $j$ minimal paths. If $j$ is strictly greater than the number of minimal components that $B$ has, then $\mu $-almost all orderings are imperfect.

A subset $X$ of a Polish group $G$ is called Haar null if there exist a Borel set $B\,\supset \,X$ and Borel probability measure $\mu$ on $G$ such that $\mu \left( g\,Bh \right)\,=\,0$ for every $g,\,h\,\in \,G$. We prove that there exist a set $X\,\subset \,\text{R}$ that is not Lebesgue null and a Borel probability measure $\mu$ such that $\mu \left( X\,+\,t \right)\,=\,0$ for every $t\,\in \,\text{R}$. This answers a question from David Fremlin’s problem list by showing that one cannot simplify the definition of a Haar null set by leaving out the Borel set $B$. (The answer was already known assuming the Continuum Hypothesis.)

This result motivates the following Baire category analogue. It is consistent with $ZFC$ that there exist an abelian Polish group $G$ and a Cantor set $C\,\subset \,G$ such that for every non-meagre set $X\,\subset \,\text{G}$ there exists a $t\in \text{G}$ such that $C\,\cap \,\left( X\,+\,t \right)$ is relatively non-meagre in $C$. This essentially generalizes results of Bartoszyński and Burke–Miller.

Let $K$ be a complex quadratic extension of $\mathbb{Q}$ and let ${{\mathbb{A}}_{K}}$ denote the adeles of $K$. We find special values at all of the critical points of twisted tensor $L$-functions attached to cohomological cuspforms on $G{{L}_{2}}\left( {{\mathbb{A}}_{K}} \right)$ and establish Galois equivariance of the values. To investigate the values, we determine the archimedean factors of a class of integral representations of these $L$-functions, thus proving a conjecture due to Ghate. We also investigate analytic properties of these $L$-functions, such as their functional equations.

Let $W$ be an infinite Coxeter group. We initiate the study of the set $E$ of limit points of “normalized” roots (representing the directions of the roots) of $\text{W}$. We show that $E$ is contained in the isotropic cone $Q$ of the bilinear form $B$ associated with a geometric representation, and we illustrate this property with numerous examples and pictures in rank 3 and 4. We also define a natural geometric action of $W$ on $E$, and then we exhibit a countable subset of $E$, formed by limit points for the dihedral reflection subgroups of $W$. We explain how this subset is built fromthe intersection with $Q$ of the lines passing through two positive roots, and finally we establish that it is dense in $E$.