We study criteria for the uniform convergence of trigonometric series with general monotone coefficients. We also obtain necessary and sufficient conditions for a given rate of convergence of partial Fourier sums of such series.

We investigate the isomorphic structure of the Cesàro spaces and their duals, the Tandori spaces. The main result states that the Cesàro function space $\text{Ces}_{\infty }$ and its sequence counterpart $\text{ces}_{\infty }$ are isomorphic. This is rather surprising since $\text{Ces}_{\infty }$ (like Talagrand’s example) has no natural lattice predual. We prove that $\text{ces}_{\infty }$ is not isomorphic to $\ell _{\infty }$ nor is $\text{Ces}_{\infty }$ isomorphic to the Tandori space $\widetilde{L_{1}}$ with the norm $\Vert f\Vert _{\widetilde{L_{1}}}=\Vert \widetilde{f}\Vert _{L_{1}}$, where $\widetilde{f}(t):=\text{ess}\,\sup _{s\geqslant t}|f(s)|$. Our investigation also involves an examination of the Schur and Dunford–Pettis properties of Cesàro and Tandori spaces. In particular, using results of Bourgain we show that a wide class of Cesàro–Marcinkiewicz and Cesàro–Lorentz spaces have the latter property.

An intrinsic quadric is a normal projective variety with a Cox ring defined by a single quadratic relation. We provide explicit descriptions of these varieties in the smooth case for small Picard numbers. As applications, we figure out in this setting the Fano examples and (affirmatively) test Fujita’s freeness conjecture.

We present the geometry behind counting twin prime polynomials in $\mathbb{F}_{q}[T]$ in general. We compute cohomology and explicitly count points by means of a twisted Lefschetz trace formula applied to these parametrizing varieties for cubic twin prime polynomials. The elliptic curve $X^{3}=Y(Y-1)$ occurs in the geometry, and thus counting cubic twin prime polynomials involves the associated modular form. In theory, this approach can be extended to higher degree twin primes, but the computations become harder.

The formula we get in degree 3 is compatible with the Hardy–Littlewood heuristic on average, agrees with the prediction for $q\equiv 2$ (mod 3), but shows anomalies for $q\equiv 1$ (mod 3).

We prove results concerning the multiplicity as well as the Cohen–Macaulay and Gorenstein properties of the special fiber ring $\mathscr{F}(E)$ of a finitely generated $R$-module $E\subsetneq R^{e}$ over a Noetherian local ring $R$ with infinite residue field. Assuming that $R$ is Cohen–Macaulay of dimension 1 and that $E$ has finite colength in $R^{e}$, our main result establishes an asymptotic length formula for the multiplicity of $\mathscr{F}(E)$, which, in addition to being of independent interest, allows us to derive a Cohen–Macaulayness criterion and to detect a curious relation to the Buchsbaum–Rim multiplicity of $E$ in this setting. Further, we provide a Gorensteinness characterization for $\mathscr{F}(E)$ in the more general situation where $R$ is Cohen–Macaulay of arbitrary dimension and $E$ is not necessarily of finite colength, and we notice a constraint in terms of the second analytic deviation of the module $E$ if its reduction number is at least three.

We establish class field theory for three-dimensional manifolds and knots. For this purpose, we formulate analogues of the multiplicative group, the idèle class group, and ray class groups in a cocycle-theoretic way. Following the arguments in abstract class field theory, we construct reciprocity maps and verify the existence theorems.

We show that if the zeros of an automorphic $L$-function are weighted by the central value of the $L$-function or a quadratic imaginary base change, then for certain families of holomorphic GL(2) newforms, it has the effect of changing the distribution type of low-lying zeros from orthogonal to symplectic, for test functions whose Fourier transforms have sufficiently restricted support. However, if the $L$-value is twisted by a nontrivial quadratic character, the distribution type remains orthogonal. The proofs involve two vertical equidistribution results for Hecke eigenvalues weighted by central twisted $L$-values. One of these is due to Feigon and Whitehouse, and the other is new and involves an asymmetric probability measure that has not appeared in previous equidistribution results for GL(2).

Suppose that N is 2-coloured. Then there are infinitely many monochromatic solutions to $x+y=z^{2}$. On the other hand, there is a 3-colouring of N with only finitely many monochromatic solutions to this equation.

Let $(M^{n},g)$ be a Riemannian manifold without boundary. We study the amount of initial regularity required so that the solution to a free Schrödinger equation converges pointwise to its initial data. Assume the initial data is in $H^{\unicode[STIX]{x1D6FC}}(M)$. For hyperbolic space, the standard sphere, and the two-dimensional torus, we prove that $\unicode[STIX]{x1D6FC}>\frac{1}{2}$ is enough. For general compact manifolds, due to the lack of a local smoothing effect, it is hard to improve on the bound $\unicode[STIX]{x1D6FC}>1$ from interpolation. We managed to go below 1 for dimension ${\leqslant}$ 3. The more interesting thing is that, for a one-dimensional compact manifold, $\unicode[STIX]{x1D6FC}>\frac{1}{3}$ is sufficient.

Let a discrete group $G$ act on a unital simple $\text{C}^{\ast }$-algebra $A$ by outer automorphisms. We establish a Galois correspondence $H\mapsto A\rtimes _{\unicode[STIX]{x1D6FC},r}H$ between subgroups of $G$ and $\text{C}^{\ast }$-algebras $B$ satisfying $A\subseteq B\subseteq A\rtimes _{\unicode[STIX]{x1D6FC},r}G$, where $A\rtimes _{\unicode[STIX]{x1D6FC},r}G$ denotes the reduced crossed product. For a twisted dynamical system $(A,G,\unicode[STIX]{x1D6FC},\unicode[STIX]{x1D70E})$, we also prove the corresponding result for the reduced twisted crossed product $A\rtimes _{\unicode[STIX]{x1D6FC},r}^{\unicode[STIX]{x1D70E}}G$.

We study the geometry of the component of the origin in the uniform spanning forest of $\mathbb{Z}^{d}$ and give bounds on the size of balls in the intrinsic metric.

We compute cup-product pairings in the integral cohomology ring of the moduli space of rank two stable bundles with odd determinant over a Riemann surface using methods of Zagier. The resulting formula is related to a generating function for certain skew Schur polynomials. As an application, we compute the nilpotency degree of a distinguished degree two generator in the mod two cohomology ring. We then give descriptions of the mod two cohomology rings in low genus, and describe the subrings invariant under the mapping-class group action.

In this article we explore the interplay between two generalizations of the Whittaker model, namely the Klyachko models and the degenerate Whittaker models, and two functorial constructions, namely base change and automorphic induction, for the class of unitarizable and ladder representations of the general linear groups.

In this paper we study spectral properties of the Dirichlet-to-Neumann map on differential forms obtained by a slight modification of the definition due to Belishev and Sharafutdinov. The resulting operator $\unicode[STIX]{x039B}$ is shown to be self-adjoint on the subspace of coclosed forms and to have purely discrete spectrum there. We investigate properties of eigenvalues of $\unicode[STIX]{x039B}$ and prove a Hersch–Payne–Schiffer type inequality relating products of those eigenvalues to eigenvalues of the Hodge Laplacian on the boundary. Moreover, non-trivial eigenvalues of $\unicode[STIX]{x039B}$ are always at least as large as eigenvalues of the Dirichlet-to-Neumann map defined by Raulot and Savo. Finally, we remark that a particular case of $p$-forms on the boundary of a $2p+2$-dimensional manifold shares many important properties with the classical Steklov eigenvalue problem on surfaces.

Fix a von Neumann algebra ${\mathcal{N}}$ equipped with a suitable trace $\unicode[STIX]{x1D70F}$. For a path of self-adjoint Breuer–Fredholm operators, the spectral flow measures the net amount of spectrum that moves from negative to non-negative. We consider specifically the case of paths of bounded perturbations of a fixed unbounded self-adjoint Breuer–Fredholm operator affiliated with ${\mathcal{N}}$. If the unbounded operator is $p$-summable (that is, its resolvents are contained in the ideal $L^{p}$), then it is possible to obtain an integral formula that calculates spectral flow. This integral formula was first proved by Carey and Phillips, building on earlier approaches of Phillips. Their proof was based on first obtaining a formula for the larger class of $\unicode[STIX]{x1D703}$-summable operators, and then using Laplace transforms to obtain a $p$-summable formula. In this paper, we present a direct proof of the $p$-summable formula that is both shorter and simpler than theirs.

We prove completeness for the main examples of infinite-dimensional Lie groups and some related topological groups. Consider a sequence $G_{1}\subseteq G_{2}\subseteq \cdots \,$ of topological groups $G_{n}$ n such that $G_{n}$ is a subgroup of $G_{n+1}$ and the latter induces the given topology on $G_{n}$, for each $n\in \mathbb{N}$. Let $G$ be the direct limit of the sequence in the category of topological groups. We show that $G$ induces the given topology on each $G_{n}$ whenever $\cup _{n\in \mathbb{N}}V_{1}V_{2}\cdots V_{n}$ is an identity neighbourhood in $G$ for all identity neighbourhoods $V_{n}\subseteq G_{n}$. If, moreover, each $G_{n}$ is complete, then $G$ is complete. We also show that the weak direct product $\oplus _{j\in J}G_{j}$ is complete for each family $(G_{j})_{j\in J}$ of complete Lie groups $G_{j}$. As a consequence, every strict direct limit $G=\cup _{n\in \mathbb{N}}G_{n}$ of finite-dimensional Lie groups is complete, as well as the diffeomorphism group $\text{Diff}_{c}(M)$ of a paracompact finite-dimensional smooth manifold $M$ and the test function group $C_{c}^{k}(M,H)$, for each $k\in \mathbb{N}_{0}\cup \{\infty \}$ and complete Lie group $H$ modelled on a complete locally convex space.

In this paper, we study the boundary quotient $\text{C}^{\ast }$-algebras associated with products of odometers. One of our main results shows that the boundary quotient $\text{C}^{\ast }$-algebra of the standard product of $k$ odometers over $n_{i}$-letter alphabets $(1\leqslant i\leqslant k)$ is always nuclear, and that it is a UCT Kirchberg algebra if and only if $\{\ln n_{i}:1\leqslant i\leqslant k\}$ is rationally independent, if and only if the associated single-vertex $k$-graph $\text{C}^{\ast }$-algebra is simple. To achieve this, one of our main steps is to construct a topological $k$-graph such that its associated Cuntz–Pimsner $\text{C}^{\ast }$-algebra is isomorphic to the boundary quotient $\text{C}^{\ast }$-algebra. Some relations between the boundary quotient $\text{C}^{\ast }$-algebra and the $\text{C}^{\ast }$-algebra $\text{Q}_{\mathbb{N}}$ introduced by Cuntz are also investigated.

In this paper, we characterize linear maps on matrix spaces that preserve matrices of local spectral radius zero at some fixed nonzero vector.

We study the percolation model on Boltzmann triangulations using a generating function approach. More precisely, we consider a Boltzmann model on the set of finite planar triangulations, together with a percolation configuration (either site-percolation or bond-percolation) on this triangulation. By enumerating triangulations with boundaries according to both the boundary length and the number of vertices/edges on the boundary, we are able to identify a phase transition for the geometry of the origin cluster. For instance, we show that the probability that a percolation interface has length $n$ decays exponentially with $n$ except at a particular value $p_{c}$ of the percolation parameter $p$ for which the decay is polynomial (of order $n^{-10/3}$). Moreover, the probability that the origin cluster has size $n$ decays exponentially if $p<p_{c}$ and polynomially if $p\geqslant p_{c}$.

The critical percolation value is $p_{c}=1/2$ for site percolation, and $p_{c}=(2\sqrt{3}-1)/11$ for bond percolation. These values coincide with critical percolation thresholds for infinite triangulations identified by Angel for site-percolation, and by Angel and Curien for bond-percolation, and we give an independent derivation of these percolation thresholds.

Lastly, we revisit the criticality conditions for random Boltzmann maps, and argue that at $p_{c}$, the percolation clusters conditioned to have size $n$ should converge toward the stable map of parameter $\frac{7}{6}$ introduced by Le Gall and Miermont. This enables us to derive heuristically some new critical exponents.