We find some examples in $\mathbf{P}^5(\mathbf{C})$ of surfaces satisfying Laplace equations. In particular, we study rational surfaces in $\mathbf{P}^{5}(\mathbf{C})$ whose hyperplane sections have genus one that satisfy a Laplace equation. Then we study monomial Togliatti systems of cubics for variety of dimension three, i.e. we find all the monomial examples of three-folds satisfying Laplace equations.

We are interested in the microlocal smoothing effect for operators of real principal type. On the initial value problem for a dispersive evolution equation, we study the fact that the sufficient decay of the initial data gives the smoothness of the solution. We develop the theory of the FBI transform in order to transform our operator of real principal type into a simple operator of first order. Since the smoothing effect is of global nature, our transformation is realized globally along the bicharacteristics defined from the principal symbol of the operator.

Elliptic equations on configurations $W = W_{1} \cup \dots \cup W_{N}$ with edge $Y$ and components $W_{j}$ of different dimension can be treated in the frame of pseudo-differential analysis on manifolds with geometric singularities, here edges. Starting from edge-degenerate operators on $W_{j}$, $j=1,\dots,N$, we construct an algebra with extra `transmission' conditions on $Y$ that satisfy an analogue of the Shapiro-Lopatinskij condition. Ellipticity refers to a two-component symbolic hierarchy with an interior and an edge part; the latter one is operator-valued, operating on the union of different dimensional model cones. We construct parametrices within our calculus, where exchange of information between the various components is encoded in Green and Mellin operators that are smoothing on $W \setminus Y$. Moreover, we obtain regularity of solutions in weighted edge spaces with asymptotics.

Let $\Phi$ be an element of the mapping class group $\mathcal{M}_{g}$ of genus $g$ ($\geq 2$) such that $\Phi$ is the isotopy class of a pseudo periodic map of negative twists. It is expected that, for each $\Phi$ which commutes with a hyperelliptic involution, there exists a hyperelliptic family whose monodromy is the conjugacy class of $\Phi$ in the mapping class group. In this paper, we give a partial solution for the conjecture in the case where $\Phi$ is a semistable element.

We investigate closed Riemannian 3-manifolds which satisfy an extremal condition. Using monopole equations and considering the action of the covering transformations, we decide the geometric structure of such 3-manifolds. As a result, we characterize the geometry of 3-manifolds with monopole classes whose dual Thurston norm is equal to one.

In this note we show that a compact solvmanifold admits a Kähler structure if and only if it is a finite quotient of a complex torus which has a structure of a complex torus bundle over a complex torus. We can show in particular that a compact solvmanifold of completely solvable type has a Kähler structure if and only if it is a complex torus, which is known as the Benson-Gordon's conjecture.

Another proof of Greenberg's theorem on automorphism groups of compact \mboxRiemann surfaces is given. Using the idea of the proof, the equivalence problem for finite Galois coverings of the compact projective line is answered affirmatively, except special type of coverings.

For all integers $n \ge 3$ we show the existence of many triples $(d,g,\rho)$ such that there is a smooth non-degenerate curve $C \subset \mathbf{P}^n$ with degree $d$, genus $g$ and index of regularity $\rho$. The curve $C$ lies in a smooth $K3$ surface $S \subset \mathbf{P}^n$.

In this paper, we study the degree of equivariant maps between Stiefel manifolds by using cohomological index theory. As applications, we have some Borsuk-Ulam type theorems on Stiefel manifolds.

In this paper we show that the natural fibrations on 3-Sasakian manifolds and on normal complex contact metric manifolds are minima of the corrected energy of the corresponding distributions.

Let $G$ be a finite group, let $k$ be an algebraically closed field of characteristic $2$ and let $\Omega:=\{g\in G\mid g^2=1_G\}$. It is shown that for a block $B$ of $kG$, the permutation module $k\Omega$ has a $B$-composition factor if and only if the Frobenius-Schur indicator of the regular character of $B$ is non-zero or equivalently if and only if $B$ is real with a strongly real defect class.

We improve our preceding result obtained in Tsujikawa and Yagi [10]. We construct the similar exponential attractors to the same adsorbate-induced phase transition model as in [10] but in a convex domain by using the compact smoothing property of corresponding nonlinear semigroup. In [10], the domain has been assumed to have $\mathcal{C}^{3}$ regularity to ensure the squeezing property of semigroup.