Related papers: The embedding conjecture and the approximation con…
We discuss the problem of embedding univalent functions into Loewner chains in higher dimension. In particular, we prove that a normalized univalent map of the ball in $\C^n$ whose image is a smooth strongly pseudoconvex domain is…
We give conditions in order to approximate locally uniformly holomorphic covering mappings of the unit ball of $\mathbb{C}^n$ with respect to an arbitrary norm, with entire holomorphic covering mappings. The results rely on a generalization…
We display four approximation theorems for manifold-valued mappings. The first one approximates holomorphic embeddings on pseudoconvex domains in $\Bbb C^n$ with holomorphic embeddings with dense images. The second theorem approximates…
In this paper we study the class of "shearing" holomorphic maps of the unit ball of the form $(z,w)\mapsto (z+g(w), w)$. Besides general properties, we use such maps to construct an example of a normalized univalent map of the ball onto a…
We construct a holomorphic embedding $\phi:\mathbb B^3\rightarrow\mathbb C^3$ such that $\phi(\mathbb B^3)$ is not Runge in any strictly larger domain. As a consequence, $\mathcal S\neq\mathcal S^1$ for $n=3$.
In this paper, we investigate holomorphic mappings $F$ on the unit ball $\mathbb{B}$ of a complex Banach space of the form $F(x)=f(x)x$, where $f$ is a holomorphic function on $\mathbb{B}$. First, we investigate criteria for univalence,…
We present a new geometric construction of Loewner chains in one and several complex variables which holds on a complete hyperbolic complex manifold M and prove that there is essentially a one-to-one correspondence between evolution…
We study two conjectures in additive combinatorics. The first is the polynomial Freiman-Ruzsa conjecture, which relates to the structure of sets with small doubling. The second is the inverse Gowers conjecture for $U^3$, which relates to…
Bilipschitz invariant theory concerns low-distortion embeddings of orbit spaces into Euclidean space. To date, embeddings with the smallest-possible distortion are known for only a few cases, to include: (a) planar rotations, (b) real phase…
In this paper, we study homological dimensions of algebras linked by recollements of derived module categories, and establish a series of new upper bounds and relationships among their finitistic or global dimensions. This is closely…
We study general properties of holomorphic isometric embeddings of complex unit balls $\mathbb B^n$ into bounded symmetric domains of rank $\ge 2$. In the first part, we study holomorphic isometries from $(\mathbb B^n,kg_{\mathbb B^n})$ to…
In this article, a new construction of derived equivalences is given. It relates different endomorphism rings and more generally cohomological endomorphism rings - including higher extensions - of objects in triangulated categories. These…
In this note we discuss some problems related to conformal slit-mappings. On the one hand, classical Loewner theory leads us to questions concerning the embedding of univalent functions into slit-like Loewner chains. On the other hand, a…
We generalise theorems of Khodorovskiy and Park-Park-Shin, and give new topological proofs of those theorems, using embedded surfaces in the 4-ball and branched double covers. These theorems exhibit smooth codimension-zero embeddings of…
We study approximation of probability measures supported on $n$-dimensional manifolds embedded in $\mathbb{R}^m$ by injective flows -- neural networks composed of invertible flows and injective layers. We show that in general, injective…
The approximability of a convex body is a number which measures the difficulty to approximate that body by polytopes. We prove that twice the approximability is equal to the volume entropy for a Hilbert geometry in dimension two end three…
This paper focuses on the embedding problem of linear fractional maps which explains when a linear fractional self-map of $B_{N}$ can be a member of a semigroup of holomorphic self-maps on the unit ball $B_{N}$ of the complex…
We introduce the notion of equiangular tight frames in real symplectic spaces and formulate a conjecture on their existence in terms of the dimension and number of vectors. Our main results shows the "symplectic Zauner's conjecture" is…
In this article, we obtain quasiconformal extensions of some classes of conformal maps defined either on the unit disc or on the exterior of it onto the extended complex plane. Some of these extensions have been obtained by constructing…
For biharmonic maps, there is a famous conjecture named Chen's conjecture. In later paper, Wang and Ou gave an affirmative partial answer to submersion version of Chen's conjecture. In this paper, we give an affirmative partial answer to…