In this paper, we prove that any complete, rotationally symmetric translating soliton in $\mathbb{C}^2$ cannot arise as finite time blow up limit of the symplectic mean curvature flow.

There are some existence problems of Jenkins-Strebel differentials on a Riemann surface. One of them is to find a Jenkins-Strebel differential whose characteristic ring domains have given positive numbers as their circumferences, for any fixed underlying Riemann surface and core curves of the ring domains. In this paper, we investigate the existence of such solutions. Our method is to use a surface of circumferences. This is an analogue of the surface of the squares of the heights introduced by Strebel to provide an existence proof for a Jenkins-Strebel differential with given ratio of the moduli. We can see some degenerations of the characteristic ring domains of Jenkins-Strebel differentials by using the surface. We also consider the behavior of the surface when the underlying Riemann surface varies.

We study semilinear elliptic equations with the fractional Laplacian in $\mathbf{R}$. The equations with single power nonlinearities have been observed by Weinstein (1987), Frank-Lenzmann (2013) and so on. We focus on the equations with double power nonlinearities and consider the existence of ground states.

Let $S$ be a locally Noetherian normal scheme and $\blacklozenge/S$ a set of properties of $S$-schemes. Then we shall write $\mathsf{Sch}_{\blacklozenge/S}$ for the full subcategory of the category of $S$-schemes $\mathsf{Sch}_{/S}$ determined by the objects $X \in \mathsf{Sch}_{\blacklozenge/S}$ that satisfy every property of $\blacklozenge/S$. In the present paper, we shall mainly be concerned with the properties "reduced", "quasi-compact over $S$", "quasi-separated over $S$", and "separated over $S$". We give a functorial category-theoretic algorithm for reconstructing $S$ from the intrinsic structure of the abstract category $\mathsf{Sch}_{\blacklozenge/S}$. This result is analogous to a result of Mochizuki [5] and may be regarded as a partial generalization of a result of de Bruyn [6] in the case where $S$ is a locally Noetherian normal scheme.

Nash's problem concerning arcs poses the question of whether it is possible to construct a bijective relationship between the minimal resolution of a surface singularity and the irreducible components within its arcs space. As a reverse question, one might inquire whether it is possible to derive a resolution from the arcs space of the given singularity. This paper focuses on non-isolated hypersurface singularities in $\mathbb{C}^3$ whose normalisations are surface in $\mathbb{C}^4$ having rational singularities of multiplicity 3. For each of these singularities, we construct a non singular refinement of its dual Newton polyhedron with valuations attached to specific irreducible components of its jet schemes. Subsequently, we get a toric embedded resolution of these singularities. To establish the minimality of this resolution, we generalize the notion of a profile of a simplicial cone, as introduced in [6]. As a corollary, we obtain that the Hilbert basis of the dual Newton polyhedron of a rational singularity with multiplicity 3 provides a minimal toric embedded resolution for our singularities.

Let $R$ be a commutative Noetherian ring and $I$ be an ideal of $R$ with $\mathrm{cd}(I,R)\leq 1$. For an $R$-module $M$, we introduce a class of prime ideals, say $\overline{\mathrm{Ass}}_R \ M$, as the set of all prime ideals $\mathfrak{p}$ of $R$ such that $\mathrm{Ann}_R (0 :_M \mathfrak{p})$ = $\mathfrak{p}$. We show that if $R$ is a Noetherian complete local ring and $M$ is an $I$-cofinite $R$-module, then $\overline{\mathrm{Ass}}_R \ M$ is finite. Also, we prove that for each $I$-cofinite $R$-module $M$, $I\not\subseteq \cup_{\mathfrak{p} \in \Lambda_R(I,M)}\ \mathfrak{p}$, where $\Lambda_R(I,M)$ is the set of all maximal elements of $\overline{\mathrm{Ass}}_R \ M\setminus V(I)$ with respect to inclusion. Subsequently, for each $a \in I$, the $R$-module $(0 :_M a)$ is finitely generated if and only if $a \not\in \cup_{\mathfrak{p}\in\Lambda_R(I,M)} \ \mathfrak{p}$.