We answer Mark Kac's famous question [13], ``can one hear the shape of a drum?'' in the negative for orbifolds that are spherical space forms. This is done by extending the techniques developed by A. Ikeda on lens spaces to the orbifold setting. Several results are proved to show that with certain restrictions on the dimensionalities of orbifold lens spaces we can obtain infinitely many pairs of isospectral non-isometric lens spaces. These results are then generalized to show that for any dimension greater than 8 we can have pairs of isospectral non-isometric orbifold lens spaces.

We apply techniques of zeta functions and regularized products theory to study the zeta determinant of a class of abstract operators with compact resolvent, and in particular the relation with other spectral functions.

In this paper we study the entire solutions of a class of periodic Allen--Cahn equations \begin{equation} -\Delta u(x,y)+a(x)W^{'}(u(x,y)) = 0{,}\quad (x,y) \in \mathbb{R}^{2}, \end{equation} where $a(x)\colon \mathbb{R} \to \mathbb{R}^{+}$ is a periodic, positive function and $W \in C^{2}(\mathbb{R}, \mathbb{R})$ is a double-well potential. We look for the entire solutions of the above equation with asymptotic conditions $u(x,y) \to \sigma_{\pm}$ as $x \to \pm\infty$ uniformly with respect to $y \in \mathbb{R}$. Via variational methods we find infinitely many solutions.

The purpose of this paper is to study the limit distribution of individual eigenvalue of 1-dimensional Schrödinger operators with random potentials derived from the derivatives of compound Poisson processes possessing purely positive jumps or purely negative jumps. The central limit theorem for ``middle eigenvalue'' is also investigated.

Let $X$ be a projective minimal Gorenstein 3-fold of general type with $\mathbb{Q}$-factorial terminal singularities. We classify minimal Gorenstein 3-folds of general type according to the birationality of 3-canonical system on $X$.

An alternate description for ribbon disk complements in the $4$-ball is provided. It is known (and reestablished) that this description is equivalent to the standard LOT description, up to $3$-deformation. Amenable to geometric arguments, the alternate description yields asphericity results for ribbon disk complements using simple graph-theoretic criteria and, later, using a relative homotopy group which arises naturally. In the course of making and modifying the description, two algorithms are given for presenting ribbon disk groups.

In this paper, we prove an analogue of Beurling's theorem for an arbitrary simply connected nilpotent Lie group extending then earlier cases.

We study the configuration space of the linkages called ``spiders''. Let $g$ be a non negative integer and $r$ be the greatest integer such that $2^{r}$ divide $g - 1$. We show that there exists a spider whose configuration space is diffeomorphic to an orientable compact surface of gender $g$ if and only if $(1/2^{r})(g - 1) \leq 6r + 12$. Afterward we give a method that allows to describe a large family of singular configuration spaces.

Résumé

On étudie les espaces des configurations des systèmes articulés appelés « araignées ». Soit $g$ un entier positif et $r$ le plus grand entier tel que $2^{r}$ divise $g - 1$. On montre qu'il existe une araignée dont une composante connexe de l'espace des configurations est difféomorphe à une surface compacte orientable de genre $g$ si et seulement si $(1/2^{r})(g - 1) \leq 6r + 12$. On donne ensuite un méthode permettant de décrire complétement une large famille d'espaces de configurations singuliers.

In this paper we give a characterization of the Gromov hyperbolicity of trains (a large class of Denjoy domains which contains the flute surfaces) in terms of the behavior of a real function. This function describes somehow the distances between some remarkable geodesics in the train. This theorem has several consequences; in particular, it allows to deduce a result about stability of hyperbolicity, even though the original surface and the modified one are not quasi-isometric. In order to obtain these results we also prove some trigonometric lemmas that are interesting by themselves, since they provide very simple estimates on some hyperbolic distances.

For an oriented link diagram $D$, the warping degree $d(D)$ is the smallest number of crossing changes which are needed to obtain a monotone diagram from $D$. We show that $d(D) + d(-D) + \mathit{sr}(D)$ is less than or equal to the crossing number of $D$, where $-D$ denotes the inverse of $D$ and $\mathit{sr}(D)$ denotes the number of components which have at least one self-crossing. Moreover, we give a necessary and sufficient condition for the equality. We also consider the minimal $d(D) + d(-D) + \mathit{sr}(D)$ for all diagrams $D$. For the warping degree and linking warping degree, we show some relations to the linking number, unknotting number, and the splitting number.

We give a sufficient condition for the girth of finitely generated groups to be infinite by using a technique to prove a so-called ping-pong lemma or table tennis lemma. We show that some convergence groups and subgroups of mapping class groups satisfy the condition. Therefore the girth of each of them is infinite.

For each integer $k \geq 2$, Johnson gave a $3$-manifold with Heegaard splittings of genera $2k$ and $2k-1$ such that any common stabilization of these two surfaces has genus at least $3k-1$. We modify his argument to produce a $3$-manifold with two Heegaard splitings of genus $2k$ such that any common stabilization of them has genus at least $3k$.

Feller property of some diffusion processes and the time changed processes is investigated. Diffusion processes treated here are skew product of one dimensional generalized diffusion processes and the spherical Brownian motion, and the time changed processes are given by additive functional associated with some underlying measure. Concrete expressions of the Dirichlet forms corresponding to time changed processes are also obtained, which may be of non-local type caused by degeneracy of the underlying measures.