arXiv subjects

Zhentao Lai

Publications and source records attributed to Zhentao Lai.

3 recordsLinked to original sources

arXiv2025

Nonsingular structural stable chaotic 3-flows of attractor-repeller type

We show that any orientable closed 3-manifold $M$ admits structurally stable non-singular flow $f^t$ whose non-wandering set $NW(f^t)$ consists of a 2-dimensional expanding attractor and finitely many repelling periodic trajectories. For $M=\mathbb{S}^3$, we prove that the set of repelling periodic trajectories can be an arbitrary link provided that this link contains the figure eight knot. When a link consists of a unique repelling periodic trajectory (not necessarily a figure eight knot), we prove that this trajectory cannot be a torus knot. For any closed 3-manifold $M$, we show that there does not admit any structurally stable non-singular flow $f^t$ whose non-wandering set $NW(f^t)$ consists of a 2-dimensional expanding attractor and a repelling periodic trajectory so that the repelling periodic trajectory is a trivial knot (i.e., it bounds a disk in $M$).

math.DS↗

arXiv2025

Note on Euler class 0 taut foliation on the Whitehead link exterior

This paper studies the existence of co-orientable taut foliations on 3-manifolds, particularly focusing on the Whitehead link exterior. We demonstrate fundamental obstructions to the existence of such foliations with certain Euler class properties, contrasting with the more permissive case of the figure-eight knot exterior. Our analysis reveals that most lattice points in the dual unit Thurston ball of the Whitehead link exterior cannot be realized as Euler classes of co-orientable taut foliations, with only limited exceptional cases. The proofs combine techniques from foliation theory, including saddle fillings and index formulas, with topological obstructions derived from Dehn surgery. These results yield general methods for studying foliations on cusped hyperbolic 3-manifolds. These techniques applied to broader classes of manifolds beyond the specific examples considered here

math.GT↗

arXiv2025

DeepMath-Creative: A Benchmark for Evaluating Mathematical Creativity of Large Language Models

To advance the mathematical proficiency of large language models (LLMs), the DeepMath team has launched an open-source initiative aimed at developing an open mathematical LLM and systematically evaluating its mathematical creativity. This paper represents the initial contribution of this initiative. While recent developments in mathematical LLMs have predominantly emphasized reasoning skills, as evidenced by benchmarks on elementary to undergraduate-level mathematical tasks, the creative capabilities of these models have received comparatively little attention, and evaluation datasets remain scarce. To address this gap, we propose an evaluation criteria for mathematical creativity and introduce DeepMath-Creative, a novel, high-quality benchmark comprising constructive problems across algebra, geometry, analysis, and other domains. We conduct a systematic evaluation of mainstream LLMs' creative problem-solving abilities using this dataset. Experimental results show that even under lenient scoring criteria -- emphasizing core solution components and disregarding minor inaccuracies, such as small logical gaps, incomplete justifications, or redundant explanations -- the best-performing model, O3 Mini, achieves merely 70% accuracy, primarily on basic undergraduate-level constructive tasks. Performance declines sharply on more complex problems, with models failing to provide substantive strategies for open problems. These findings suggest that, although current LLMs display a degree of constructive proficiency on familiar and lower-difficulty problems, such performance is likely attributable to the recombination of memorized patterns rather than authentic creative insight or novel synthesis.

cs.AI↗