We construct a number of analytic cycles on the moduli space of stable curves by using three moduli spaces: the moduli space of tori with one marked point, that of spheres with four marked points, and that of tori with two marked points. We then prove the linear independence of the cycles in the rational homology groups in order to improve Wolpert's estimates for even degree Betti numbers of the moduli space of stable curves.

The spectra of the quadratic Hamiltonians on the twodimensional Euclidean space are determined completely by using the theory of the metaplectic representation. In some cases, the corresponding heat kernels are studied in connection with the well-definedness of the Wiener integrations. A proof of the Lévy formula for the stochastic area and a relation between the real and complex Hermite polynomials are given in our framework.

We assume that a factor is equipped with outer actions of two finite groups, and consider a factor-subfactor pair consisting of the crossed product by one group and the fixed point algebra by the other. For this inclusion of factors, the group of one-dimensional bimodules appearing at even levels of the associated graph and that of non-strongly outer automorphisms for the subfactor (i.e., centrally trivial ones in the strongly amenable case) are determined.

Using the covering game, we prove that every (lightface)${\Pi }_2^1$-set of positive Lebesgue measure contains a member which is arithmetical in $0♯$. This result generalizes a result for ${\Pi }_1^1$ due to Sacks and Tanaka.

The class of self-similar additive processes is an important subclass of stochastically continuous processes with independent increments which are not assumed to be time-homogeneous. It is shown that this process is transient if it is proper and the dimension is three or more. Furthermore sufficient conditions for transience are given in one- or two-dimensional cases.

Generalizing the transfer maps concerned with the projective spaces, we study some fundamental properties of transfer maps for sphere bundles. We show that their cofibers are represented by Thom spectra, which enables us to calculate the $e$-invariants of the transfer maps. We give some concrete formula for the $e$-invariants of them and its application.

We consider homogeneous random Sierpinski carpets, a class of infinitely ramified random fractals which have spatial symmetry but which do not have exact self-similarity. For a fixed environment we construct "natural" diffusion processes on the fractal and obtain upper and lower estimates of the transition density for the process that are up to constants best possible. By considering the random case, when the environment is stationary and ergodic, we deduce estimates of Aronson type.

We show that the real Seifert form determines the weights for nondegenerate quasihomogeneous polynomials in $C^3$. Consequently the real Seifert form determines the spectrum for semiquasihomogeneous hypersurface singularities in $C^3$. As a corollary, we obtain the topological invariance of weights for nondegenerate quasihomogeneous polynomials in $C^3$, which has already been proved by the author [Sael] and independently by Xu and Yau [Yal], [Ya2], [XY1], [XY2]. The method in this paper is totally different from their approaches and gives some new results, as corollaries, about holomorphic function germs in $C^3$ which are connected by $\mu$-constant deformations to nondegenerate quasihomogeneous polynomials. For example, we show that two semiquasihomogeneous functions of three complex variables have the same topological type if and only if they are connected by a $\mu$-constant deformation.

We show that a real analytic family of real analytic functions admits a modified analytic trivialization (MAT) via an appropriate toric modification, if their weighted initial forms define isolated singularities near zero.

Let $\mathcal{F}$ be a codimension-one foliation, transversally oriented, of class $C^r(r\ge 0)$ on a connected closed manifold $M$. The class of a leaf $F$ of $\mathcal{F}$ is defined to be the union of all leaves $G$ with $\overline{F}=\overline{G}$. Let $X$ be the space of classes of leaves in $M$ and let $X_0$ be the union of open subsets of $X$ which are homeomorphic to $R$ or to $S^1$. In this paper we prove that if the level of $\mathcal{F}$ is well defined (in the sense of [12]), then $X-X_0$ is a spectral space.

Generalizing a well-known group-theoretical notion we define transversals for (association) schemes. Two results on transversals of schemes are offered. Firstly, we show that a closed subset in a scheme possesses a factor scheme if it possesses a transversal. Secondly, we characterize the Coxeter schemes in terms of transversals. (Coxeter schemes are exactly those schemes which can be identified with the buildings in the sense of Tits). The second result may be viewed as a "thick version" of the characterization of Coxeter groups by the existence of "minimal coset representatives". On the other hand, the characterizing conditions given in this result are similar to the well-known "gate property" defined for chamber systems having a Coxeter matrix as type. Thus, our second main result may be viewed as a unified treatment of these two results.