Related papers: Approximation of holomorphic maps with a lower bou…

Conic quasi-linear maps are nonlinear operators from $C_0(X)$ to a normed linear space $E$ which preserve nonnegative linear combinations on positive cones generated by single functions; quasi-linear maps are linear on singly generated…

In this paper, we prove that unital homomorphisms from continuous functions on a compact metric space to matrices over a C*-algebra with tracial rank at most one are approximately diagonalizable. We also consider some generalizations of…

The Lichnerowicz conjecture asserts that all harmonic manifolds are either flat or locally symmetric spaces of rank 1. This conjecture has been proved by Z.I. Szabo for harmonic manifolds with compact universal cover. E. Damek and F. Ricci…

Let $\cP_n$ be the space of homogeneous polynomials of degree $n$ on $\bbR^{m+1}$. We consider the asymptotic behavior of some coefficients relating to the decomposition of $\cP_n$ into the sum of $\SO(m+1)$-irreducible components. Using…

We consider the problem of approximating a given matrix by a low-rank matrix so as to minimize the entrywise $\ell_p$-approximation error, for any $p \geq 1$; the case $p = 2$ is the classical SVD problem. We obtain the first provably good…

Quasiregular maps are differentiable almost everywhere maps which are analogous to holomorphic maps in the plane for higher real dimensions. Introduced by Gutlyanskii et al, the infinitesimal space is a generalization of the notion of…

In this paper, we shall prove that a harmonic map from $\mathbb{C}^{n}$ ($n\geq2$) to any Kahler manifold must be holomorphic under an assumption of energy density. It can be considered as a complex analogue of the Liouville type theorem…

We prove that the classical Oka property of a complex manifold Y, concerning the existence and homotopy classification of holomorphic mappings from Stein manifolds to Y, is equivalent to a Runge approximation property for holomorphic maps…

Let $K$ be an algebraically closed field of arbitrary characteristic and let $X$ be an irreducible projective variety over $K$. Let $G\subseteq\text{Bir}(X)$ be a bounded-degree subgroup. We prove that there exists an irreducible projective…

Let $f \colon X \to X$ be a surjective endomorphism of a normal projective surface. When $\operatorname{deg} f \geq 2$, applying an (iteration of) $f$-equivariant minimal model program (EMMP), we determine the geometric structure of $X$.…

A continous map $f: \mathbb{C}^n \rightarrow \mathbb{C}^N$ is $k$-regular if the image of any $k$ points spans a $k$-dimensional subspace. It is an important problem in topology and interpolation theory, going back to Borsuk and Chebyshev,…

We generalize a version of Lavrent\'ev's theorem which says that a function that is continuous on a compact set K with connected complement and without interior points can be uniformly approximated as closely as desired by a polynomial…

Given a number field $K$ and a finite set $S$ of places of $K$, the first main result of this paper shows that the quadratic rational maps $\phi:{\mathbb P}^1\to{\mathbb P}^1$ defined over $K$ which have good reduction at all places outside…

We study minimal harmonic maps $g: {\mathbb{C}} \to SO(3) \backslash SL(3,{\mathbb{R}})$, parameterized by polynomial cubic differentials $P$ in the plane. The asymptotic structure of such a $g$ is determined by a convex polygon $Y(P)$ in…

Let X be quasi-isometric to either the mapping class group equipped with the word metric, or to Teichmuller space equipped with either the Teichmuller metric or the Weil-Petersson metric. We introduce a unified approach to study the coarse…

This survey paper is devoted to Riemannian manifolds with special holonomy. To any Riemannian manifold of dimension n is associated a closed subgroup of SO(n), the holonomy group; this is one of the most basic invariants of the metric. A…

Let $M$ be complete nonpositively curved Riemannian manifold of finite volume whose fundamental group $\Gamma$ does not contain a finite index subgroup which is a product of infinite groups. We show that the universal cover $\tilde M$ is a…

We introduce two new algebraic invariants, the (co)homological distances between continuous maps, which provide computable lower bounds for the homotopic distance and strictly refine the classical cup-length estimates. We then define the…

The problem of approximating the infinite dimensional space of all continuous maps from an algebraic variety $X$ to an algebraic variety $Y$ by finite dimensional spaces of algebraic maps arises in several areas of geometry and mathematical…

Given a projective variety X and a smooth projective curve C one may consider the moduli space of maps C --> X. This space admits certain compactification whose points are called quasi-maps. In the last decade it has been discovered that in…