geometry
-
Manifolds: Transforming Mathematical Views of Space
Read Full Article: Manifolds: Transforming Mathematical Views of Space
Manifolds, a fundamental concept in mathematics, have revolutionized the way mathematicians perceive and understand space. These mathematical structures allow for the examination of complex, high-dimensional spaces by breaking them down into simpler, more manageable pieces that resemble familiar, flat surfaces. This approach has been instrumental in advancing fields such as topology, geometry, and even theoretical physics, providing insights into the nature of the universe. Understanding manifolds is crucial as they form the backbone of many modern mathematical theories and applications, impacting both theoretical research and practical problem-solving.
-
ModelCypher: Exploring LLM Geometry
Read Full Article: ModelCypher: Exploring LLM Geometry![ModelCypher: A toolkit for the geometry of LLMs (open source) [P]](https://www.tweakedgeek.com/wp-content/uploads/2025/12/featured-article-6349-300x171.png)
ModelCypher is an open-source toolkit designed to explore the geometry of small language models, challenging the notion that these models are inherently black boxes. It features cross-architecture adapter transfer and jailbreak detection using entropy divergence, implementing methods from over 46 recent research papers. Although the hypothesis that Wierzbicka's "Semantic Primes" would show unique geometric invariance was disproven, the toolkit reveals that distinct concepts have a high convergence across different models. The tools are documented with analogies to aid understanding, though they primarily provide raw metrics rather than user-friendly outputs. This matters because it provides a new way to understand and potentially improve language models by examining their geometric properties.