If $X$ is a connected complex manifold with ${{d}_{X}}\,=\,2$ that admits a (connected) Lie group $G$ acting transitively as a group of holomorphic transformations, then the action extends to an action of the complexification $\widehat{G}$ of $G$ on $X$ except when either the unit disk in the complex plane or a strictly pseudoconcave homogeneous complex manifold is the base or fiber of some homogeneous fibration of $X$ .

In this paper we study connections between topological games such as Rothberger, Menger, and compact-open games, and we relate these games to properties involving covers by ${{G}_{\delta }}$ subsets. The results include the following: (1) If TWO has a winning strategy in theMenger game on a regular space $X$ , then $X$ is an Alster space. (2) If TWO has a winning strategy in the Rothberger game on a topological space $X$ , then the ${{G}_{\delta }}$ -topology on $X$ is Lindelöf. (3) The Menger game and the compact-open game are (consistently) not dual.

We describe a general setting where the monodromy action on the first cohomology group of the Milnor fiber of a hyperplane arrangement is the identity.

It is known that the normalized standard generators of the free orthogonal quantum group $O_{N}^{+}$ converge in distribution to a free semicircular system as $N\,\to \,\infty$ . In this note, we substantially improve this convergence result by proving that, in addition to distributional convergence, the operator normof any non-commutative polynomial in the normalized standard generators of $O_{N}^{+}$ converges as $N\,\to \,\infty$ to the operator norm of the corresponding non-commutative polynomial in a standard free semicircular system. Analogous strong convergence results are obtained for the generators of free unitary quantum groups. As applications of these results, we obtain a matrix-coefficient version of our strong convergence theorem, and we recover a well-known ${{\mathcal{L}}^{2}}\,-\,{{\mathcal{L}}^{\infty }}$ norm equivalence for noncommutative polynomials in free semicircular systems.

We classify integral modular categories of dimension $p{{q}^{4}}$ and ${{p}^{2}}{{q}^{2}}$ , where $p$ and $q$ are distinct primes. We show that such categories are always group-theoretical, except for categories of dimension $4{{q}^{2}}$ . In these cases there are well-known examples of non-group-theoretical categories, coming from centers of Tambara–Yamagami categories and quantum groups. We show that a non-grouptheoretical integral modular category of dimension $4{{q}^{2}}$ is either equivalent to one of these well-known examples or is of dimension 36 and is twist-equivalent to fusion categories arising froma certain quantum group.

We describe all finite dimensional uniserial representations of a commutative associative (resp. abelian Lie) algebra over a perfect (resp. sufficiently large perfect) field. In the Lie case the size of the field depends on the answer to following question, considered and solved in this paper. Let $K/F$ be a finite separable field extension and let $x,\,y\,\in \,K$ . When is $F\left[ x,\,y \right]\,=\,F\left[ \alpha x\,+\,\beta y \right]$ for some nonzero elements $\alpha ,\,\beta \,\in \,F?$

We describe double Hurwitz numbers as intersection numbers on the moduli space of curves ${{\overline{M}}_{g,n}}$ Using a result on the polynomiality of intersection numbers of psi classes with the Double Ramification Cycle, our formula explains the polynomiality in chambers of double Hurwitz numbers and the wall-crossing phenomenon in terms of a variation of correction terms to the $\varphi$ classes. We interpret this as suggestive evidence for polynomiality of the Double Ramification Cycle (which is only known in genera 0 and 1).

We consider the Finsler space $\left( {{\overline{M}}^{3}},\,\overline{F} \right)$ obtained by perturbing the Euclidean metric of ${{\mathbb{R}}^{3}}$ by a rotation. It is the open region of ${{\mathbb{R}}^{3}}$ bounded by a cylinder with a Randers metric. Using the Busemann–Hausdorff volume form, we obtain the differential equation that characterizes the helicoidal minimal surfaces in ${{\overline{M}}^{3}}$ . We prove that the helicoid is a minimal surface in ${{\overline{M}}^{3}}$ only if the axis of the helicoid is the axis of the cylinder. Moreover, we prove that, in the Randers space $\left( {{\overline{M}}^{3}},\,\overline{F} \right)$ , the only minimal surfaces in the Bonnet family with fixed axis $O{{\overline{x}}^{3}}$ are the catenoids and the helicoids.

Previous results by the author on the connection between three measures of noncompactness obtained for ${{\mathcal{L}}_{p}}$ are extended to regular spaces of measurable functions. An example is given of the advantages of some cases in comparison with others. Geometric characteristics of regular spaces are determined. New theorems for $\left( k,\,\beta\right)$ -boundedness of partially additive operators are proved.

We give some new characterizations for compactness of weighted composition operators $u{{C}_{\varphi }}$ acting on Bloch-type spaces in terms of the power of the components of $\varphi$ , where $\varphi$ is a holomorphic self-map of the polydisk ${{\mathbb{D}}^{n}}$ , thus generalizing the results obtained by Hyvärinen and Lindström in 2012.

Let $L\left( X \right)$ be the free locally convex space over a Tychonoff space $X$ . Then $L\left( X \right)$ is a $k$ -space if and only if $X$ is a countable discrete space. We prove also that $L\left( D \right)$ has uncountable tightness for every uncountable discrete space $D$ .

We show that if $E$ is a separable reflexive space, and $L$ is a weak-star closed linear subspace of $L\left( E \right)$ such that $L\cap K\left( E \right)$ is weak-star dense in $L$ , then $L$ has a unique isometric predual. The proof relies on basic topological arguments.

A formula is provided to explicitly describe global dimensions of all kinds of tree type finite dimensional $k$ -algebras for $k$ an algebraic closed field. In particular, it is pointed out that if the underlying tree type quiver has $n$ vertices, then the maximum global dimension is $n\,-\,1$ .

In this paper we give a characterization of a real hypersurface of Type $\left( A \right)$ in complex two-plane Grassmannians ${{G}_{2}}\left( {{\mathbb{C}}^{m+2}} \right)$ , which means a tube over a totally geodesic ${{G}_{2}}\left( {{\mathbb{C}}^{m+1}} \right)$ in ${{G}_{2}}\left( {{\mathbb{C}}^{m+2}} \right)$ , by means of the Reeb parallel structure Jacobi operator ${{\nabla }_{\xi }}{{R}_{\xi }}\,=\,0$ .

We study ${{L}^{p}}\to {{L}^{r}}$ restriction estimates for algebraic varieties $V$ in the case when restriction operators act on radial functions in the finite field setting. We show that if the varieties $V$ lie in odd dimensional vector spaces over finite fields, then the conjectured restriction estimates are possible for all radial test functions. In addition, assuming that the varieties $V$ are defined in even dimensional spaces and have few intersection points with the sphere of zero radius, we also obtain the conjectured exponents for all radial test functions.

Let $f$ be a modular form that is non-ordinary at $p$ . Loeffler has recently constructed four two-variable $p$ -adic $L$ -functions associated with $f$ . In the case where ${{a}_{p}}\,=\,0$ , he showed that, as in the one-variable case, Pollack’s plus and minus splitting applies to these new objects. In this article, we show that such a splitting can be generalised to the case where ${{a}_{p}}\ne 0$ using Sprung’s logarithmic matrix.

Let $X$ be a compact Hausdorff space. In this paper, we give an example to show that there is $u\,\in \,C\left( X \right)\,\otimes \,{{M}_{n}}$ with $\det \left( u\left( x \right) \right)\,=\,1$ for all $x\,\in \,X$ and $u{{\tilde{\ }}_{h}}1$ such that the ${{C}^{*}}$ exponential length of $u$ (denoted by $\text{cel}\left( u \right)$ ) cannot be controlled by $\pi$ . Moreover, in simple inductive limit ${{C}^{*}}$ -algebras, similar examples also exist.

It is a theorem of Bers that any closed hyperbolic surface admits a pants decomposition consisting of curves of bounded length where the bound only depends on the topology of the surface. The question of the quantification of the optimal constants has been well studied, and the best upper bounds to date are linear in genus, due to a theorem of Buser and Seppälä. The goal of this note is to give a short proof of a linear upper bound that slightly improves the best known bound.

We prove some results concerning convolutions, additive energies, and sumsets of convex sets and their generalizations. In particular, we show that if a set $A\,=\,{{\{{{a}_{1}},\,.\,.\,.\,,\,{{a}_{n}}\}}_{<}}\,\subseteq \,\mathbb{R}$ has the property that for every fixed $1\,\le \,d\,<\,n$ , all differences ${{a}_{i}}\,-\,{{a}_{i-d}},\,d\,<\,i\,<n$ , are distinct, then $\left| A\,+\,A \right|\,\gg \,{{\left| A \right|}^{3/2+c}}$ for a constant $c\,>\,0$ .

Let $A$ be a subgroup of a finite group $G$ and $\sum \,=\,\{{{G}_{0}}\,\le \,{{G}_{1}}\,\le \,.\,.\,.\,\le \,{{G}_{n}}\}$ some subgroup series of $G$ . Suppose that for each pair $\left( K,\,H \right)$ such that $K$ is a maximal subgroup of $H$ and ${{G}_{i-1}}\,\le \,K\,<\,H\,\le \,{{G}_{i}}$ , for some i, either $A\,\cap \,H\,=\,A\,\cap \,K\,\text{or}\,\text{AH}\,\text{=}\,\text{AK}$ . Then $A$ is said to be $\sum$ -embedded in $G$ . And $A$ is said to be $m$ -embedded in $G$ if $G$ has a subnormal subgroup $T$ and $a\,\{1\,\le \,G\}$ -embedded subgroup $C$ in $G$ such that $G\,=\,AT$ and $T\cap A\,\le \,C\,\le \,A$ . In this article, some sufficient conditions for a finite group $G$ to be $p$ -nilpotent are given whenever all subgroups with order ${{p}^{k}}$ of a Sylow $p$ -subgroup of $G$ are $m$ -embedded for a given positive integer $k$ .