We show that every $3$-strand braid has a representative word of a given form, and furthermore, this form allows us, in most cases, to deduce positivity (or negativity) in the $\sigma$-ordering of $B_{3}$. The $\sigma$-ordering of $B_{n}$ was introduced by Patrick Dehornoy in the late 1990's, however, other (equivalent) orderings were discovered soon after by Fenn, Greene, Rolfsen, et al.

We introduce and study local moves for links, called simple ribbon moves. We also introduce a complexity of links, called the $h$-complexity, which coincides with the genus in the case of knots, and we show that simple ribbon moves never reduce the $h$-complexities of links.

In this paper we use a knot invariant, namely the Tristram--Levine signature, to study deformations of singular points of plane curves. We bound, in some cases, the difference between the $M$-number of the singularity of the central fiber and the sum of $M$-numbers of the generic fiber.

We give an obstruction to unknotting a knot by adding a twisted band, derived from Heegaard Floer homology.

Recently Taylor and Socolar introduced an aperiodic mono-tile. The associated tiling can be viewed as a substitution tiling. We use the substitution rule for this tiling and apply the algorithm of [1] to check overlap coincidence. It turns out that the tiling has overlap coincidence. So the tiling dynamics has pure point spectrum and we can conclude that this tiling has a quasicrystalline structure.

We prove the global existence of analytic solutions to the Cauchy problem for nonlinear Schrödinger equations in two dimensions, where the nonlinearity behaves as a cubic power at the origin and the Cauchy data are small and decay exponentially at infinity.

We prove a scale-invariant boundary Harnack principle in inner uniform domains in the context of non-symmetric local, regular Dirichlet spaces. For inner uniform Euclidean domains, our results apply to divergence form operators that are not necessarily symmetric, and complement earlier results by H. Aikawa and A. Ancona.

We investigate unbranched Riemann domains $p\colon X \to \tilde{\mathbb{C}}^{n+1}$ over the blow-up of $\mathbb{C}^{n+1}$ at the origin in the case when $p$ is a Stein morphism. We prove that such a domain is Stein if and only if it does not contain an open set $G \subset X$ such that $p|_{G}$ is injective and $p(G)$ contains a subset of the form $W \setminus A$, where $A$ is the exceptional divisor of $\tilde{\mathbb{C}}^{n+1}$ and $W$ is an open neighborhood of $A$.

We determine the minimum number of generators of the homological Goldman Lie algebra of a surface consisting of elements of the first homology group of the surface.

In this paper, for a compact special Legendrian submanifold with smooth boundary of contact Calabi--Yau manifolds we study the deformation of it with boundary confined in an appropriately chosen contact submanifold of codimension two which we also call a scafford (Definition 2.3) by analogy with Butsher [1]. Our first result shows that it cannot be deformed, and the second claims that deformations of such a special Legendrian submanifold forms a one-dimensional smooth manifold under suitably weaker boundary confinement conditions. They may be viewed as supplements of the closed case considered by Tomassini and Vezzoni [17].

Let $\mathcal{A}$ be the class of functions \begin{equation*} f(z)=z+\sum_{n=2}^{\infty}a_{n}z^{n} \end{equation*} which are analytic in the unit disk $\mathbb{D}=\{z\colon \lvert z\rvert <1\}$. Let $C(r)$ be the closed curve which is the image of the circle $\lvert z\rvert =r<1$ under the mapping $w=f(z)$, $L(r)$ the length of $C(r)$, and let $A(r)$ be the area enclosed by the curve $C(r)$. It was shown in [13] that if $f\in\mathcal{A}$, $f$ is starlike with respect to the origin, and for $0\leq r<1$, $A(r)<A$, an absolute constant, then \begin{equation} \label{eq0.1} L(r)=\mathcal{O}\left(\log\frac{1}{1-r}\right) \quad\text{as}\quad r\to 1. \end{equation} It is the purpose of this work to prove, using a modified methods than that in [13], a strengthened form of (0.1) for Bazilevi\u{c} functions, strongly starlike functions and for close-to-convex functions.

P.D. Hislop and F. Klopp proved a Wegner estimate for Schrödinger operators with nonsign definite potentials for each fixed position of impurities [12]. In this paper, a similar estimate is proven treating also the position of impurities as random variables.

A general construction of braid group actions on coherent sheaves via M. Saito's theory of mixed Hodge modules is given.

This paper discusses a new notion of quadratic variation and covariation for Banach space valued processes (not necessarily semimartingales) and related Itô formula. If $\mathbb{X}$ and $\mathbb{Y}$ take respectively values in Banach spaces $B_{1}$ and $B_{2}$ and $\chi$ is a suitable subspace of the dual of the projective tensor product of $B_{1}$ and $B_{2}$ (denoted by $(B_{1} \hatotimes_{\pi}B_{2})^{*}$), we define the so-called $\chi$-covariation of $\mathbb{X}$ and $\mathbb{Y}$. If $\mathbb{X} = \mathbb{Y}$, the $\chi$-covariation is called $\chi$-quadratic variation. The notion of $\chi$-quadratic variation is a natural generalization of the one introduced by Métivier--Pellaumail and Dinculeanu which is too restrictive for many applications. In particular, if $\chi$ is the whole space $(B_{1} \hatotimes_{\pi}B_{1})^{*}$ then the $\chi$-quadratic variation coincides with the quadratic variation of a $B_{1}$-valued semimartingale. We evaluate the $\chi$-covariation of various processes for several examples of $\chi$ with a particular attention to the case $B_{1} = B_{2} = C([-\tau, 0])$ for some $\tau>0$ and $\mathbb{X}$ and $\mathbb{Y}$ being window processes. If $X$ is a real valued process, we call window process associated with $X$ the $C([-\tau, 0])$-valued process $\mathbb{X} := X({}\cdot{})$ defined by $X_{t}(y) = X_{t+y}$, where $y \in [-\tau, 0]$. The Itô formula introduced here is an important instrument to establish a representation result of Clark--Ocone type for a class of path dependent random variables of type $h = H(X_{T}({}\cdot{}))$, $H\colon C([-T, 0])\to\mathbb{R}$ for not-necessarily semimartingales $X$ with finite quadratic variation. This representation will be linked to a function $u\colon [0, T]\times C([-T, 0])\to \mathbb{R}$ solving an infinite dimensional partial differential equation.

We define the induction and restriction functors for cyclotomic $q$-Schur algebras, and study some properties of them. As an application, we categorify a higher level Fock space by using the module categories of cyclotomic $q$-Schur algebras.

We give a new proof of the characterization of self-bumping points on the space of Kleinian once-punctured torus groups of Ito [8], based on some recent work of Bromberg [4].