We discuss various questions of the following kind: for a continuous map $X \to Y$ from a compact metric space to a simplicial complex, can one guarantee the existence of a fiber large in the sense of Urysohn width? The $d$-width measures how well a space can be approximated by a $d$-dimensional complex. The results of this paper include the following. 1) Any piecewise linear map $f: [0,1]^{m+2} \to Y^m$ from the unit euclidean $(m+2)$-cube to an $m$-polyhedron must have a fiber of $1$-width at least $\frac{1}{2βm +m^2 + m + 1}$, where $β= \sup_y \text{ rk } H_1(f^{-1}(y))$ measures the topolo…

2021-07-25math.MG

稀疏映射

Sparse maps

该研究针对从单纯复形到欧氏空间的映射，推广了柯尔莫哥洛夫-巴尔兹丁定理，引入稀疏映射概念并探讨了具有受控1-腰的单纯复形的映射，完善了相关理论。

2026-08-03math.MG

Gromov's waist of non-radial Gaussian measures and radial non-Gaussian measures

We study the Gromov waist in the sense of $t$-neighborhoods for measures in the Euclidean space, motivated by the famous theorem of Gromov about the waist of radially symmetric Gaussian measures. In particular, it turns our possible to extend Gromov's original result to the case of not necessarily radially symmetric Gaussian measure. We also provide examples of measures having no $t$-neighborhood waist property, including a rather wide class of compactly supported radially symmetric measures and their maps into the Euclidean space of dimension at least 2. We use a simpler form of Gromov's panc…

2019-12-26math.MGmath.PR

Area preserving maps and volume preserving maps between a class of polyhedrons and a sphere

For a class of polyhedrons denoted $\mathbb K_n(r,\varepsilon)$, we construct a bijective continuous area preserving map from $\mathbb K_n(r,\varepsilon)$ to the sphere $\mathbb S^{2}(r)$, together with its inverse. Then we investigate for which polyhedrons $\mathbb K_n(r',\varepsilon)$ the area preserving map can be used for constructing a bijective continuous volume preserving map from $\bar{\mathbb K}_n(r',\varepsilon)$ to the ball $\bar{\mathbb S^{2}}(r)$. These maps can be further used in constructing uniform and refinable grids on the sphere and on the ball, starting from uniform and ref…

2015-04-07math.MG

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圆与球面间的二次映射

Circles and Quadratic Maps Between Spheres

考虑向量空间中0的邻域到欧几里得空间的解析映射。假设这个映射把所有通过0的线的芽都取为圆的芽。这样的映射称为舍入。我们引入了一个自然等价关系，并证明了任何舍入，其微分在0的秩至少为2，是等价于一个分数二次舍入。分数阶二次映射就是二次映射与二次多项式之比。我们还表明，任何四舍五入产生一个二次映射之间的领域。球面间二次映射的已知结果对球面间二次映射有一些有趣的意义。

2004-12-11math.DGmath.MG

从球面到圆映射中的正交对

Orthogonal Pairs in Maps from the Sphere to the Circle

University of Victoria（维多利亚大学）

本文证明对任意从球面到圆的映射，存在正交向量其像点间最短弧长不超过$\pi/2+\varepsilon$，从而证实了实正交图圆形色数为四的猜想。

2026-09-22math.COmath.MG

Expanding Thurston Maps

We study the dynamics of Thurston maps under iteration. These are branched covering maps $f$ of 2-spheres $S^2$ with a finite set $\mathop{post}(f)$ of postcritical points. We also assume that the maps are expanding in a suitable sense. Every expanding Thurston map $f\: S^2 \to S^2$ gives rise to a type of fractal geometry on the underlying sphere $S^2$. This geometry is represented by a class of \emph{visual metrics} $\varrho$ that are associated with the map. Many dynamical properties of the map are encoded in the geometry of the corresponding {\em visual sphere}, meaning $S^2$ equipped with…

2017-10-10math.CVmath.DSmath.MG