We present a determinant expression for a family of classical transcendental solutions of the Painlevé V and the Painlevé VI equation. Degeneration of these solutions along the process of coalescence for the Painlevé equations is discussed.

In 1973, Lawson and Simons conjectured that there are no stable currents in any compact, simply connected Riemannian manifold $M^m$ which is $1/4$-pinched. In this paper, we regard $M^m$ as a submanifold immersed in a Euclidean space and prove the conjecture under some pinched conditions about the sectional curvatures and the principal curvatures of $M^m$. We also show that there is no stable $p$-current in a submanifold of $M^m$ and the $p$-th homology group vanishes when the shape operator of the submanifold satisfies certain conditions.

We generalize Roberts' counterexample to the fourteenth problem of Hilbert, and give a sufficient condition for certain invariant rings not to be finitely generated. It shows that there exist a lot of counterexamples of this type. We also determine the initial algebra of Roberts' counterexample for some monomial order

The Euler class, which lies in the second cohomology of the group of orientation preserving homeomorphisms of the circle, is pulled back to the "smooth'' Euler class in the cohomology of the group of orientation preserving smooth diffeomorphisms of the circle. Suppose a surface group $\Gamma$ (of genus $> 1$) is a normal subgroup of a group $G$, so that we have an extension of $Q = G/\Gamma$ by $\Gamma$. We prove that if the canonical outer action of $Q$ on $\Gamma$ is finite, then there is a canonical second cohomology class of $G$ restricting to the Euler class on $\Gamma$ which is smoothly representable, that is, it is pulled back from the smooth Euler class by a representation from $G$ to the group of diffeomorphisms. Also, we prove that if the above outer action is infinite, then any second cohomology class of $G$ restricting to the Euler class on $\Gamma$ is not smoothly representable

We evaluate the intersection numbers of loaded cycles associated with an $n$-fold Selberg-type integral. We proceed inductively using high-dimensional local systems.

We define the extremal length of horizontal vector measures on a Carnot group and study capacities associated with linear sub-elliptic equations. The coincidence between the definition of the $p$-module of horizontal vector measure system and two different definitions of the $p$-capacity is proved. We show the continuity property of a $p$-module generated by a family of horizontal vector measures. Reciprocal relations between the $p$-capacity and $q$-module $(1/p+1/q=1)$ of horizontal vector measures are obtained. A peculiarity of our approach consists of the study of the above mentioned notions in domains with an intrinsic metric.

In this paper we study normal subgroups of Kleinian groups as well as discrepancy groups (d-groups), that are Kleinian groups for which the exponent of convergence is strictly less than the Hausdorff dimension of the limit set. We show that the limit set of a d-group always contains a range of fractal subsets, each containing the set of radial limit points and having Hausdorff dimension strictly less than the Hausdorff dimension of the whole limit set. We then consider normal subgroups $G$ of an arbitrary non-elementary Kleinian group $H$, and show that the exponent of convergence of $G$ is bounded from below by half of the exponent of convergene of $H$. Finally, we give a discussion of various examples of d-groups.

Deligne, Kazhdan and Vignéras proved that, for an inner form of $GL_n$ over a zero characteristic $p$-adic field, the induced representation from a square integrable irreducible representation is irrreducible. Here we prove the case of non-zero characteristic.

We study different notions of blow-up of a scheme $X$ along a subscheme $Y$, depending on the datum of an embedding of $X$ into an ambient scheme. The two extremes in this theory are the ordinary blow-up, corresponding to the identity, and the 'quasi-symmetric blow-up', corresponding to the embedding of $X$ into a nonsingular variety. We prove that this latter blow-up is intrinsic of $Y$ and $X$, and is universal with respect to the requirement of being embedded as a subscheme of the ordinary blow-up of some ambient space along~$Y$.

We consider these notions in the context of the theory of characteristic classes of singular varieties. We prove that if $X$ is a hypersurface in a nonsingular variety and $Y$ is its 'singularity subscheme', these two extremes embody respectively the conormal and characteristic cycles of $X$. Consequently, the first carries the essential information computing Chern-Mather classes, and the second is likewise a carrier for Chern-Schwartz-MacPherson classes. In our approach, these classes are obtained from Segre class-like invariants, in precisely the same way as other intrinsic characteristic classes such as those proposed by Fulton, and by Fulton and Johnson.

We also identify a condition on the singularities of a hypersurface under which the quasi-symmetric blow-up is simply the linear fiber space associated with a coherent sheaf.