topology

This arXiv paper proves conditions under which a spectral single-linkage hierarchy built from diffusion distances is topologically equivalent to the density cluster tree, up to monotone rescaling of merge heights, including a finite-sample k-NN graph result.

该论文基于等变度数理论（equivariant degree theory）提出了一种通用的拓扑对称性指数，用于刻画神经网络损失景观中临界轨道的各向同性及其净拓扑贡献，适用于任意网络架构并给出软激活 MLP、Transformer 和等变神经网络的计算示例。

The paper identifies a theoretical incompatibility between 3D point cloud masked autoencoding and SE(3)-equivariant networks, proving that standard masking triggers a topological collapse, and introduces SIEVE, an equivariant self-supervised framework using a spherical projection mask to preserve global pose while obfuscating local semantics, improving downstream generalization to unseen shape categories.

The paper presents TopoPlanner, a topology-consistent planning framework that lifts tool dependency graphs into cellular workflow complexes to guide LLM tool planning, enabling handling of verification-correction loops, branch merging, and reusable intermediate states beyond sequential or DAG-like workflows, with consistent gains over baselines on four benchmarks.

SILSA is a topology-aware 3D generation framework that replaces expensive voxel tokens with compact sliding-window slice latents and single-stage rectified-flow generation, using persistence-diagram/Betti-based slice-level topology supervision. It improves structural fidelity (PSNR +8.7%, Betti error -9.2%) while cutting tokens by 70%, memory by 40.4%, and inference time by 58.5%.

The paper introduces TopoSIGN, a topology-guided graph pre-training and prompt learning framework for signed graphs, which combines structural encoding and persistent homology to improve transfer learning in tasks like link prediction and node classification.

This paper introduces topology as a distinct generalization axis for neural operators in PDE solving, using Hodge heat flow and the TopoBox-3D framework to show its influence across the Hodge spectrum beyond geometric compatibility.

This article describes a rendering technique for creating hand-hatched, non-photorealistic images of topological surfaces, inspired by Francis' book and various academic methods in computer graphics.

This paper investigates the existence or properties of a complex structure on the six-dimensional sphere S^6.

Microsoft Research introduces MindTopo, a benchmark that tests whether multimodal AI models understand topological relationships like connectivity, enclosure, and knots. Current models perform well on static recognition but struggle with interactive planning tasks, revealing a key gap for robotics and interactive environments.

Presents ArchEGraph, a large-scale graph dataset for building energy modeling with aligned geometry, topology, weather, and thermal loads, along with benchmark tasks for graph reconstruction and load prediction.

This paper proves that the Burau representation of the braid group B_4 is faithful, settling the final unknown case in a long-standing problem in low-dimensional topology.

Introduces TopoTuner, a topology-guided fine-tuning framework that selectively freezes attention projection matrices by measuring topological drift via Wasserstein distances between persistence diagrams. It achieves competitive performance to full fine-tuning while training only 1-2% of parameters and outperforms LoRA in most settings.

Cotal is an open standard for AI agent coordination that allows configurable topologies (peer-to-peer, supervised, hierarchical, or hybrid) using NATS messaging, complementing existing protocols like MCP and A2A.

An open-source 2.5D diagram engine in Go that separates topology from geometry to enable LLMs to generate clean architecture diagrams without spatial hallucinations.

This thread argues that standard transformers have a topological flaw: once a state representation reaches the top layer, they cannot update beliefs over time, causing collapse as depth increases.

This paper introduces TopVAE, a topology-optimized VAE that reduces 'dark areas' in molecular latent diffusion by making the decoder internalize structural and chemical constraints, achieving significant improvements in molecular generation quality.

This paper introduces Topological Neural Operators, which lift neural operators from point-only domains to cell complexes, embedding geometry and topology to reduce the learning burden. It demonstrates that operator learning improves when geometry is not an afterthought, though the topology remains prescribed.

This paper introduces a categorical framework for transfer learning using Kan extensions, defining a transfer discrepancy that compares target invariants against those forced by a prescribed task transformation. It proves finite cokernel formulas for chain complexes and persistence modules and validates the approach on neural latent point clouds.

This article delves into the division of labor design in multi-agent systems, including trigger mechanisms, topology structures, and call chains, analyzing the engineering practices of systems such as Codex, Claude Code, OpenClaw, and Hermes Agent.