In this paper, we develop the theory of “cuspidalizations” of the étale fundamental group of a proper hyperbolic curve over a finite or nonarchimedean mixed-characteristic local field. The ultimate goal of this theory is the group-theoretic reconstruction of the étale fundamental group of an arbitrary open subscheme of the curve from the étale fundamental group of the full proper curve. We then apply this theory to show that a certain absolute $p$-adic version of the Grothendieck Conjecture holds for hyperbolic curves “of Belyi type”. This includes, in particular, affine hyperbolic curves over a nonarchimedean mixed-characteristic local field which are defined over a number field and isogenous to a hyperbolic curve of genus zero. Also, we apply this theory to prove the analogue for proper hyperbolic curves over finite fields of the version of the Grothendieck Conjecture that was shown in [Tama].

A new homotopy fibration is constructed at the prime 3 which shows that the quotient group $E_{7}/F_{4}$ is spherically resolved. This is then used to show that the 3-primary homotopy exponent of $E_{7}$ is bounded above by $3^{23}$, which is at most four powers of 3 from being optimal.

For an algebraically closed field $K$ with $\mathrm{ch}(K)\neq 2$, let $\mathcal{O}M(1, SO(n,K))$ denote the moduli space of holomorphic bundles on $\mathbb{P}^{2}$ with the structure group $SO(n,K)$ and half the first Pontryagin index being equal to 1, each of which is trivial on a fixed line $l_{\infty}$ and has a fixed holomorphic trivialization there. In this paper we determine the Chow ring of $\mathcal{O}M(1, SO(n,K))$.

Unprojection is an effort, initiated by Miles Reid, to develop an algebraic language for the study of birational geometry. [Ki] contains motivation and examples, and poses the problem of developing a general theory of unprojection. The main purpose of the present work is to suggest a general definition of unprojection, and to show that it indeed generalizes previous work done in the topic. In addition, in Section 6 we present an unprojection analysis of an example of Reid and K. Suzuki, and Section 7 contains more examples.

We consider asymptotic stability of a small solitary wave to supercritical 2-dimensional nonlinear Schrödinger equations \[ \begin{array}{cc} iu_{t} +\Delta u = V u \pm |u|^{p-1}u & \textrm{for } (x, t) \in \mathbb{R}^{2}\times \mathbb{R}, \end{array} \] in the energy class. This problem was studied by Gustafson-Nakanishi-Tsai [14] in the $n$-dimensional case ($n\geq 3$) by using the endpoint Strichartz estimate. Since the endpoint Strichartz estimate fails in 2-dimensional case, we use a time-global local smoothing estimate of Kato type to prove the asymptotic stability of a solitary wave.

We consider Murre’s conjectures on Chow groups for a fourfold which is a product of two curves and a surface. We give a result which concerns Conjecture D:the kernel of a certain projector is equal to the homologically trivial part of the Chow group. We also give a proof of Conjecture B for a product of two surfaces.

For each k ∈ N, we describe a mapping $f_{k}:\mathbb{C} \longrightarrow E_{k}$ into a suitable non-complete complex locally convex space $E_{k}$ such that $f_{k}$ is $k$ times continuously complex differentiable (i.e., a $C^{k}_{\mathbb{C}}$-map) but not $C^{k+1}_{\mathbb{C}}$ and hence not complex analytic. We also describe a complex analytic map from $\ell^{1}$ to a suitable complete complex locally convex space $E$ which is unbounded on each non-empty open subset of $\ell^{1}$. Finally, we present a smooth map $\mathbb{R} \longrightarrow E$ into a non-complete locally convex space which is not real analytic although it is given locally by its Taylor series around each point.

We consider the exact controllability of a Timoshenko beam system, clamped at one end and attached at the other end to a rigid antenna. Such a system is governed by two partial differential equations and two ordinary differential equations. Using the HUM method, we show that the system is exactly controllable in the usual energy space.

Let $M$ be an $n$-dimensional complete simply connected Riemannian manifold with nonpositive sectional curvature for $n = 2,3$ and $4$. We prove the following Faber-Krahn type inequality for the first eigenvalue $\lambda _{1}$ of the mixed boundary problem. A domain $\Omega$ outside a closed convex subset $C$ in $M$ satisfies \[ \lambda _{1}(\Omega )\geq \lambda _{1}(\Omega ^{*}) \] with equality if and only if $\Omega$ is isometric to the half ball $\Omega$ in $\mathbb{R}_{n}$, whose volume is equal to that of $\Omega$. We also prove the Sobolev type inequality outside a closed convex set $C$ in $M$.