In this paper, we investigate germs of smooth real hypersurfaces in $\mathbb{C}^{n}$. We show that if the hypersurface is of infinite D’Angelo type at a point, then there exists a formal complex curve in the hypersurface through that point.

Beiglböck, Bergelson and Fish proved that if subsets $A$, $B$ of a countable discrete amenable group $G$ have positive Banach densities $\alpha$ and $\beta$ respectively, then the product set $AB$ is piecewise syndetic, that is, there exists $k$ such that the union of $k$-many left translates of $AB$ is thick. Using nonstandard analysis, we give a shorter alternative proof of this result that does not require $G$ to be countable and moreover yields the explicit bound $k\le1/\alpha\beta$. We also prove with similar methods that if $\{A_{i}\}_{i=1}^{n}$ are finitely many subsets of $G$ having positive Banach densities $\alpha_{i}$ and $G$ is countable, then there exists a subset $B$ whose Banach density is at least $\prod_{i=1}^{n}\alpha_{i}$ and such that $BB^{-1}\subseteq\bigcap_{i=1}^{n}A_{i}A_{i}^{-1}$. In particular, the latter set is piecewise Bohr.

We compute the BNS-invariant for the pure symmetric automorphism groups of right-angled Artin groups. We use this calculation to show that the pure symmetric automorphism group of a right-angled Artin group is itself not a right-angled Artin group provided that its defining graph contains a separating intersection of links.

In this paper, we study $G$-Higgs bundles over an elliptic curve when the structure group $G$ is a classical complex reductive Lie group. Modifying the notion of family, we define a new moduli problem for the classification of semistable $G$-Higgs bundles of a given topological type over an elliptic curve and we give an explicit description of the associated moduli space as a finite quotient of a product of copies of the cotangent bundle of the elliptic curve. We construct a bijective morphism from this new moduli space to the usual moduli space of semistable $G$-Higgs bundles, proving that the former is the normalization of the latter. We also obtain an explicit description of the Hitchin fibration for our (new) moduli space of $G$-Higgs bundles and we study the generic and non-generic fibres.

Let $X$ be a separable real or complex $L_{1}$-predual such that its dual unit ball $B_{X^{*}}$ has the set $\operatorname{ext}B_{X^{*}}$ of its extreme points of type $F_{\sigma}$. We identify intrinsic Baire classes of $X$ with the spaces of odd or homogeneous Baire functions on $\operatorname{ext}B_{X^{*}}$. Further, we answer a question of S. A. Argyros, G. Godefroy and H. P. Rosenthal by showing that there exists a separable $C^{*}$-algebra $X$ (the so-called CAR-algebra) for which the second intrinsic Baire class of $X^{**}$ does not coincide with the second Baire class of $X^{**}$.

A question about Ricci flow is when the diameters of the manifold under the evolving metrics stay finite and bounded away from $0$. Topping (Comm. Anal. Geom. 13 (2005) 1039–1055) addresses the question with an upper bound that depends on the $L^{(n-1)/2}$ bound of the scalar curvature, volume and a local version of Perelman’s $\nu$ invariant. Here $n$ is the dimension. His result is sharp when Perelman’s F entropy is positive. In this note, we give a direct proof that for all compact manifolds, the diameter bound depends just on the $L^{(n-1)/2}$ bound of the scalar curvature, volume and the Sobolev constants (or positive Yamabe constant). This bound seems directly computable in large time for some Ricci flows. In addition, since the result in its most general form is independent of Ricci flow, further applications may be possible.

A generally sharp lower bound for the diameters is also given, which depends only on the initial metric, time and $L^{\infty}$ bound of the scalar curvature. These results imply that, in finite time, the Ricci flow can neither turn the diameter to infinity nor zero, unless the scalar curvature blows up.

For each positive integer $n$, let $s(n)$ denote the sum of the proper divisors of $n$. If $s(n)>0$, put $s_{2}(n)=s(s(n))$, and define the higher iterates $s_{k}(n)$ similarly. In 1976, Erdős proved the following theorem: For each $\delta>0$ and each integer $K\geq2$, we have

\[-\delta<\frac{s_{k+1}(n)}{s_{k}(n)}-\frac{s(n)}{n}\] for all $1\leq k<K$, except for a set of $n$ of asymptotic density zero. He also conjectured that

\[\frac{s_{k+1}(n)}{s_{k}(n)}-\frac{s(n)}{n}<\delta\] for all $1\leq k<K$ and all $n$ outside of a set of density zero. This conjecture has proved rather recalcitrant and is known only when $K=2$, a 1990 result of Erdős, Granville, Pomerance, and Spiro. We reprove their theorem in quantitative form, by what seems to be a simpler and more transparent argument.

Similar techniques are used to investigate the arithmetic properties of the sum of the distinct prime divisors of $n$, which we denote by $\beta(n)$. We show that for a randomly chosen $n$, the integer $\beta(n)$ is squarefree with the same probability as $n$ itself. We also prove the same result with “squarefree” replaced by “abundant”.

Finally, we prove that for either of the functions $f(n)=s(n)$ or $f(n)=\beta(n)$, the number of $n\le x$ for which $f(n)$ is prime is $O(x/\log{x})$.

Let $X$ be a supermartingale starting from $0$ which has only nonnegative jumps. For each $0<p<1$ we determine the best constants $c_{p}$, $C_{p}$ and $\mathfrak{c}_{p}$ such that

\begin{eqnarray*}\sup_{t\geq0}\big\|X_{t}\big\|_{p}&\leq&C_{p}\big\|-\inf_{t\geq0}X_{t}\big\|_{p},\\\phantom{\Vert }\Vert \sup_{t\geq0}X_{t}\Vert _{p}&\leq&c_{p}\|-\inf_{t\geq0}X_{t}\|_{p}\end{eqnarray*} and

\[\Vert \sup_{t\geq0}\vert X_{t}\vert \Vert _{p}\leq\mathfrak{c}_{p}\|-\inf_{t\geq0}X_{t}\|_{p}.\] The estimates are shown to be sharp if $X$ is assumed to be a stopped one-dimensional Brownian motion. The inequalities are deduced from the existence of special functions, enjoying certain majorization and convexity-type properties. Some applications concerning harmonic functions on Euclidean domains are indicated.

In this paper, we consider a stochastic Burgers equation driven by Lévy noise and study the transition semigroup of the solution to the initial value problem for the equation in the space of continuous functions weighted by a proper potential. We show that the infinitesimal generator is the closure of the Kolmogorov operator associated to the equation in a suitable topology. We also prove existence and uniqueness results for the associated Fokker–Planck equation.

Let $Z(s)$ be the Selberg zeta-function associated with a compact Riemann surface. We prove that the imaginary parts of the nontrivial $a$-points of $Z(s)$ are uniformly distributed modulo one. We also consider the question whether the eigenvalues of the corresponding Laplacian are uniformly distributed modulo one.

Let $X$ be an algebraic subvariety of $\mathbb{C}^{n}$ and $\overline{X}$ be its closure in $\mathbb{P}^{n}$. In their paper (J. Reine Angew. Math. 676 (2013), 33–49), Coman, Guedj and Zeriahi proved that any plurisubharmonic function with logarithmic growth on $X$ extends to a plurisubharmonic function with logarithmic growth on $\mathbb{C}^{n}$ when the germs $(\overline{X},a)$ in $\mathbb{P}^{n}$ are irreducible for all $a\in\overline{X}\setminus X$. In this paper we consider $X$ for which the germ $(\overline{X},a)$ is reducible for some $a\in\overline{X}\setminus X$ and we give a necessary and sufficient condition for $X$ so that any plurisubharmonic function with logarithmic growth on $X$ extends to a plurisubharmonic function with logarithmic growth on $\mathbb{C}^{n}$.

For each $k\geq2$, we construct a family of surfaces in $\Sigma(\kappa)\times\mathbb{R}$ with constant mean curvature $H\in[0,1/2]$, where $\kappa+4H^{2}\leq0$ and $\Sigma(\kappa)$ is a two-dimensional space form. The surfaces are invariant under $2\pi/k$-rotations about a vertical fiber of $\Sigma(\kappa)\times\mathbb{R}$, have genus zero, and $2k$ ends. Each surface arises as the sister surface of a minimal graph in a homogeneous $3$-manifold. The domain of the graph is non-convex. We use the sisters of a generalization of Jorge–Meeks-$k$-noids in homogeneous $3$-manifolds as barriers in the conjugate Plateau construction.

Let $\mathcal{E}$ be a Dirichlet form and $\mu$ a positive Kato measure. We define criticality and subcriticality for the Schrödinger form, $\mathcal{E}(\cdot,\cdot)+(\cdot,\cdot)_{\mu}$, through $h$-transform. For a certain potential $\mu$, an analytic characterization of these properties is given in terms of the bottom of spectrum.

In 1952, Littlewood stated a conjecture about the average growth of spherical derivatives of polynomials, and showed that it would imply that for entire function of finite order, “most” preimages of almost all points are concentrated in a small subset of the plane. In 1988, Lewis and Wu proved Littlewood’s conjecture. Using techniques from complex dynamics, we construct entire functions of finite order with a bounded set of singular values for which the set of exceptional preimages is infinite, with logarithmically growing cardinality.