Proper Holomorphic Embeddings of Riemann Surfaces with Arbitrary Topology into ℂ2

We prove that given an open Riemann surface $\mathcal{N}$ of arbitrary (finite or infinite) topology, there exists an open domain $\mathcal{M}\subset \mathcal{N}$ homeomorphic to $\mathcal{N}$ which properly holomorphically embeds in ?². Furthermore, $\mathcal{M}$ can be chosen with hyperbolic conformal type. In particular, any open orientable surface M admits a complex structure $\mathcal{C}$ such that $(M,\mathcal{C})$ can be properly holomorphically embedded into ?².     

Topological Brandt semigroups

In the present paper, the properties of a locally compact Hausdorff topological Brandt semigroup, and the relation between its semigroup algebras and ? ¹-Munn algebras over group algebras are investigated. It is proved that for each locally compact Hausdorff topological group G, and each index set I, there exists a locally compact Hausdorff topological Brandt semigroup S=B(G,I) such that the Banach algebras $\mathcal {LM}_{I}(M(G))$ and $\mathcal{LM}_{I}(L^{1}(G))$ are isometrically isomorphic to M(S)/? ¹({0}) and M _a (S)/? ¹({0}), respectively.     

Proper/Residually-Finite Idempotent Semirings

An idempotent semiring (= ISR) is called L-E if its underlying additive Abelian semigroup is generated by join-primes. Not all ISRs are L-E; not even when finite. The submodule of “linear-recognizable” elements of an ISR $\mathcal {M}$ is denoted $\mathcal {C}\mathcal {M}$ , and $\mathcal {M}$ is called proper if there are enough elements in $\mathcal {C}\mathcal {M}$ to separate points. If there are enough finite-index congruences to separate points, $\mathcal {M}$ is called residually-finite. Finite and proper ISRs are always residually-finite, but finite ISRs are not always proper, unless they are L-E. For certain classes of ISRs, conditions are given to guarantee proper and residual-finiteness. Among these is one which requires that the compact elements of the linear dual of $\mathcal {M}$ belong to $\mathcal {C}\mathcal {M}$ . Another condition requires that the recognizable subsets of a certain underlying monoid remain recognizable under the closure operator relative to a certain natural topology. These conditions are automatic for any finite L-E ISR, or any L-E ISR arising from a bounded, distributive lattice. Thus, a large class of proper/residually-finite ISRs exists. Moreover, the theorem of Malcev for semigroups (finitely-generated, commutative implies residually-finite) is shown to fail for ISRs in general.     

Completely continuous and weakly completely continuous abstract Segal algebras

Let $\mathcal{A}$ be a Banach algebra. It is obtained a necessary and sufficient condition for the complete continuity and also weak complete continuity of symmetric abstract Segal algebras with respect to $\mathcal{A}$ , under the condition of the existence of an approximate identity for $\mathcal{B}$ , bounded in $\mathcal{A}$ . In addition, a necessary condition for the weak complete continuity of $\mathcal{A}$ is given. Moreover, the applications of these results about some group algebras on locally compact groups are obtained.     

Finiteness of the basic intersection cohomology of a Killing foliation

We prove that the basic intersection cohomology ${{\mathbb H}^{*}_{\overline{p}}({M / \mathcal{F}})}$ , where ${\mathcal F}$ is the singular foliation determined by an isometric action of a Lie group G on the compact manifold M, is finite dimensional.     

Characterization of Lie higher derivations on triangular algebras

Let $\mathcal{A}$ and $\mathcal{B}$ be unital rings, and $\mathcal{M}$ be an $\left( {\mathcal{A},\mathcal{B}} \right)$ -bimodule, which is faithful as a left $\mathcal{A}$ -module and also as a right $\mathcal{B}$ -module. Let $\mathcal{U} = Tri\left( {\mathcal{A},\mathcal{M},\mathcal{B}} \right)$ be the triangular algebra. In this paper, we give some different characterizations of Lie higher derivations on $\mathcal{U}$ .     

Positive definite superfunctions and unitary representations of lie supergroups

For a broad class of Fréchet-Lie supergroups $ \mathcal{G} $ , we prove that there exists a correspondence between positive definite smooth (resp., analytic) superfunctions on $ \mathcal{G} $ and matrix coefficients of smooth (resp., analytic) unitary representations of the Harish-Chandra pair (G, $ \mathfrak{g} $ ) associated to $ \mathcal{G} $ . As an application, we prove that a smooth positive definite superfunction on $ \mathcal{G} $ is analytic if and only if it restricts to an analytic function on the underlying manifold of $ \mathcal{G} $ . When the underlying manifold of $ \mathcal{G} $ is 1-connected we obtain a necessary and sufficient condition for a linear functional on the universal enveloping algebra U( $ {{\mathfrak{g}}_{\mathbb{C}}} $ ) to correspond to a matrix coefficient of a unitary representation of (G, $ \mathfrak{g} $ ). The class of Lie supergroups for which the aforementioned results hold is characterised by a condition on the convergence of the Trotter product formula. This condition is strictly weaker than assuming that the underlying Lie group of $ \mathcal{G} $ is a locally exponential Fréchet-Lie group. In particular, our results apply to examples of interest in representation theory such as mapping supergroups and diffeomorphism supergroups.     

On topological diversity of rectilinear representations of countably infinite graphs

We prove that, for each simple graph G whose set of vertices is countably infinite, there is a family ${\varvec{\mathcal{R}}(\varvec{G})}$ of the cardinality of the continuum of graphs such that (1) each graph ${\varvec{H} \in \varvec{\mathcal{R}}(\varvec{G})}$ is isomorphic to G, all vertices of H are points of the Euclidean space E ³, all edges of H are straight line segments (the ends of each edge are the vertices joined by it), the intersection of any two edges of H is either their common vertex or empty, and any isolated vertex of H does not belong to any edge of H; (2) all sets ${\varvec{\mathcal{B}}(\varvec{H})}$ ( ${\varvec{H} \in \varvec{\mathcal{R}}(\varvec{G})}$ ), where ${\varvec{\mathcal{B}}(\varvec{H})\subset \mathbf{E}^3}$ is the union of all vertices and all edges of H, are pairwise not homeomorphic; moreover, for any graphs ${\varvec{H}_1 \in \varvec{\mathcal{R}}(\varvec{G})}$ and ${\varvec{H}_2 \in \varvec{\mathcal{R}}(\varvec{G})}$ , ${\varvec{H}_1 \ne \varvec{H}_2}$ , and for any finite subsets ${\varvec{S}_i \subset \varvec{\mathcal{B}}(\varvec{H}_i)}$ (i = 1, 2), the sets ${\varvec{\mathcal{B}}(\varvec{H}_1){\setminus} \varvec{S}_1}$ and ${\varvec{\mathcal{B}}(\varvec{H}_2){\setminus} \varvec{S}_2}$ are not homeomorphic.     

Quantitative Stratification and the Regularity of Mean Curvature Flow

Let ${\mathcal{M}}$ be a Brakke flow of n-dimensional surfaces in ${\mathbb{R}^N}$ . The singular set ${\mathcal{S} \subset \mathcal{M}}$ has a stratification ${\mathcal{S}^0 \subset \mathcal{S}^1 \subset \cdots \mathcal{S}}$ , where ${X \in \mathcal{S}^j}$ if no tangent flow at X has more than j symmetries. Here, we define quantitative singular strata ${\mathcal{S}^j_{\eta, r}}$ satisfying ${\cup_{\eta>0} \cap_{0<r} \mathcal{S}^j_{\eta, r} = \mathcal{S}^j}$ . Sharpening the known parabolic Hausdorff dimension bound ${{\rm dim} \mathcal{S}^j \leq j}$ , we prove the effective Minkowski estimates that the volume of r-tubular neighborhoods of ${\mathcal{S}^j_{\eta, r}}$ satisfies ${{\rm Vol} (T_r(\mathcal{S}^j_{\eta, r}) \cap B_1) \leq Cr^{N + 2 - j-\varepsilon}}$ . Our primary application of this is to higher regularity of Brakke flows starting at k-convex smooth compact embedded hypersurfaces. To this end, we prove that for the flow of k-convex hypersurfaces, any backwards selfsimilar limit flow with at least k symmetries is in fact a static multiplicity one plane. Then, denoting by ${\mathcal{B}_r \subset \mathcal{M}}$ the set of points with regularity scale less than r, we prove that ${{\rm Vol}(T_r(\mathcal{B}_r)) \leq C r^{n+4-k-\varepsilon}}$ . This gives L ^p -estimates for the second fundamental form for any p < n + 1 ? k. In fact, the estimates are much stronger and give L ^p -estimates for the reciprocal of the regularity scale. These estimates are sharp. The key technique that we develop and apply is a parabolic version of the quantitative stratification method introduced in Cheeger and Naber (Invent. Math., (2)191 2013), 321–339) and Cheeger and Naber (Comm. Pure. Appl. Math, arXiv:1107.3097v1, 2013).     

{\mathbb{G}_a^ M} degeneration of flag varieties

Let ${\mathcal{F}_\lambda}$ be a generalized flag variety of a simple Lie group G embedded into the projectivization of an irreducible G-module V _λ . We define a flat degeneration ${\mathcal{F}_\lambda^a}$ , which is a ${\mathbb{G}^M_a}$ variety. Moreover, there exists a larger group G ^a acting on ${\mathcal{F}_\lambda^a}$ , which is a degeneration of the group G. The group G ^a contains ${\mathbb{G}^M_a}$ as a normal subgroup. If G is of type A, then the degenerate flag varieties can be embedde‘d into the product of Grassmannians and thus to the product of projective spaces. The defining ideal of ${\mathcal{F}_\lambda}$ is generated by the set of degenerate Plücker relations. We prove that the coordinate ring of ${\mathcal{F}_\lambda^a}$ is isomorphic to a direct sum of dual PBW-graded ${\mathfrak{g}}$ -modules. We also prove that there exists bases in multi-homogeneous components of the coordinate rings, parametrized by the semistandard PBW-tableux, which are analogs of semistandard tableaux.     

A Topologist’s View of Chu Spaces

For a symmetric monoidal-closed category $\mathcal{X}$ and any object K, the category of K-Chu spaces is small-topological over $\mathcal{X}$ and small cotopological over $\mathcal{X}^{{{\text{op}}}}$ . Its full subcategory of $\mathcal{M}$ -extensive K-Chu spaces is topological over $\mathcal{X}$ when $\mathcal{X}$ is $\mathcal{M}$ -complete, for any morphism class $\mathcal{M}$ . Often this subcategory may be presented as a full coreflective subcategory of Diers’ category of affine K-spaces. Hence, in addition to their roots in the theory of pairs of topological vector spaces (Barr) and their connections with linear logic (Seely), the Dialectica categories (Hyland, de Paiva), and with the study of event structures for modeling concurrent processes (Pratt), Chu spaces seem to have a less explored link with algebraic geometry. We use the Zariski closure operator to describe the objects of the *-autonomous category of $\mathcal{M}$ -extensive and $\mathcal{M}$ -coextensive K-Chu spaces in terms of Zariski separation and to identify its important subcategory of complete objects.     

Differentiable vectors and unitary representations of Fréchet–Lie supergroups

A locally convex Lie group G has the Trotter property if, for every $x_1, x_2 \in \mathfrak{g }$ , $$\begin{aligned} \exp _G(t(x_1 + x_2))=\lim _{n \rightarrow \infty } \left(\exp _G\left(\frac{t}{n}x_1\right)\exp _G\left(\frac{t}{n}x_2\right)\right)^n \end{aligned}$$ holds uniformly on compact subsets of $\mathbb{R }$ . All locally exponential Lie groups have this property, but also groups of automorphisms of principal bundles over compact smooth manifolds. A key result of the present article is that, if G has the Trotter property, $\pi : G \rightarrow {\mathrm{GL}}(V)$ is a continuous representation of G on a locally convex space, and $v \in V$ is a vector such that $\overline{\mathtt{d}\pi }(x)v :=\frac{d}{dt}|_{t=0} \pi (\exp _G(tx))v$ exists for every $x \in \mathfrak{g }$ , then the map $\mathfrak{g }\rightarrow V,x \mapsto \overline{\mathtt{d}\pi }(x)v$ is linear. Using this result we conclude that, for a representation of a locally exponential Fréchet–Lie group G on a metrizable locally convex space, the space of $\mathcal{C }^{k}$ -vectors coincides with the common domain of the k-fold products of the operators $\overline{\mathtt{d}\pi }(x)$ . For unitary representations on Hilbert spaces, the assumption of local exponentiality can be weakened to the Trotter property. As an application, we show that for smooth (resp., analytic) unitary representations of Fréchet–Lie supergroups $(G,\mathfrak{g })$ where G has the Trotter property, the common domain of the operators of $\mathfrak{g }=\mathfrak{g }_{\overline{0}}\oplus \mathfrak{g }_{\overline{1}}$ can always be extended to the space of smooth (resp., analytic) vectors for G.     

Symmetric diameter two graphs with affine-type vertex-quasiprimitive automorphism group

We study the class of G-symmetric graphs Γ with diameter 2, where G is an affine-type quasiprimitive group on the vertex set of Γ. These graphs arise from normal quotient analysis as basic graphs in the class of symmetric diameter 2 graphs. It is known that ${G \cong V \rtimes G_0}$ , where V is a finite-dimensional vector space over a finite field and G ₀ is an irreducible subgroup of GL (V), and Γ is a Cayley graph on V. In particular, we consider the case where ${V = \mathbb {F}_p^d}$ for some prime p and G ₀ is maximal in GL (d, p), with G ₀ belonging to the Aschbacher classes ${\mathcal {C}_2, \mathcal {C}_4, \mathcal {C}_6, \mathcal {C}_7}$ , and ${\mathcal {C}_8}$ . For ${G_0 \in \mathcal {C}_i, i = 2,4,8}$ , we determine all diameter 2 graphs which arise. For ${G_0 \in \mathcal {C}_6, \mathcal {C}_7}$ we obtain necessary conditions for diameter 2, which restrict the number of unresolved cases to be investigated, and in some special cases determine all diameter 2 graphs.     

Analytic Toeplitz algebras and the Hilbert transform associated with a subdiagonal algebra

Let M be aσ-finite von Neumann algebra and let AM be a maximal subdiagonal algebra with respect to a faithful normal conditional expectationΦ.Based on the Haagerup’s noncommutative Lpspace Lp(M)associated with M,we consider Toeplitz operators and the Hilbert transform associated with A.We prove that the commutant of left analytic Toeplitz algebra on noncommutative Hardy space H2(M)is just the right analytic Toeplitz algebra.Furthermore,the Hilbert transform on noncommutative Lp(M)is shown to be bounded for 1p∞.As an application,we consider a noncommutative analog of the space BMO and identify the dual space of noncommutative H1(M)as a concrete space of operators.     

Towards a classification of modular compactifications of \mathcal {M}_{g,n}

The moduli space of smooth curves admits a beautiful compactification $\mathcal{M}_{g,n} \subset \overline{\mathcal{M}}_{g,n}$ by the moduli space of stable curves. In this paper, we undertake a systematic classification of alternate modular compactifications of $\mathcal{M}_{g,n}$ . Let $\mathcal{U}_{g,n}$ be the (non-separated) moduli stack of all n-pointed reduced, connected, complete, one-dimensional schemes of arithmetic genus g. When g=0, $\mathcal{U}_{0,n}$ is irreducible and we classify all open proper substacks of $\mathcal{U}_{0,n}$ . When g≥1, $\mathcal{U}_{g,n}$ may not be irreducible, but there is a unique irreducible component $\mathcal{V}_{g,n} \subset\mathcal{U}_{g,n}$ containing $\mathcal{M}_{g,n}$ . We classify open proper substacks of $\mathcal {V}_{g,n}$ satisfying a certain stability condition.     

Some Critical Almost Hermitian Structures

Let M be an even dimensional compact smooth manifold admitting an almost complex structure. Let ${{(\lambda, \mu)} \in \mathbb{R}^2 - (0,0)}$ . We discuss the critical points of the functional ${\mathcal {F}_{\lambda, \mu} (J, g) = \int_M (\lambda \tau + \mu \tau^* ) dv_g}$ on the space of all almost Hermitian structures ${\mathcal{AH}(M)}$ on M and its subspace ${{\mathcal{AH}_{c}(M)}}$ with a certain positive constant c, where τ and τ ^* are the scalar curvature and the *-scalar curvature of (J, g), respectively. Further, we provide some examples illustrating our arguments.     

Note on Group Distance Magic Graphs G[C 4]

A group distance magic labeling or a ${\mathcal{G}}$ -distance magic labeling of a graph G =  (V, E) with ${|V | = n}$ is a bijection f from V to an Abelian group ${\mathcal{G}}$ of order n such that the weight ${w(x) = \sum_{y\in N_G(x)}f(y)}$ of every vertex ${x \in V}$ is equal to the same element ${\mu \in \mathcal{G}}$ , called the magic constant. In this paper we will show that if G is a graph of order n =  2 ^p (2k + 1) for some natural numbers p, k such that ${\deg(v)\equiv c \mod {2^{p+1}}}$ for some constant c for any ${v \in V(G)}$ , then there exists a ${\mathcal{G}}$ -distance magic labeling for any Abelian group ${\mathcal{G}}$ of order 4n for the composition G[C ₄]. Moreover we prove that if ${\mathcal{G}}$ is an arbitrary Abelian group of order 4n such that ${\mathcal{G} \cong \mathbb{Z}_2 \times\mathbb{Z}_2 \times \mathcal{A}}$ for some Abelian group ${\mathcal{A}}$ of order n, then there exists a ${\mathcal{G}}$ -distance magic labeling for any graph G[C ₄], where G is a graph of order n and n is an arbitrary natural number.     

A group theoretic characterization of classical unitals

Let G be the group of projectivities stabilizing a unital $\mathcal{U}$ in PG(2,q ²). In?this paper, we prove that $\mathcal{U}$ is a classical unital if and only if there are two points in $\mathcal{U}$ such that the stabilizer of these two points in G has order?q ²?1.     

Floyd maps for relatively hyperbolic groups

Let ${\mathbf{delta}_{\mathcal S,\lambda}}$ denote the Floyd metric on a discrete group G generated by a finite set ${\mathcal S}$ with respect to the scaling function f _n ?= λ ⁿ for a positive λ <?1. We prove that if G is relatively hyperbolic with respect to a collection ${\mathcal P}$ of subgroups then there exists λ such that the identity map ${G\to G}$ extends to a continuous equivariant map from the completion with respect to ${\mathbf{\delta}_{\mathcal S,\lambda}}$ to the Bowditch completion of G with respect to ${\mathcal P}$ . In order to optimize the proof and the usage of the map theorem we propose two new definitions of relative hyperbolicity equivalent to the other known definitions. In our approach some “visibility” conditions in graphs are essential. We introduce a class of “visibility actions” that contains the class of relatively hyperbolic actions. The convergence property still holds for the visibility actions. Let a locally compact group G act on a compactum Λ with convergence property and on a locally compact Hausdorff space Ω properly and cocomactly. Then the topologies on Λ and Ω extend uniquely to a topology on the direct union ${T=\Lambda{\sqcup}\Omega}$ making T a compact Hausdorff space such that the action ${G{\curvearrowright}T}$ has convergence property. We call T the attractor sum of Λ and Ω.