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Xuezhi Zhao

Publications and source records attributed to Xuezhi Zhao.

At least 19 recordsLinked to original sources
arXiv2026

A Classification of Self-Maps of Generalized Grassmannians

Generalized Grassmannians form a fundamental class of flag manifolds associated with Lie groups. The purpose of this paper is to classify self-maps of generalized Grassmannians of nonzero degree in terms of their induced actions on cohomology. We prove a rigidity theorem showing that, despite the rich and intricate structure of their cohomology rings, the induced cohomology endomorphisms fall into only two natural types: Adams operations determined by the degree and Dynkin symmetries arising from automorphisms of the Dynkin diagram. Combining the geometry of root and weight systems, the actions of Weyl and Dynkin symmetries on Schubert classes, and Bott--Samelson desingularizations of Schubert varieties, our approach applies uniformly to generalized Grassmannians of all Lie types.

math.AT↗
arXiv2025

Word Length Formulae and Normal Forms of Conjugacy Classes in Surface Groups

In this paper, we primarily investigate the following symmetric presentation of the surface group $π_1(Σ_g)=\left\langle c_1,\dots, c_{2g}\mid c_1\cdots c_{2g}c_1^{-1}\cdots c_{2g}^{-1}\right\rangle$. For every nontrivial element $x\in π_1(Σ_g)$, we obtain a uniform representation of the normal forms of $x^k$ under the length-lexicographical order. Based on this, we find a new relation among these normal forms, and then derive the following three formulae related to the word length: $|x^2|>|x|$; $|x^k|=(k-1)(|x^2|-|x|)+|x|$; $\lim_{k\to\infty}\frac{|x^k|}{k}=|x^2|-|x|$. Moreover, we extend these results to obtain analogous but less precise formulae for every minimal geometric presentation. Then, we define the normal forms of conjugacy classes in $π_1(Σ_g)$ and give a criterion for determining the conjugacy of elements. As a consequence, we give efficient algorithms for solving the root-finding and conjugacy problems. Finally, we present applications concerning the computation of some growth rates.

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arXiv2025

The multiple points of maps from sphere to Euclidean space

In this paper, we obtain some sufficient conditions to guarantee the existence of multiple points of maps from $S^m$ to $\mathbb{R}^d$. Our main tool is the ideal-valued index of $G$-space defined by E. Fadell and S. Husseini. We obtain more detailed relative positional relationship of multiple points. It is proved that for a continuous real value function $f: S^m\rightarrow \mathbb{R}$ such that $f(-p)=-f(p)$, if $m+1$ is a power of $2$, then there are $m+1$ points $p_1, \ldots, p_{m+1}$ in $S^m$ such that $f(p_1)=\cdots=f(p_{m+1})$, where $p_1, \ldots, p_{m+1}$ are linearly dependent and any $m$ points of $p_1, \ldots, p_{m+1}$ are linearly independent. As a generalization of Hopf's theorem, we also prove that for any continuous map $f: S^m\rightarrow \mathbb{R}^d$, if $m> d$, then there exists a pair of mutually orthogonal points having the same image in addition to the antipodal points.

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arXiv2025

Q-Mirror: Unlocking the Multi-Modal Potential of Scientific Text-Only QA Pairs

High-quality, multi-modal benchmarks are crucial for advancing scientific reasoning in large models yet their manual creation is costly and unscalable. To address this bottleneck, we explore the potential for transforming Text-Only QA Pairs (TQAs) into high-quality Multi-Modal QA Pairs (MMQAs), which include three parts: 1) Task Definition \& Evaluation Rubric: We develop a TQA-to-MMQA framework and establish a comprehensive, multi-dimensional MMQA quality rubric that provides principles for the transformation. 2) Benchmark Construction: Then we construct two extensive benchmarks to rigorously evaluate state-of-the-art generation \& understanding models on the distinct tasks of MMQA generation \& MMQA quality evaluation. 3) Preliminary Solution: We develop an agentic system (Q-Mirror), which operationalizes our framework by integrating MMQA generation and evaluation into a closed loop for iterative refinement. Our experiments show that while state-of-the-art models can generate MMQAs, their outputs still leave substantial gaps, underscoring the need for reliable evaluation. We further demonstrate that top-tier understanding models align closely with human judgment in MMQA quality assessment. Leveraging both insights, the Q-Mirror agent raises average scores from 78.90 to 85.22 and pass rates from 72\% to 95\%, offering a practical path to large-scale scientific benchmarks.

cs.CL↗
arXiv2025

On Jiang's Bounded Index Property for products of nilmanifolds

In 2023, Zhang and Zhao presented the first examples of aspherical manifolds lacking the Bounded Index Property (BIP) for fixed points. This answered a question posed by Jiang in 1998 in the negative. In this paper, we first extend the notion of BIP to that of iterates of selfmaps (BIP$_k$), and then demonstrate BIP$_k$ for certain products $M\times N$ of nilmanifolds. Finally, we give characterizations for the Lefschetz number, the Nielsen number, and the minimal number of fixed points of self-homotopy equivalences of $M\times N$.

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arXiv2025

A Large-Scale Referring Remote Sensing Image Segmentation Dataset and Benchmark

Referring Remote Sensing Image Segmentation is a complex and challenging task that integrates the paradigms of computer vision and natural language processing. Existing datasets for RRSIS suffer from critical limitations in resolution, scene diversity, and category coverage, which hinders the generalization and real-world applicability of refer segmentation models. To facilitate the development of this field, we introduce NWPU-Refer, the largest and most diverse RRSIS dataset to date, comprising 15,003 high-resolution images (1024-2048px) spanning 30+ countries with 49,745 annotated targets supporting single-object, multi-object, and non-object segmentation scenarios. Additionally, we propose the Multi-scale Referring Segmentation Network (MRSNet), a novel framework tailored for the unique demands of RRSIS. MRSNet introduces two key innovations: (1) an Intra-scale Feature Interaction Module (IFIM) that captures fine-grained details within each encoder stage, and (2) a Hierarchical Feature Interaction Module (HFIM) to enable seamless cross-scale feature fusion, preserving spatial integrity while enhancing discriminative power. Extensive experiments conducte on the proposed NWPU-Refer dataset demonstrate that MRSNet achieves state-of-the-art performance across multiple evaluation metrics, validating its effectiveness. The dataset and code are publicly available at https://github.com/CVer-Yang/NWPU-Refer.

cs.CV↗
arXiv2023

Persistence of sub-chain groups

In this work, we present a generalization of extended persistent homology to filtrations of graded sub-groups by defining relative homology in this setting. Our work provides a more comprehensive and flexible approach to get an algebraic invariant overcoming the limitations of the standard approach. The main contribution of our work is the development of a stability theorem for extended persistence modules using an extension of the definition of interleaving and the rectangle measure. This stability theorem is a crucial property for the application of mathematical tools in data analysis. We apply the stability theorem to extended persistence modules obtained from extended path homology of directed graphs and extended homology of hypergraphs, which are two important examples in topological data analysis.

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arXiv2023

On the Whitney disks in Heegaard Floer homology theory

The Whitney disks play a central role in defining Heegaard Floer homology of a $3$-dimensional manifold. We use Nielsen theory to a simple criterion to the existence of Whitney disks, connecting two given intersections.

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arXiv2022

Geometric intersections of loops on surfaces

Based on Nielsen fixed point theory and Gröbner-Shirshov basis, we obtain a simple method to compute geometric intersection numbers and self-intersection geometric numbers of loops on surfaces.

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arXiv2022

Schubert calculus and Intersection theory of Flag manifolds

Hilbert's 15th problem called for a rigorous foundation of Schubert's calculus, in which a long standing and challenging part is Schubert's problem of characteristics. In the course of securing the foundation of algebraic geometry, Van der Waerden and André Weil attributed the problem to the determination of the intersection theory of flag manifolds. This article surveys the background, content, and resolution of the problem of characteristics. Our main results are a unified formula for the characteristics, and a system description for the intersection rings of flag manifolds. We illustrate the effectiveness of the formula and the algorithm via explicit examples.

math.AG↗
arXiv2021

Estimate of number of simplices of triangulations of Lie groups

We present estimates of number of simplices of given dimension of classical compact Lie groups. As in the previous work \cite{GMP2} the approach is a combination of an estimate of number of vertices with a use of valuation of the covering type by cohomological argument of \cite{GMP} and application of the recent versions of the Lower Bound Theorem of combinatorial topology. For the case of exceptional Lie groups we made a complete calculation using the description of their cohomology rings given by the first and third author. For infinite increasing series of Lie groups of growing dimension $d$ the rate of growth of number of simplices of highest dimension is given which extends onto the case of simplices of (fixed) codimension $d-i$.

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arXiv2020

Simple Smale flows and their templates on $S^3$

The embedded template is a geometric tool in dynamics being used to model knots and links as periodic orbits of $3$-dimensional flows. We prove that for an embedded template in $S^3$ with fixed homeomorphism type, its boundary as a trivalent spatial graph is a complete isotopic invariant. Moreover, we construct an invariant of embedded templates by Kauffman's invariant of spatial graphs, which is a set of knots and links. As application, the isotopic classification of simple Smale flows on $S^3$ is discussed.

math.GT↗
arXiv2020

On Schubert's Problem of Characteristics

The Schubert varieties on a flag manifold G/P give rise to a cell decomposition on G/P whose Kronecker duals, known as the Schubert classes on G/P, form an additive base of the integral cohomology of G/P. The Schubert's problem of characteristics asks to express a monomial in the Schubert classes as a linear combination in the Schubert basis. We present a unified formula expressing the characteristics of a flag manifold G/P as polynomials in the Cartan numbers of the group G. As application we develop a direct approach to our recent works on the Schubert presentation of the cohomology rings of flag manifolds G/P.

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arXiv2020

Fixed point indices and fixed words at infinity of selfmaps of graphs

Indices of fixed point classes play a central role in Nielsen fixed point theory. Jiang-Wang-Zhang proved that for selfmaps of graphs and surfaces, the index of any fixed point class has an upper bound called its characteristic. In this paper, we study the difference between the index and the characteristic for selfmaps of graphs. First, for free groups, we extend attracting fixed words at infinity of automorphisms into that of injective endomorphisms. Then, by using relative train track technique, we show that the difference mentioned above is quite likely to be the number of equivalence classes of attracting fixed words of the endomorphism induced on the fundamental group. Since both of attracting fixed words and the existed characteristic are totally determined by endomorphisms themselves, we give a new algebraic approach to estimate indices of fixed point classes of graph selfmaps. As consequence, we obtain an upper bound for attracting fixed words of injective endomorphisms of free groups, generalizing the one for automorphisms due to Gaboriau-Jaeger-Levitt-Lustig. Furthermore, we give a simple approach to roughly detecting whether fixed words exist or not.

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arXiv2019

Isotopy Uniqueness of Self-diffeomorphism of Handlebodies

The mapping class group $MCG(Σ_g)$ of a surface of genus $g$ has a long-history in topology and group theory. More recently, the mapping class group $MCG(V_g)$ of a handlebody $V_g$ of genus $g$ has become an interesting topic in the study of $3$ manifolds, largely thanks to Heegaard splitting. While $MCG(V_g)$ can be regarded naturally as a sub group of $MCG(Σ_g)$, we could not find any complete proof of this fundamental theorem. It is the purpose of this paper that we give a rigorous proof of embedding of $MCG(V_g)$ into $MCG(V_g)$. The key step is: Any self-homeomorphism $f$ of handlebody $V_g$ of genus $g$ is ambient isotopic to identity if the restriction $f|_{\partial V_g}$ is isotopic to identity.

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arXiv2019

On generalized configuration space and its homotopy groups

Let $M$ be a subset of vector space or projective space. The authors define the \emph{generalized configuration space} of $M$ which is formed by $n$-tuples of elements of $M$ where any $k$ elements of each $n$-tuple are linearly independent. The \emph{generalized configuration space} gives a generalization of the classical configuration space defined by E.Fadell. Denote the \emph{generalized configuration space} of $M$ by $W_{k,n}(M)$. The authors are mainly interested in the calculation about the homotopy groups of generalized configuration space. This article gives the fundamental groups of generalized configuration spaces of $\mathbb{R}P^m$ for some special cases, and the connections between the homotopy groups of generalized configuration spaces of $S^m$ and the homotopy groups of Stiefel manifolds. It is also proved that the higher homotopy groups of generalized configuration spaces $W_{k,n}(S^m)$ and $W_{k,n}(\mathbb{R}P^m)$ are isomorphic.

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