We consider the semiclassical Schrödinger operator with a well in an island potential, on which we assume smoothness only, except near infinity. We give the asymptotic expansion of the imaginary part of the shape resonance at the bottom of the well. This is a generalization of the result by Helffer and Sjöstrand [HeSj1] in the globally analytic case. We use an almost-analytic extension in order to continue the WKB solution coming from the well beyond the caustic set, and, for the justification of the accuracy of this approximation, we develop some refined microlocal arguments in h-dependent small regions.

We introduce the notion of symmetric triad, which is a generalization of the notion of irreducible root system, and study its fundamental properties. Applying these results, we study the orbit spaces of Hermann actions on compact symmetric spaces.

In this paper, we are considering the Baire property of the eventually different topology as a regularity property for sets of reals and investigate the logical strength of the statements "Every $\Delta_2^1$ set has the Baire property in the eventually different topology" and "Every $\Sigma_2^1$ set has the Baire property in the eventually different topology". The latter statement turns out to be equivalent to "$\omega_1$ is inaccessible by reals".

For a given closed surface, we study height functions with three critical values associated with immersions of the surface into 3-space, where the critical points may not be non-degenerate. We completely characterize the numbers of critical points corresponding to the three critical values that can be realized by such height functions. We also study the cases where the immersion can be replaced by an embedding or the critical points are all non-degenerate. Similar problems are studied for distance functions as well.

We consider the spectrum of the Stokes operator with and without rotation effect for the whole space and exterior domains in $L^q$-spaces. Based on similar results for the Dirichlet-Laplacian on $mathbf{R}^n$, $n \geq 2$, we prove in the whole space case that the spectrum as a set in $\mathbf{c}$ does not change with $q \in (1,\infty)$, but it changes its type from the residual to the continuous or to the point spectrum with $q$. The results for exterior domains are less complete, but the spectrum of the Stokes operator with rotation mainly is an essential one, consisting of infinitely many equidistant parallel half-lines in the left complex half-plane. The tools are strongly based on Fourier analysis in the whole space case and on stability properties of the essential spectrum for exterior domains.

For a plane curve, a point in the projective plane is said to be Galois when the point projection induces a Galois extension of function fields. We completely classify plane curves with infinitely many outer Galois points.

We generalize the notion of modular forms mod p to the case of Hermitian modular forms. Moreover we determine the structure of the algebra of degree 2 Hermitian modular forms mod p in the cases that the corresponding quadratic field is Q($\sqrt{-1}$) and Q($\sqrt{-3}$).

We study the time decay of scattering solutions to one-dimensional Schrödinger equations and prove a weighted dispersive estimate with stronger time decay than the case of unweighted estimates for the non-resonant state. Furthermore asymptotic expansions in time of scattering solutions are given. The key of the proof is the study of the Fourier properties of the Jost functions. We improve the Fourier properties of the Jost functions obtained by D'Ancona and Fanelli [2].

A sublinear elliptic equation whose coefficient is singular on the boundary is studied in any bounded domain Ω under the zero Dirichlet boundary condition. It is proved that the equation has a unique positive solution and infinitely many sign-changing solutions which belong to C¹($\overline{\Omega}$) or C²($\overline{\Omega}$). Moreover, it is proved that the solutions have the higher order regularity corresponding to the smoothness of the coefficient.

Let $(X,d,\mu)$ be a metric measure space endowed with a distance $d$ and a nonnegative Borel doubling measure $\mu$. Let $L$ be a non-negative self-adjoint operator on $L^2(X)$. Assume that the semigroup $e^{-tL}$ generated by $L$ satisfies the Davies-Gaffney estimates. Let $H_L^p(X)$ be the Hardy space associated with $L$. We prove a Hörmander-type spectral multiplier theorem for $L$ on $H_L^p(X)$ for $0 < p <\infty:$ the operator $m(L)$ is bounded from $H_L^p(X)$ to $H_L^p(X)$ if the function $m$ possesses $s$ derivatives with suitable bounds and $s > n(1/p - 1/2)$ where $n$ is the "dimension" of $X$. By interpolation, $m(L)$ is bounded on $H_L^p(X)$ for all $0 < p < \infty$ if $m$ is infinitely differentiable with suitable bounds on its derivatives. We also obtain a spectral multiplier theorem on $L^p$ spaces with appropriate weights in the reverse Hölder class.

Let $N$ be a connected, simply connected nilpotent Lie group with Lie algebra $\mathfrak{n}$ and let $\mathscr{W}$ be a submanifold of $\mathfrak{n}^*$ such that the dimension of all polarizations associated to elements of $\mathscr{W}$ is fixed. We choose $(\mathfrak{p}(w))_{w \in \mathscr{W}}$ and $(\mathfrak{p}'(w))_{w \in \mathscr{W}}$ two smooth families of polarizations in $\mathfrak{n}$. Let $\pi_w = \mathsf{ind}_{P(w)}^N \chi_w$ and $\pi'_w = \mathsf{ind}_{P'(w)}^N \chi_w$ be the corresponding induced representations, which are unitary and irreducible. It is well known that $\pi_w$ and $\pi'_w$ are unitary equivalent. In this paper, we prove the existence of a smooth family of intertwining operator $(T_w)_w$ for theses representations, where $w$ runs through an appropriate non-empty relatively open subset of $\mathscr{W}$. The intertwining operators are given by an explicit formula.