The space $\Omega(G)$ of all based loops in a compact simply connected Lie group $G$ has an action of the maximal torus $T \subset G$ (by pointwise conjugation) and of the circle $S^{1}$ (by rotation of loops). Let $\mu\colon \Omega(G) \to (\mathfrak{t} \times i\mathbb{R})^{*}$ be a moment map of the resulting $T \times S^{1}$ action. We show that all levels (that is, pre-images of points) of $\mu$ are connected subspaces of $\Omega(G)$ (or empty). The result holds if in the definition of $\Omega(G)$ loops are of class $C^{\infty}$ or of any Sobolev class $H^{s}$, with $s \ge 1$ (for loops of class $H^{1}$ connectedness of regular levels has been proved by Harada, Holm, Jeffrey, and the author in [3]).

In this paper we extend the concept of weakly polymatroidal ideals to monomial ideals which are not necessarily generated in one degree, and show that any ideal in this class has linear quotients. As an application we study some vertex cover ideals of weighted hypergraphs.

In this note we prove a product property for the pluricomplex energy, and then give some applications.

In this paper, we continue our study of the pro-$\Sigma$ fundamental groups of configuration spaces associated to a hyperbolic curve, where $\Sigma$ is either the set of all prime numbers or a set consisting of a single prime number, begun in an earlier paper. Our main result may be regarded either as a combinatorial, partially bijective generalization of an injectivity theorem due to Matsumoto or as a generalization to arbitrary hyperbolic curves of injectivity and bijectivity results for genus zero curves due to Nakamura and Harbater--Schneps. More precisely, we show that if one restricts one's attention to outer automorphisms of such a pro-$\Sigma$ fundamental group of the configuration space associated to a(n) affine (respectively, proper) hyperbolic curve which are compatible with certain ``fiber subgroups'' (i.e., groups that arise as kernels of the various natural projections of a configuration space to lower-dimensional configuration spaces) as well as with certain cuspidal inertia subgroups, then, as one lowers the dimension of the configuration space under consideration from $n+1$ to $n \ge 1$ (respectively, $n \ge 2$), there is a natural injection between the resulting groups of such outer automorphisms, which is a bijection if $n \ge 4$. The key tool in the proof is a combinatorial version of the Grothendieck conjecture proven in an earlier paper by the author, which we apply to construct certain canonical sections.

In the paper [5] we obtained explicit examples of Moishezon twistor spaces of some compact self-dual four-manifolds admitting a non-trivial Killing field, and also determined their moduli space. In this note we investigate minitwistor spaces associated to these twistor spaces. We determine their structure, minitwistor lines and also their moduli space, by using a double covering structure of the twistor spaces. In particular, we find that these minitwistor spaces have different properties in many respects, compared to known examples of minitwistor spaces. Especially, we show that the moduli space of the minitwistor spaces is identified with the configuration space of different 4 points on a circle divided by the standard $\mathrm{PSL}(2,\mathbf{R})$-action.

Let $M$ be a compact complex manifold and let $(L,H)$ be a holomorphic Hermitian line bundle over $M$ such that the curvature form of $h$ is nondegenerate and splits into the difference $\Theta_{+} - \Theta_{-}$ of two semipositive forms $\Theta_{+}$ and $\Theta_{-}$ whose null spaces define mutually transverse holomorphic foliations $\mathcal{F}_{-}$ and $\mathcal{F}_{+}$, respectively. Then $L^{m}$ admits, for sufficiently large $m \in \mathbb{N}$, $C^{\infty}$ sections whose ratio embeds $M$ into $\mathbb{CP}^{N}$ holomorphically (resp. antiholomorphically) along $\mathcal{F}_{+}$ (resp. along $\mathcal{F}_{-}$).

We establish continuity and Schatten--von Neumann properties for Fourier integral operators with amplitudes in weighted modulation spaces, when acting on modulation spaces themselves. The phase functions are non smooth and admit second order derivatives again in suitable classes of modulation spaces.

There are several types of equation of motion of elastic wires. In this paper, we treat an equation taking account of the thickness of wire. The equation was introduced by Caflisch and Maddocks on plane curves, and they proved the existence of solutions. We will prove the existence of solutions for any dimensional Euclidean space. Note that, in the case of plane, the equation can be explicitly written in terms of polar coordinates. For higher dimensional case, we use covariant differentiation on the unit sphere.

In [3] Dade generalized the Glauberman character correspondence. In [13] Tasaka showed that the Dade correspondence induces an isotypy between blocks of finite groups under some assumptions. In this paper we obtain a generalization of [13], Theorem 5.5.

We study the reflectional symmetry of a generically embedded $2$-dimensional surface $M$ in the hyperbolic or de Sitter $3$-dimensional spaces. This symmetry is picked up by the singularities of folding maps that are defined and studied here. We also define the evolute and symmetry set of $M$ and prove duality results that relate them to the bifurcation sets of the family of folding maps.

A pseudo diagram of a spatial graph is a spatial graph projection on the $2$-sphere with over/under information at some of the double points. We introduce the trivializing (resp. knotting) number of a spatial graph projection by using its pseudo diagrams as the minimum number of the crossings whose over/under information lead the triviality (resp. nontriviality) of the spatial graph. We determine the set of non-negative integers which can be realized by the trivializing (resp. knotting) numbers of knot and link projections, and characterize the projections which have a specific value of the trivializing (resp. knotting) number.