A region crossing change is a local transformation on a knot or link diagram. We show that a region crossing change on a knot diagram is an unknotting operation, and we define the region unknotting number for a knot diagram and a knot.

In this paper, we first use the method of Colding and Minicozzi II [7] to show that K. Smoczyk's classification theorem [25] for complete self-shrinkers in higher codimension also holds under a weaker condition. Then as an application, we give some rigidity results for self-shrinkers in arbitrary codimension.

We study the Gottlieb group of a wedge sum of suspension spaces. We give necessary and sufficient conditions, in terms of Hopf invariants, for an element of a homotopy group to be in the Gottlieb group. We apply our results to wedge sums of spheres and to Moore spaces.

The paper contains the study of weak-type constants of Fourier multipliers resulting from modulation of the jumps of Léevy processes. We exhibit a large class of functions $m: {\mathbb R}^d\to {\mathbb C}$, for which the corresponding multipliers $T_m$ satisfy the estimates

$ \|T_m f\|_{L^{p,\infty}({\mathbb R}^d)} \leq \bigg[ \frac{1}{2}\Gamma \bigg( \frac{2p-1}{p-1} \bigg) \bigg]^{(p-1)/p} \|f\|_{L^p({\mathbb R}^d)} $

for 1 < $p$ < 2, and

$ \|T_m f\|_{L^{p,\infty}({\mathbb R}^d)} \leq \bigg[ \frac{p^{p-1}}{2} \bigg]^{1/p} \|f\|_{L^p({\mathbb R}^d)} $

for $2 \leq p$ < $\infty$. The proof rests on a novel duality method and a new sharp inequality for differentially subordinated martingales. We also provide lower bounds for the weak-type constants by constructing appropriate examples for the Beurling-Ahlfors operator on ${\mathbb C}$.

We investigate the long-time asymptotics for the defocusing integrable discrete nonlinear Schrödinger equation of Ablowitz-Ladik by means of the inverse scattering transform and the Deift-Zhou nonlinear steepest descent method. The leading part is a sum of two terms that oscillate with decay of order $t^{-1/2}$.

Let $f:{\mathbb R}^2\longrightarrow {\mathbb R}^2$ be a generic polynomial mapping. There are constructed quadratic forms whose signatures determine the number of positive and negative cusps of $f$.

Let $\Omega^3(SU(n))$ be the Lie group of based mappings from $S^3$ to $SU(n)$. We construct a Lie group extension of $\Omega^3(SU(n))$ for $n\geq 3$ by the abelian group $\exp 2\pi i {\cal A}_3^{\ast}$, where ${\cal A}_3^{\ast}$ is the affine dual of the space of $SU(n)$-connections on $S^3$. J. Mickelsson in 1987 constructed a similar Lie group extension. In this article we give several improvement of his results, especially we give a precise description of the extension of those components that are not the identity component. We also correct several argument about the extension of $\Omega^3(SU(2))$ which seems not to be exact in Mickelsson's work, though his observation about the fact that the extension of $\Omega^3(SU(2))$ reduces to the extension by Z$_2$ is correct. Then we shall investigate the adjoint representation of the Lie group extension of $\Omega^3(SU(n))$ for $n\geq 3$.

This paper is devoted to a systematic study of the geometry of nondegenerate ${\mathbb R}^n$-actions on $n$-manifolds. The motivations for this study come from both dynamics, where these actions form a special class of integrable dynamical systems and the understanding of their nature is important for the study of other Hamiltonian and non-Hamiltonian integrable systems, and geometry, where these actions are related to a lot of other geometric objects, including reflection groups, singular affine structures, toric and quasi-toric manifolds, monodromy phenomena, topological invariants, etc. We construct a geometric theory of these actions, and obtain a series of results, including: local and semi-local normal forms, automorphism and twisting groups, the reflection principle, the toric degree, the monodromy, complete fans associated to hyperbolic domains, quotient spaces, elbolic actions and toric manifolds, existence and classification theorems.

In this article we prove resolvent estimates for the Laplace-Beltrami operator or more general elliptic Fourier multipliers on symmetric spaces of noncompact type. Then the Kato theory implies time-global smoothing estimates for corresponding dispersive equations, especially the Schrödinger evolution equation. For low-frequency estimates, a pseudo-dimension appears as an upper bound of the order of elliptic Fourier multipliers. A key of the proof is to show a weighted $L^{2}$-continuity of the modified Radon transform and fractional integral operators.

Let $v_d^{(s)}$ denote the set of coefficient vectors of contractive polynomials of degree $d$ with $2s$ non-real zeros.We prove that $v_d^{(s)}$ can be computed by a multiple integral, which is related to the Selberg integral and its generalizations. We show that the boundary of the above set is the union of finitely many algebraic surfaces. We investigate arithmetical properties of $v_d^{(s)}$ and prove among others that they are rational numbers. We will show that within contractive polynomials, the ‘probability’ of picking a totally real polynomial decreases rapidly when its degree becomes large.

We prove a sharp lower bound on the curvatures of non-flat ASD connections over the cylinder.

For a $\sigma$-finite measure $\mu$ and a Banach space $Y$ we study the Bishop-Phelps-Bollobás property (BPBP) for bilinear forms on $L_1(\mu)\times Y$, that is, a (continuous) bilinear form on $L_1(\mu)\times Y$ almost attaining its norm at $(f_0,y_0)$ can be approximated by bilinear forms attaining their norms at unit vectors close to $(f_0,y_0)$. In case that $Y$ is an Asplund space we characterize the Banach spaces $Y$ satisfying this property. We also exhibit some class of bilinear forms for which the BPBP does not hold, though the set of norm attaining bilinear forms in that class is dense.

We determine the singular locus of the holonomic system of differential equations annihilating Lauricella's hypergeometric function $F_C$ by the theory of $D$-modules and of Gröbner bases. We also study the $A$-hypergeometric system associated to $F_C$.

We give explicit equations of smooth Jacobian Kummer surfaces of degree 8 in $\mathbb P^5$ by theta functions. As byproducts, we can write down Rosenhain's 80 hyperplanes and 32 lines on these Kummer surfaces explicitly. Moreover we study the fibration of Kummer surfaces over the Satake compactification of the Siegel modular 3-fold of level (2,4). The total space is a smooth projective 5-fold which is regarded as a higher-dimensional analogue of Shioda's elliptic modular surfaces.

The paper considers nonlinear partial differential equations of the form $t (\partial u/\partial t) = F(t,x,u, \partial u/\partial x)$, with independent variables $(t,x) \in \mathbb{R} \times \mathbb{C}$, and where $F(t,x,u,v)$ is a function continuous in $t$ and holomorphic in the other variables. It is shown that the equation has a unique solution in a sectorial domain centered at the origin under the condition that $F(0,x,0,0)=0$, Re$F_u(0,0,0,0)$ < 0, and $F_v(0,x,0,0)=x^{p+1}\gamma(x)$, where $\gamma(0) \neq 0$ and $p$ is any positive integer. In this case, the equation has a Fuchsian singularity at $t=0$ and an irregular singularity at $x=0$.