On établit quelques résultats généraux relatifs à la taille du groupe d'automorphismes de l'espace de Douady de points sur une surface, puis on étudie quelques propriétés des automorphismes provenant d'un automorphisme de la surface, en particulier leur action sur la cohomologie et la classification de leurs points fixes.

We consider the problem of minimizing the energy of $N$ points repelling each other on curves in $\mathbb{R}^d$ with the potential $|x-y|^{-s}$, $s\geq 1$, where $|\, \cdot\, |$ is the Euclidean norm. For a sufficiently smooth, simple, closed, regular curve, we find the next order term in the asymptotics of the minimal $s$-energy. On our way, we also prove that at least for $s\geq 2$, the minimal pairwise distance in optimal configurations asymptotically equals $L/N$, $N\to\infty$, where $L$ is the length of the curve.

We consider the Randers space $(V^n,F_b)$ obtained by perturbing the Euclidean metric by a translation, $F_b=\alpha+\beta$, where $\alpha$ is the Euclidean metric and $\beta$ is a $1$-form with norm $b$, $0\leq b\lt 1$. We introduce the concept of a hypersurface with constant mean curvature in the direction of a unitary normal vector field. We obtain the ordinary differential equation that characterizes the rotational surfaces $(V^3,F_b)$ of constant mean curvature (cmc) in the direction of a unitary normal vector field. These equations reduce to the classical equation of the rotational cmc surfaces in Euclidean space, when $b=0$. It also reduces to the equation that characterizes the minimal rotational surfaces in $(V^3,F_b)$ when $H=0$, obtained by M. Souza and K. Tenenblat. Although the differential equation depends on the choice of the normal direction, we show that both equations determine the same rotational surface, up to a reflection. We also show that the round cylinders are cmc surfaces in the direction of the unitary normal field. They are generated by the constant solution of the differential equation. By considering the equation as a nonlinear dynamical system, we provide a qualitative analysis, for $0\lt b\lt \frac{\sqrt{3}}{3}$. Using the concept of stability and considering the linearization around the single equilibrium point (the constant solution), we verify that the solutions are locally asymptotically stable spirals. This is proved by constructing a Lyapunov function for the dynamical system and by determining the basin of stability of the equilibrium point. The surfaces of rotation generated by such solutions tend asymptotically to one end of the cylinder.

Let $E$ be an elliptic curve over $\mathbb Q$ without complex multiplication, and for each prime $p$ of good reduction, let $n_E(p) = | E(\mathbb F_p) |$. For any integer $b$, we consider elliptic pseudoprimes to the base $b$. More precisely, let $Q_{E,b}(x)$ be the number of primes $p \leq x$ such that $b^{n_E(p)} \equiv b\,({\rm mod}\,n_E(p))$, and let $\pi_{E, b}^{\operatorname{pseu}}(x)$ be the number of compositive $n_E(p)$ such that $b^{n_E(p)} \equiv b\,({\rm mod}\,n_E(p))$ (also called elliptic curve pseudoprimes). Motivated by cryptography applications, we address the problem of finding upper bounds for $Q_{E,b}(x)$ and $\pi_{E, b}^{\operatorname{pseu}}(x)$, generalising some of the literature for the classical pseudoprimes to this new setting.

We introduce a flow of a spatial graph and see how invariants for spatial graphs and handlebody-links are derived from those for flowed spatial graphs. We define a new quandle (co)homology by introducing a subcomplex of the rack chain complex. Then we define quandle colorings and quandle cocycle invariants for spatial graphs and handlebody-links.

In this article, we study the correspondence between the geometry of del Pezzo surfaces $S_{r}$ and the geometry of the $r$-dimensional Gosset polytopes $(r-4)_{21}$. We construct Gosset polytopes $(r-4)_{21}$ in $\operatorname{Pic} S_{r}\otimes\mathbb{Q}$ whose vertices are lines, and we identify divisor classes in $\operatorname{Pic} S_{r}$ corresponding to $(a-1)$-simplexes ($a\leq r$), $(r-1)$-simplexes and $(r-1)$-crosspolytopes of the polytope $(r-4)_{21}$. Then we explain how these classes correspond to skew $a$-lines($a\leq r$), exceptional systems, and rulings, respectively. As an application, we work on the monoidal transform for lines to study the local geometry of the polytope $(r-4)_{21}$. And we show that the Gieser transformation and the Bertini transformation induce a symmetry of polytopes $3_{21}$ and $4_{21}$, respectively.

Fix an elliptic curve $E/\mathbb{Q}$ and assume the Riemann Hypothesis for the $L$-function $L(E_D, s)$ for every quadratic twist $E_D$ of $E$ by $D\in\mathbb{Z}$. We combine Weil's explicit formula with techniques of Heath-Brown to derive an asymptotic upper bound for the weighted moments of the analytic rank of $E_D$. We derive from this an upper bound for the density of low-lying zeros of $L(E_D, s)$ that is compatible with the random matrix models of Katz and Sarnak. We also show that for any unbounded increasing function $f$ on $\mathbb{R}$, the analytic rank and (assuming in addition the Birch and Swinnerton-Dyer conjecture) the number of integral points of $E_D$ are less than $f(D)$ for almost all $D$.

We study negative powers of Laguerre differential operators in $\mathbb{R}^d$, $d\ge1$. For these operators we prove two-weight $L^p-L^q$ estimates with ranges of $q$ depending on $p$. The case of the harmonic oscillator (Hermite operator) has recently been treated by Bongioanni and Torrea by using a straightforward approach of kernel estimates. Here these results are applied in certain Laguerre settings. The procedure is fairly direct for Laguerre function expansions of Hermite type, due to some monotonicity properties of the kernels involved. The case of Laguerre function expansions of convolution type is less straightforward. For half-integer type indices $\alpha$ we transfer the desired results from the Hermite setting and then apply an interpolation argument based on a device we call the <it> convexity principle </it> to cover the continuous range of $\alpha\in[-1/2,\infty)^d$. Finally, we investigate negative powers of the Dunkl harmonic oscillator in the context of a finite reflection group acting on $\mathbb{R}^d$ and isomorphic to $\mathbb Z^d_2$. The two weight $L^p-L^q$ estimates we obtain in this setting are essentially consequences of those for Laguerre function expansions of convolution type.

In this paper, we establish the regularity of strong solutions to nondivergence parabolic equations with BMO coefficients in nondoubling weighted spaces.

Our main result is that many triangles of Baumslag-Solitar groups collapse to finite groups, generalizing a famous example of Hirsch and other examples due to several authors. A triangle of Baumslag-Solitar groups means a group with three generators, cyclically ordered, with each generator conjugating some power of the previous one to another power. There are six parameters, occurring in pairs, and we show that the triangle fails to be developable whenever one of the parameters divides its partner, except for a few special cases. Furthermore, under fairly general conditions, the group turns out to be finite and solvable of derived length $\leq3$. We obtain a lot of information about finite quotients, even when we cannot determine developability.

We fix a mistake in the proof of Theorem 1.6 in the paper in the title.

In this paper we characterize the compactness of the commutator $[b,T]$ for the singular integral operator on the Morrey spaces $L^{p,\lambda}(\mathbb R^n)$. More precisely, we prove that if $b\in \operatorname{VMO}(\mathbb R^n)$, the $\operatorname {BMO} (\mathbb R^n)$-closure of $C_c^\infty(\mathbb R^n)$, then $[b,T]$ is a compact operator on the Morrey spaces $L^{p,\lambda}(\mathbb R^n)$ for $1\lt p\lt \infty$ and $0\lt \lambda\lt n$. Conversely, if $b\in \operatorname{BMO}(\mathbb R^n)$ and $[b,T]$ is a compact operator on the $L^{p,\,\lambda}(\mathbb R^n)$ for some $p\ (1\lt p\lt \infty)$, then $b\in \operatorname {VMO}(\mathbb R^n)$. Moreover, the boundedness of a rough singular integral operator $T$ and its commutator $[b,T]$ on $L^{p,\,\lambda}(\mathbb R^n)$ are also given. We obtain a sufficient condition for a subset in Morrey space to be a strongly pre-compact set, which has interest in its own right.

We discuss a clean level lowering theorem modulo prime powers for weight $2$ cusp forms. Furthermore, we illustrate how this can be used to completely solve certain twisted Fermat equations $ax^n+by^n+cz^n=0$.

We formulate an analog of Hermite's constant for function fields over a finite field and state a conjectural value for this analog. We prove our conjecture in many cases, and prove slightly weaker results in all other cases.

We consider cubic polynomials $f(z)=z^3+az+b$ defined over $\mathbb{C}(\lambda)$, with a marked point of period $N$ and multiplier $\lambda$. In the case $N=1$, there are infinitely many such objects, and in the case $N\geq 3$, only finitely many (subject to a mild assumption). The case $N=2$ has particularly rich structure, and we are able to describe all such cubic polynomials defined over the field $\bigcup_{n\geq 1}\mathbb{C}(\lambda^{1/n})$.

We present a general construction of Salem numbers via rational functions whose zeros and poles mostly lie on the unit circle and satisfy an interlacing condition. This extends and unifies earlier work. We then consider the ``obvious'' limit points of the set of Salem numbers produced by our theorems and show that these are all Pisot numbers, in support of a conjecture of Boyd. We then show that all Pisot numbers arise in this way. Combining this with a theorem of Boyd, we produce all Salem numbers via an interlacing construction.

We define the filtrated K-theory of a $\mathrm{C}^*$-algebra over a finite topological space \(X\) and explain how to construct a spectral sequence that computes the bivariant Kasparov theory over \(X\) in terms of filtrated K-theory. For finite spaces with a totally ordered lattice of open subsets, this spectral sequence becomes an exact sequence as in the Universal Coefficient Theorem, with the same consequences for classification. We also exhibit an example where filtrated K-theory is not yet a complete invariant. We describe two $\mathrm{C}^*$-algebras over a space \(X\) with four points that have isomorphic filtrated K-theory without being $\mathrm{KK}(X)$-equivalent. For this space \(X\), we enrich filtrated K-theory by another K-theory functor to a complete invariant up to $\mathrm{KK}(X)$-equivalence that satisfies a Universal Coefficient Theorem.

Let $\rho \colon G \to \operatorname{GL}(V)$ be a rational finite dimensional complex representation of a reductive linear algebraic group $G$, and let $\sigma_1,\dots,\sigma_n$ be a system of generators of the algebra of invariant polynomials $\mathbb C[V]^G$. We study the problem of lifting mappings $f\colon \mathbb R^q \supseteq U \to \sigma(V) \subseteq \mathbb C^n$ over the mapping of invariants $\sigma=(\sigma_1,\dots,\sigma_n) \colon V \to \sigma(V)$. Note that $\sigma(V)$ can be identified with the categorical quotient $V /\!\!/ G$ and its points correspond bijectively to the closed orbits in $V$. We prove that if $f$ belongs to a quasianalytic subclass $\mathcal C \subseteq C^\infty$ satisfying some mild closedness properties that guarantee resolution of singularities in $\mathcal C$, e.g., the real analytic class, then $f$ admits a lift of the same class $\mathcal C$ after desingularization by local blow-ups and local power substitutions. As a consequence we show that $f$ itself allows for a lift that belongs to $\operatorname{SBV}_{\operatorname{loc}}$, i.e., special functions of bounded variation. If $\rho$ is a real representation of a compact Lie group, we obtain stronger versions.

An extension theorem for holomorphic mappings between two domains in $\mathbb C^2$ is proved under purely local hypotheses.

We demonstrate how most common cardinal invariants associated with a von Neumann algebra $\mathcal M$ can be computed from the decomposability number, $\operatorname{dens}(\mathcal M)$, and the minimal cardinality of a generating set, $\operatorname{gen}(\mathcal M)$. Applications include the equivalence of the well-known generator problem, ``Is every separably-acting von Neumann algebra singly-generated?", with the formally stronger questions, ``Is every countably-generated von Neumann algebra singly-generated?" and ``Is the $\operatorname{gen}$ invariant monotone?" Modulo the generator problem, we determine the range of the invariant $\bigl( \operatorname{gen}(\mathcal M), \operatorname{dens}(\mathcal M) \bigr)$, which is mostly governed by the inequality $\operatorname{dens}(\mathcal M) \leq \mathfrak C^{\operatorname{gen}(\mathcal M)}$.

In this article we study some Sobolev-type inequalities on polynomial volume growth Lie groups. We show in particular that improved Sobolev inequalities can be extended to this general framework without the use of the Littlewood-Paley decomposition.

Dans cet article, on énonce une variante du lemme fondamental pondéré d'Arthur pour le groupe métaplectique de Weil, qui sera un ingrédient indispensable de la stabilisation de la formule des traces. Pour un corps de caractéristique résiduelle suffisamment grande, on en donne une démonstration à l'aide de la méthode de descente, qui est conditionnelle: on admet le lemme fondamental pondéré non standard sur les algèbres de Lie. Vu les travaux de Chaudouard et Laumon, on s'attend à ce que cette condition soit ultérieurement vérifiée.

A K-theoretic classification is given of the simple inductive limits of finite direct sums of the type I $C^*$-algebras known as splitting interval algebras with dimension drops. (These are the subhomogeneous $C^*$-algebras, each having spectrum a finite union of points and an open interval, and torsion $\textrm{K}_1$-group.)

We introduce the fundamental group ${\mathcal F}(A)$ of a simple $\sigma$-unital $C^*$-algebra $A$ with unique (up to scalar multiple) densely defined lower semicontinuous trace. This is a generalization of ``Fundamental Group of Simple $C^*$-algebras with Unique Trace I and II'' by Nawata and Watatani. Our definition in this paper makes sense for stably projectionless $C^*$-algebras. We show that there exist separable stably projectionless $C^*$-algebras such that their fundamental groups are equal to $\mathbb{R}_+^\times$ by using the classification theorem of Razak and Tsang. This is a contrast to the unital case in Nawata and Watatani. This study is motivated by the work of Kishimoto and Kumjian.

In this article we refine the method of Bertolini and Darmon and prove several finiteness results for anticyclotomic Selmer groups of Hilbert modular forms of parallel weight two.

We classify all genuine unitary representations of the metaplectic group whose infinitesimal character is real and at least as regular as that of the oscillator representation. In a previous paper we exhibited a certain family of representations satisfying these conditions, obtained by cohomological induction from the tensor product of a one-dimensional representation and an oscillator representation. Our main theorem asserts that this family exhausts the genuine omega-regular unitary dual of the metaplectic group.

It is shown that simplicity of the crossed product of a unital AH-algebra with slow dimension growth by an endomorphism implies that the algebra is also purely infinite, provided only that the endomorphism leaves no trace state invariant and takes the unit to a full projection.

We revisit with another view point the construction by R. Brylinski and B. Kostant of minimal representations of simple Lie groups. We start from a pair $(V,Q)$, where $V$ is a complex vector space and $Q$ a homogeneous polynomial of degree 4 on $V$. The manifold $\Xi $ is an orbit of a covering of ${\rm Conf}(V,Q)$, the conformal group of the pair $(V,Q)$, in a finite dimensional representation space. By a generalized Kantor-Koecher-Tits construction we obtain a complex simple Lie algebra $\mathfrak g$, and furthermore a real form ${\mathfrak g}_{\mathbb R}$. The connected and simply connected Lie group $G_{\mathbb R}$ with ${\rm Lie}(G_{\mathbb R})={\mathfrak g}_{\mathbb R}$ acts unitarily on a Hilbert space of holomorphic functions defined on the manifold $\Xi $.

We study projections in the corona algebra of $C(X)\otimes K$, where K is the $C^*$-algebra of compact operators on a separable infinite dimensional Hilbert space and $X=[0,1],[0,\infty),(-\infty,\infty)$, or $[0,1]/\{ 0,1 \}$. Using BDF's essential codimension, we determine conditions for a projection in the corona algebra to be liftable to a projection in the multiplier algebra. We also determine the conditions for two projections to be equal in $K_0$, Murray-von Neumann equivalent, unitarily equivalent, or homotopic. In light of these characterizations, we construct examples showing that the equivalence notions above are all distinct.

We study the geometry of non-reductive $4$-dimensional homogeneous spaces. In particular, after describing their Levi-Civita connection and curvature properties, we classify homogeneous Ricci solitons on these spaces, proving the existence of shrinking, expanding and steady examples. For all the non-trivial examples we find, the Ricci operator is diagonalizable.

Considering quantum random walks, we construct discrete-time approximations of the eigenvalues processes of minors of Hermitian Brownian motion. It has been recently proved by Adler, Nordenstam, and van Moerbeke that the process of eigenvalues of two consecutive minors of a Hermitian Brownian motion is a Markov process; whereas, if one considers more than two consecutive minors, the Markov property fails. We show that there are analog results in the noncommutative counterpart and establish the Markov property of eigenvalues of some particular submatrices of Hermitian Brownian motion.

We introduce a $q,t$-enumeration of Dyck paths that are forced to touch the main diagonal at specific points and forbidden to touch elsewhere and conjecture that it describes the action of the Macdonald theory $\nabla$ operator applied to a Hall--Littlewood polynomial. Our conjecture refines several earlier conjectures concerning the space of diagonal harmonics including the ``shuffle conjecture" (Duke J. Math. $\mathbf {126}$ ($2005$), 195-232) for $\nabla e_n[X]$. We bring to light that certain generalized Hall--Littlewood polynomials indexed by compositions are the building blocks for the algebraic combinatorial theory of $q,t$-Catalan sequences, and we prove a number of identities involving these functions.

We study the topology of tropical varieties that arise from a certain natural class of varieties. We use the theory of tropical degenerations to construct a natural, ``multiplicity-free'' parameterization of $\operatorname{Trop}(X)$ by a topological space $\Gamma_X$ and give a geometric interpretation of the cohomology of $\Gamma_X$ in terms of the action of a monodromy operator on the cohomology of $X$. This gives bounds on the Betti numbers of $\Gamma_X$ in terms of the Betti numbers of $X$ which constrain the topology of $\operatorname{Trop}(X)$. We also obtain a description of the top power of the monodromy operator acting on middle cohomology of $X$ in terms of the volume pairing on $\Gamma_X$.

In this paper we study the balayage of semi-Dirichlet forms. We present new results on balayaged functions and balayaged measures of semi-Dirichlet forms. Some of the results are new even in the Dirichlet forms setting.

Let $({\mathcal X}, d, \mu)$ be a separable metric measure space satisfying the known upper doubling condition, the geometrical doubling condition, and the non-atomic condition that $\mu(\{x\})=0$ for all $x\in{\mathcal X}$. In this paper, we show that the boundedness of a Calderón-Zygmund operator $T$ on $L^2(\mu)$ is equivalent to that of $T$ on $L^p(\mu)$ for some $p\in (1, \infty)$, and that of $T$ from $L^1(\mu)$ to $L^{1,\,\infty}(\mu).$ As an application, we prove that if $T$ is a Calderón-Zygmund operator bounded on $L^2(\mu)$, then its maximal operator is bounded on $L^p(\mu)$ for all $p\in (1, \infty)$ and from the space of all complex-valued Borel measures on ${\mathcal X}$ to $L^{1,\,\infty}(\mu)$. All these results generalize the corresponding results of Nazarov et al. on metric spaces with measures satisfying the so-called polynomial growth condition.

The regularity of solutions to optimal transportation problems has become a hot topic in current research. It is well known by now that the optimal measure may not be concentrated on the graph of a continuous mapping unless both the transportation cost and the masses transported satisfy very restrictive hypotheses (including sign conditions on the mixed fourth-order derivatives of the cost function). The purpose of this note is to show that in spite of this, the optimal measure is supported on a Lipschitz manifold, provided only that the cost is $C^{2}$ with non-singular mixed second derivative. We use this result to provide a simple proof that solutions to Monge's optimal transportation problem satisfy a change of variables equation almost everywhere.

In his last letter to Hardy, Ramanujan defined 17 functions $F(q)$, $|q|\lt 1$, which he called mock $\theta$-functions. He observed that as $q$ radially approaches any root of unity $\zeta$ at which $F(q)$ has an exponential singularity, there is a $\theta$-function $T_\zeta(q)$ with $F(q)-T_\zeta(q)=O(1)$. Since then, other functions have been found that possess this property. These functions are related to a function $H(x,q)$, where $x$ is usually $q^r$ or $e^{2\pi i r}$ for some rational number $r$. For this reason we refer to $H$ as a ``universal'' mock $\theta$-function. Modular transformations of $H$ give rise to the functions $K$, $K_1$, $K_2$. The functions $K$ and $K_1$ appear in Ramanujan's lost notebook. We prove various linear relations between these functions using Appell-Lerch sums (also called generalized Lambert series). Some relations (mock theta ``conjectures'') involving mock $\theta$-functions of even order and $H$ are listed.

We study the densities of uniform random walks in the plane. A special focus is on the case of short walks with three or four steps and less completely those with five steps. As one of the main results, we obtain a hypergeometric representation of the density for four steps, which complements the classical elliptic representation in the case of three steps. It appears unrealistic to expect similar results for more than five steps. New results are also presented concerning the moments of uniform random walks and, in particular, their derivatives. Relations with Mahler measures are discussed.

We consider the problem of constructing Poisson brackets on smooth manifolds $M$ with prescribed Casimir functions. If $M$ is of even dimension, we achieve our construction by considering a suitable almost symplectic structure on $M$, while, in the case where $M$ is of odd dimension, our objective is achieved by using a convenient almost cosymplectic structure. Several examples and applications are presented.

In this article, we show to which extent one can improve a theorem of Bombieri, Friedlander and Iwaniec by using Hooley's variant of the divisor switching technique. We also give an application of the theorem in question, which is a Bombieri-Vinogradov type theorem for the Tichmarsh divisor problem in arithmetic progressions.

In this paper we study the extension problem, the averaging problem, and the generalized Erdős-Falconer distance problem associated with arbitrary homogeneous varieties in three dimensional vector spaces over finite fields. In the case when the varieties do not contain any plane passing through the origin, we obtain the best possible results on the aforementioned three problems. In particular, our result on the extension problem modestly generalizes the result by Mockenhaupt and Tao who studied the particular conical extension problem. In addition, investigating the Fourier decay on homogeneous varieties enables us to give complete mapping properties of averaging operators. Moreover, we improve the size condition on a set such that the cardinality of its distance set is nontrivial.

A body moves in a rarefied medium composed of point particles at rest. The particles make elastic reflections when colliding with the body surface, and do not interact with each other. We consider a generalization of Newton's minimal resistance problem: given two bounded convex bodies $C_1$ and $C_2$ such that $C_1 \subset C_2 \subset \mathbb{R}^3$ and $\partial C_1 \cap \partial C_2 = \emptyset$, minimize the resistance in the class of connected bodies $B$ such that $C_1 \subset B \subset C_2$. We prove that the infimum of resistance is zero; that is, there exist "almost perfectly streamlined" bodies.

We consider the stochastic difference equation $$\eta _k = \xi _k \phi (\eta _{k-1}), \quad k \in \mathbb Z $$ on a locally compact group $G$ where $\phi$ is an automorphism of $G$, $\xi _k$ are given $G$-valued random variables and $\eta _k$ are unknown $G$-valued random variables. This equation was considered by Tsirelson and Yor on one-dimensional torus. We consider the case when $\xi _k$ have a common law $\mu$ and prove that if $G$ is a distal group and $\phi$ is a distal automorphism of $G$ and if the equation has a solution, then extremal solutions of the equation are in one-one correspondence with points on the coset space $K\backslash G$ for some compact subgroup $K$ of $G$ such that $\mu$ is supported on $Kz= z\phi (K)$ for any $z$ in the support of $\mu$. We also provide a necessary and sufficient condition for the existence of solutions to the equation.

In this paper we first generalize to the case of partial flags a result proved both by Spaltenstein and by Steinberg that relates the relative position of two complete flags and the irreducible components of the flag variety in which they lie, using the Robinson-Schensted-Knuth correspondence. Then we use this result to generalize the mirabolic Robinson-Schensted-Knuth correspondence defined by Travkin, to the case of two partial flags and a line.

Darmon cycles are a higher weight analogue of Stark--Heegner points. They yield local cohomology classes in the Deligne representation associated with a cuspidal form on $\Gamma _{0}( N) $ of even weight $k_{0}\geq 2$. They are conjectured to be the restriction of global cohomology classes in the Bloch--Kato Selmer group defined over narrow ring class fields attached to a real quadratic field. We show that suitable linear combinations of them obtained by genus characters satisfy these conjectures. We also prove $p$-adic Gross--Zagier type formulas, relating the derivatives of $p$-adic $L$-functions of the weight variable attached to imaginary (resp. real) quadratic fields to Heegner cycles (resp. Darmon cycles). Finally we express the second derivative of the Mazur--Kitagawa $p$-adic $L$-function of the weight variable in terms of a global cycle defined over a quadratic extension of $\mathbb{Q}$.

Extending the work of Larson and Todorcevic, we show there is a model of set theory in which normal spaces are collectionwise Hausdorff if they are either first countable or locally compact, and yet there are no first countable $L$-spaces or compact $S$-spaces. The model is one of the form PFA$(S)[S]$, where $S$ is a coherent Souslin tree.

Let $(n_k)_{k \geq 1}$ be an increasing sequence of positive integers, and let $f(x)$ be a real function satisfying \begin{equation} \tag{1} f(x+1)=f(x), \qquad \int_0^1 f(x) ~dx=0,\qquad \operatorname{Var_{[0,1]}} f \lt \infty. \end{equation} If $\lim_{k \to \infty} \frac{n_{k+1}}{n_k} = \infty$ the distribution of \begin{equation} \tag{2} \frac{\sum_{k=1}^N f(n_k x)}{\sqrt{N}} \end{equation} converges to a Gaussian distribution. In the case $$ 1 \lt \liminf_{k \to \infty} \frac{n_{k+1}}{n_k}, \qquad \limsup_{k \to \infty} \frac{n_{k+1}}{n_k} \lt \infty $$ there is a complex interplay between the analytic properties of the function $f$, the number-theoretic properties of $(n_k)_{k \geq 1}$, and the limit distribution of (2). In this paper we prove that any sequence $(n_k)_{k \geq 1}$ satisfying $\limsup_{k \to \infty} \frac{n_{k+1}}{n_k} = 1$ contains a nontrivial subsequence $(m_k)_{k \geq 1}$ such that for any function satisfying (1) the distribution of $$ \frac{\sum_{k=1}^N f(m_k x)}{\sqrt{N}} $$ converges to a Gaussian distribution. This result is best possible: for any $\varepsilon\gt 0$ there exists a sequence $(n_k)_{k \geq 1}$ satisfying $\limsup_{k \to \infty} \frac{n_{k+1}}{n_k} \leq 1 + \varepsilon$ such that for every nontrivial subsequence $(m_k)_{k \geq 1}$ of $(n_k)_{k \geq 1}$ the distribution of (2) does not converge to a Gaussian distribution for some $f$. Our result can be viewed as a Ramsey type result: a sufficiently dense increasing integer sequence contains a subsequence having a certain requested number-theoretic property.

Let $\Delta$ be an Euclidean quiver. We prove that the closures of the maximal orbits in the varieties of representations of $\Delta$ are normal and Cohen--Macaulay (even complete intersections). Moreover, we give a generalization of this result for the tame concealed-canonical algebras.

In this paper, we generalize a conjecture due to Darmon and Logan in an adelic setting. We study the relation between our construction and Kudla's works on cycles on orthogonal Shimura varieties. This relation allows us to conjecture a Gross-Kohnen-Zagier theorem for Darmon's points.

In this paper we study systems of weakly coupled Hamilton-Jacobi equations with implicit obstacles that arise in optimal switching problems. Inspired by methods from the theory of viscosity solutions and weak KAM theory, we extend the notion of Aubry set for these systems. This enables us to prove a new result on existence and uniqueness of solutions for the Dirichlet problem, answering a question of F. Camilli, P. Loreti and N. Yamada.

Let $C$ and $D$ be digraphs. A mapping $f\colon V(D)\to V(C)$ is a $C$-colouring if for every arc $uv$ of $D$, either $f(u)f(v)$ is an arc of $C$ or $f(u)=f(v)$, and the preimage of every vertex of $C$ induces an acyclic subdigraph in $D$. We say that $D$ is $C$-colourable if it admits a $C$-colouring and that $D$ is uniquely $C$-colourable if it is surjectively $C$-colourable and any two $C$-colourings of $D$ differ by an automorphism of $C$. We prove that if a digraph $D$ is not $C$-colourable, then there exist digraphs of arbitrarily large girth that are $D$-colourable but not $C$-colourable. Moreover, for every digraph $D$ that is uniquely $D$-colourable, there exists a uniquely $D$-colourable digraph of arbitrarily large girth. In particular, this implies that for every rational number $r\geq 1$, there are uniquely circularly $r$-colourable digraphs with arbitrarily large girth.

We study properties of composition operators induced by symbols acting from the unit disk to the polydisk. This result will be involved in the investigation of weighted composition operators on the Hardy space on the unit disk and moreover be concerned with composition operators acting from the Bergman space to the Hardy space on the unit disk.

We present a new construction of the entropy-maximizing, invariant probability measure on a Smale space (the Bowen measure). Our construction is based on points that are unstably equivalent to one given point, and stably equivalent to another: heteroclinic points. The spirit of the construction is similar to Bowen's construction from periodic points, though the techniques are very different. We also prove results about the growth rate of certain sets of heteroclinic points, and about the stable and unstable components of the Bowen measure. The approach we take is to prove results through direct computation for the case of a Shift of Finite type, and then use resolving factor maps to extend the results to more general Smale spaces.

In 1960, Sobolev proved that for a finite reflection group $G$, a $G$-invariant cubature formula is of degree $t$ if and only if it is exact for all $G$-invariant polynomials of degree at most $t$. In this paper, we find some observations on invariant cubature formulas and Euclidean designs in connection with the Sobolev theorem. First, we give an alternative proof of theorems by Xu (1998) on necessary and sufficient conditions for the existence of cubature formulas with some strong symmetry. The new proof is shorter and simpler compared to the original one by Xu, and moreover gives a general interpretation of the analytically-written conditions of Xu's theorems. Second, we extend a theorem by Neumaier and Seidel (1988) on Euclidean designs to invariant Euclidean designs, and thereby classify tight Euclidean designs obtained from unions of the orbits of the corner vectors. This result generalizes a theorem of Bajnok (2007) which classifies tight Euclidean designs invariant under the Weyl group of type $B$ to other finite reflection groups.

Using ideas from Shelah's recent proof that a completely separable maximal almost disjoint family exists when $\mathfrak{c} \lt {\aleph}_{\omega}$, we construct a weakly tight family under the hypothesis $\mathfrak{s} \leq \mathfrak{b} \lt {\aleph}_{\omega}$. The case when $\mathfrak{s} \lt \mathfrak{b}$ is handled in $\mathrm{ZFC}$ and does not require $\mathfrak{b} \lt {\aleph}_{\omega}$, while an additional PCF type hypothesis, which holds when $\mathfrak{b} \lt {\aleph}_{\omega}$ is used to treat the case $\mathfrak{s} = \mathfrak{b}$. The notion of a weakly tight family is a natural weakening of the well studied notion of a Cohen indestructible maximal almost disjoint family. It was introduced by Hrušák and García Ferreira, who applied it to the Katétov order on almost disjoint families.

This article gives an existence theory for weak solutions of second order non-elliptic linear Dirichlet problems of the form \begin{align*} \nabla'P(x)\nabla u +{\bf HR}u+{\bf S'G}u +Fu &= f+{\bf T'g} \text{ in }\Theta \\ u&=\varphi\text{ on }\partial \Theta. \end{align*} The principal part $\xi'P(x)\xi$ of the above equation is assumed to be comparable to a quadratic form ${\mathcal Q}(x,\xi) = \xi'Q(x)\xi$ that may vanish for non-zero $\xi\in\mathbb{R}^n$. This is achieved using techniques of functional analysis applied to the degenerate Sobolev spaces $QH^1(\Theta)=W^{1,2}(\Theta,Q)$ and $QH^1_0(\Theta)=W^{1,2}_0(\Theta,Q)$ as defined in previous works. Sawyer and Wheeden give a regularity theory for a subset of the class of equations dealt with here.

Analytical study to the regularization of the Boussinesq system is performed in frequency space using Fourier theory. Existence and uniqueness of weak solution with minimum regularity requirement are proved. Convergence results of the unique weak solution of the regularized Boussinesq system to a weak Leray-Hopf solution of the Boussinesq system are established as the regularizing parameter $\alpha$ vanishes. The proofs are done in the frequency space and use energy methods, Arselà-Ascoli compactness theorem and a Friedrichs like approximation scheme.