Let $D$ be a strictly convex domain of $\mathbb{C}^{n}$, $f_{1}$ and $f_{2}$ be two holomorphic functions defined on a neighbourhood of $\overline{D}$ and set $X_{l}=\{z,f_{l}(z)=0\}$, $l=1,2$. Suppose that $X_{l}\cap bD$ is transverse for $l=1$ and $l=2$, and that $X_{1}\cap X_{2}$ is a complete intersection. We give necessary conditions when $n\geq2$ and sufficient conditions when $n=2$ under which a function $g$ can be written as $g=g_{1}f_{1}+g_{2}f_{2}$ with $g_{1}$ and $g_{2}$ in $L^{q}(D)$, $q\in[1,+\infty)$, or $g_{1}$ and $g_{2}$ in $\operatorname{BMO}(D)$. In order to prove the sufficient condition, we explicitly write down the functions $g_{1}$ and $g_{2}$ using integral representation formulae and new residue currents.

The total absolute torsion of smooth curves in $E^{3}$ is defined as the total integral of the absolute value of the torsion. This notion is extended to piecewise smooth curves. We study the infimum of the total absolute torsion in a certain set of curves, where the endpoints, the osculating planes at the endpoints and the length are all prescribed. We show how the infimum is calculated from the boundary data.

Randers metrics are natural and important Finsler metrics. This paper shows a non-existence theorem for non-homothetic conformal fields on a non-Riemannian Randers manifold with isotropic $S$-curvature.

We study minimal annuli in $\mathbb{S}^{2}\times\mathbb{R}$ of finite type by relating them to harmonic maps $\mathbb{C} \to\mathbb{S} ^{2}$ of finite type. We rephrase an iteration by Pinkall–Sterling in terms of polynomial Killing fields. We discuss spectral curves, spectral data and the geometry of the isospectral set. We consider polynomial Killing fields with zeroes and the corresponding singular spectral curves, bubbletons and simple factors. We investigate the differentiable structure on the isospectral set of any finite type minimal annulus. We apply the theory to a 2-parameter family of embedded minimal annuli foliated by horizontal circles.

The discrete version of Morse theory due to Robin Forman is a powerful tool utilized in the study of topology, combinatorics, and mathematics involving the overlap of these fields. Inspired by the success of discrete Morse theory, we take the first steps in defining a discrete version of the Lusternik–Schnirelmann category suitable for simplicial complexes. This invariant is based on collapsibility as opposed to contractibility, and is defined in the spirit of the geometric category of a topological space. We prove some basic results of this theory, showing where it agrees and differs from that of the smooth case. Our work culminates in a discrete version of the Lusternik–Schnirelmann theorem relating the number of critical points of a discrete Morse function to its discrete category.

We define a trivolution on a complex algebra $A$ as a non-zero conjugate-linear, anti-homomorphism $\tau$ on $A$, which is a generalized inverse of itself, that is, $\tau^{3}=\tau$. We obtain characterizations of trivolutions and show with examples that they appear naturally on many Banach algebras, particularly those arising from group algebras. We give several results on the existence or non-existence of involutions on the dual of a topologically introverted space. We investigate conditions under which the dual of a topologically introverted space admits trivolutions.

We prove a boundedness criterion for a class of dyadic multilinear forms acting on two-dimensional functions. Their structure is more general than the one of classical multilinear Calderón–Zygmund operators as several functions can now depend on the same one-dimensional variable. The study of this class is motivated by examples related to the two-dimensional bilinear Hilbert transform and to bilinear ergodic averages. This paper is a sequel to a prior paper by the first author.

Let $\mathbf{R}$ be an o-minimal expansion of a real closed field. Given definable continuous functions $f:U\rightarrow R$ and $\epsilon:U\rightarrow(0,+\infty)$, where $U$ is an open subset of $R^{n}$, we construct a definable $C^{m}$-function $g:U\to R$ with $\vert g(x)-f(x)\vert <\epsilon(x)$ for all $x\in U$. Moreover, we show that if $f$ is uniformly continuous, then $g$ can also chosen to be uniformly continuous.

We introduce the notion of a $G$-family of quandles which is an algebraic system whose axioms are motivated by handlebody-knot theory, and use it to construct invariants for handlebody-knots. Our invariant can detect the chiralities of some handlebody-knots including unknown ones.

We establish a result on the existence of nontrivial solution for the following class of biharmonic elliptic equation

\[(\mathrm{P})\quad \Bigl\{\begin{array}{l@{\quad}l}\Delta^{2}u+V(x)u=K(x)f(u)&\mbox{in }R^{N},\\u\neq0,&\mbox{in }R^{N},u\in\mathcal{D}^{2,2}(R^{N}),\end{array}\]

where $\Delta^{2}u=\Delta(\Delta u)$, $V$ and $K$ are nonnegative potentials. $K$ vanishes at infinity and $f$ has a subcritical growth at infinity. The technique used here is the variational approach.

In this paper, we study the behavior of the bounds of matrix-valued maximal inequality in $\mathbb{R}^{n}$ for large $n$. The main result of this paper is that the $L_{p}$-bounds ($p>1$) can be taken to be independent of $n$, which is a generalization of Stein and Strömberg’s result in the scalar-valued case. We also show that the weak type $(1,1)$ bound has similar behavior as Stein and Stömberg’s.

Let $I$ be a weakly polymatroidal ideal or a squarefree monomial ideal of a polynomial ring $S$. In this paper, we provide a lower bound for the Stanley depth of $I$ and $S/I$. In particular, we prove that if $I$ is a squarefree monomial ideal which is generated in a single degree, then $\operatorname{sdepth} (I)\geq n-\ell(I)+1$ and $\operatorname{sdepth} (S/I)\geq n-\ell(I)$, where $\ell(I)$ denotes the analytic spread of $I$. This proves a conjecture of the author in a special case.

We relate a continuous-time version of the geometric RSK correspondence to the Toda lattice, in a way which can be viewed as a semi-classical limit of a recent result by the author which relates the continuous-time geometric RSK mapping, with Brownian motion as input, to the quantum Toda lattice.