We consider a two sided noetherian ring $R$ such that the character modules of Gorenstein injective left $R$-modules are Gorenstein flat right $R$-modules. We then prove that the class of Gorenstein flat right $R$-modules is preenveloping. We also show that the class of Gorenstein flat complexes of right $R$-modules is preenevloping in $\mathit{Ch}(R)$. In the second part of the paper we give examples of rings with the property that the character modules of Gorenstein injective modules are Gorenstein flat. We prove that any two sided noetherian ring $R$ with $\mathop{\mathit{i.d.}}_{R^{\mathrm{op}}} R < \infty$ has the desired property. We also prove that if $R$ is a two sided noetherian ring with a dualizing bimodule ${}_{R}V_{R}$ and such that $R$ is left $n$-perfect for some positive integer $n$, then the character modules of Gorenstein injective modules are Gorenstein flat.

We apply Forelli--Rudin construction and Nakazawa's hodograph transformation to prove a graph theoretic closed formula for invariant theoretic coefficients in the asymptotic expansion of the Szegö kernel on strictly pseudoconvex complete Reinhardt domains. The formula provides a structural analogy between the asymptotic expansion of the Bergman and Szegö kernels. It can be used to effectively compute the first terms of Fefferman's asymptotic expansion in CR invariants. Our method also works for the asymptotic expansion of the Sobolev--Bergman kernel introduced by Hirachi and Komatsu.

Let $S$ be a smooth minimal surface of general type with a (rational) pencil of hyperelliptic curves of minimal genus $g$. We prove that if $K_{S}^{2} < 4\chi(\mathcal{O}_{S})-6$, then $g$ is bounded. The surface $S$ is determined by the branch locus of the covering $S \to S/i$, where $i$ is the hyperelliptic involution of $S$. For $K_{S}^{2} < 3\chi(\mathcal{O}_{S})-6$, we show how to determine the possibilities for this branch curve. As an application, given $g > 4$ and $K_{S}^{2}-3\chi(\mathcal{O}_{S}) < -6$, we compute the maximum value for $\chi(\mathcal{O}_{S})$. This list of possibilities is sharp.

In this paper we prove that if $\Sigma$ is a properly embedded constant mean curvature $H = 1/2$ surface which is asymptotic to a horocylinder $C \subset \widetilde{\mathit{PSL}}_{2}(\mathbb{R}, \tau)$, in one side of $C$, such that the mean curvature vector of $\Sigma$ has the same direction as that of the $C$ at points of $\Sigma$ converging to $C$, then $\Sigma$ is a subset of $C$.

We classify the dynamical action of matrices in $\mathrm{SU}(p, q)$ using the coefficients of their characteristic polynomial. This generalises earlier work of Goldman for $\mathrm{SU}(2, 1)$ and the classical result for $\mathrm{SU}(1, 1)$, which is conjugate to $\mathrm{SL}(2, \mathbb{R})$. As geometrical applications, we show how this enables us to classify automorphisms of real and complex hyperbolic space and anti de Sitter space.

We show that if each of $K_{1}$ and $K_{2}$ is a trefoil knot or a figure eight knot, the homology $3$-sphere defined by the Kirby diagram which is a simple link of $K_{1}$ and $K_{2}$ with framing ($0$, $n$) is represented by an $n$-twisted Whitehead double of $K_{2}$.

Let $G$ be a compact Lie group and $X$ be a compact smooth $G$-manifold with finitely many $G$-fixed points. We show that if $X$ admits a $G$-equivariant hyperbolic diffeomorphism having a certain convergence property, there exists an open covering of $X$ indexed by the $G$-fixed points so that each open set is $G$-invariant and $G$-equivariantly diffeomorphic to the tangential $G$-representation at the corresponding $G$-fixed point. We also show that the converse is also true in case of holomorphic torus actions.

In this article, we shall prove that for any finite solvable group $G$, there exist infinitely many abelian extensions $K/\mathbb{Q}$ and Galois extensions $M/\mathbb{Q}$ such that the Galois group $\Gal(M/K)$ is isomorphic to $G$ and $M/K$ is unramified. The difference between our result and [3, 4, 6, 7, 13] is that we have a base field $K$ which is not only Galois over $\mathbb{Q}$, but also has very small degree compared to their results. We will also get another proof of Nomura's work [9], which gives us a base field of smaller degree than Nomura's. Finally for a given finite nonabelian simple group $G$, we will show there exists an unramified extension $M/K'$ such that the Galois group is isomorphic to $G$ and $K'$ has relatively small degree.

The main result of this note gives an explicit presentation of the $S^{1}$-equivariant cohomology ring of the $(n-k, k)$ Springer variety (in type $A$) as a quotient of a polynomial ring by an ideal $I$, in the spirit of the well-known Borel presentation of the cohomology of the flag variety.

In this paper, we determine nonminimal pseudohermitian biminimal Legendre surfaces in the unit 5-sphere $S^{5}$. In fact, the product of a circle and a helix of order $4$ is realized as a nonminimal pseudohermitian biminimal Legendre immersion into $S^{5}$. In addition, we obtain that there exist no nonminimal pseudohermitian biminimal Legendre surfaces in a 5-dimensional Sasakian space form of non-positive constant holomorphic sectional curvature for the Tanaka--Webster connection.

This paper extends the work of Nikkuni [4] finding an explicit relationship for the graph $K_{3, 3, 1}$ between knotting and linking, which relates the sum of the squares of linking numbers of links in the embedding and the second coefficient of the Conway polynomial of certain cycles in the embedding. Then we use this and other similar relationships to better understand the relationship between knotting and linking in the Petersen family. The Petersen family is the set of minor minimal intrinsically linked graphs. We prove that if such a spatial graph is complexly algebraically linked then it is knotted.

Let $(G/H, \sigma)$ be a compact $4$-symmetric space of inner and exceptional type. Suppose that the dimension of the center of $H$ is one and $H$ is not a centralizer of a toral subgroup of $G$. In this paper we shall classify the involution $\tau$ of $G$ satisfying $\tau \circ \sigma = \sigma \circ \tau$.

We introduce a method which can be used to study maximal inequalities for martingales of bounded mean oscillation. As an application, we establish sharp $\Phi$-inequalities and tail inequalities for the one-sided maximal function of a BMO martingale. The results can be regarded as BMO counterparts of the classical maximal estimates of Doob.

From radial curvature geometry's standpoint, we prove a few sphere theorems of the Grove--Shiohama type for certain classes of compact Finsler manifolds.

In the first part, this paper studies the characteristic rank of the canonical oriented $k$-plane bundle over the Grassmann manifold $\tilde{G}_{n, k}$ of oriented $k$-planes in Euclidean $n$-space. It presents infinitely many new exact values if $k=3$ or $k=4$, as well as new lower bounds for the number in question if $k\geq 5$. In the second part, these results enable us to improve on the general upper bounds for the $\mathbb{Z}_{2}$-cup-length of $\tilde{G}_{n, k}$. In particular, for $\tilde{G}_{2^{t}, 3}$ ($t\geq 3$) we prove that the cup-length is equal to $2^{t}-3$, which verifies the corresponding claim of Tomohiro Fukaya's conjecture from 2008.

In this paper, we investigate a relation between finite graphs, simplicial flag complexes and right-angled Coxeter groups, and we provide a class of reconstructible finite graphs. We show that if $\Gamma$ is a finite graph which is the 1-skeleton of some simplicial flag complex $L$ which is a homology manifold of dimension $n \ge 1$, then the graph $\Gamma$ is reconstructible.