geometry

Otary, a Python library for image and geometry processing, now has tutorials available to help users understand its capabilities.

This paper introduces Statistically Meaningful Geometry (SMG), a geometric framework for modeling over-parameterized learning systems as infinite-dimensional non-parametric Orlicz fiber bundles. It proposes that under out-of-distribution stimuli, the system undergoes a gauge symmetry break, leading to the emergence of new causal axes that can distinguish genuine scientific discovery from hallucinations.

The article summarizes the core learning methodology of world-class geometer and quantitative king James Simons: incubation by the subconscious, stripping away noise, and an extreme aversion to rote memorization, emphasizing deep thinking, leaving mental space, and cross-disciplinary learning through leveraging networks of genius.

This paper observes that token embeddings in small language models condense into a narrow cone-like subspace, a phenomenon termed embedding condensation, and proposes a dispersion loss to counteract it, improving generalization.

This paper studies the geometric properties of chain-of-thought trajectories in the hidden state space of transformers, introducing effective dimension and kinematic features to predict task hardness and solution correctness from early tokens.

Discussed the current state and future of AI in mathematics. Citing an example, ChatGPT 5.5 Pro autonomously solved the farthest pair problem in high-dimensional computational geometry, which had been stuck for years, demonstrating AI's potential in mathematical discovery.

An interactive web-based shader visualization that generates living geometry from a field equation, allowing users to explore and animate the forms.

Gorden_Sun has open-sourced the Math2GGB skill, which converts geometry math problems into interactive GeoGebra files. It supports three modes: static reproduction, clean interaction, and dynamic teaching, helping teachers digitize problems and assisting students in comprehension.

This paper introduces a normal-fan geometry for finite-horizon adversarial MDPs with fixed transitions, developing a face-crossing price that separates consequential from harmless non-stationarity. It shows that dynamic regret decomposes into intrinsic priced face motion plus within-face selection error.

This paper introduces SD-GPS, a solver-driven framework for geometry problem solving that uses autoformalization guided by solver feedback and verified theorem proposing to overcome bottlenecks in neuro-symbolic systems.

This paper investigates the geometric relationship between directions in language model activations that detect a behavior versus those that control it, finding that for hallucination detection they are nearly orthogonal (cosine ~0.12), while for output format they align perfectly, challenging a common assumption in mechanistic interpretability.

The tweet highlights a research finding that signals of physical plausibility can be extracted from the geometry of frozen image encoders without video training or physics supervision.

This paper provides a theoretical explanation for why diffusion models can generate clean samples without explicit noise-level conditioning, attributing it to high-dimensional geometry and analyzing why some model parameterizations succeed while others collapse.

Introduces FllumaOne, a multimodal CAD dataset of 100,000 samples generated by executable Python programs in the Flluma CAD system, providing feature trees, STEP geometry, point clouds, and language descriptions to support editable CAD research.

VeriGeo introduces a controllable geometry question generation framework that uses verification-guided reflection to ensure numerical and analytical consistency. The method produces high-quality synthetic data, achieving state-of-the-art results on GeoQA and strong performance on PGPS9K and MathVista-GPS.

This paper introduces SGR-BIM, a graph-driven semantic reasoning framework that dynamically aligns regulatory intent with BIM geometry to automate geometry-intensive compliance checks, achieving 84.3% accuracy on fire safety code queries.

This paper from the Axiom team investigates the rarity of lattice triangles, presenting a mathematical result on the distribution of convex lattice polygons.

This paper analyzes on-policy distillation (OPD), finding that OPD updates are sparse, distributed across layers and FFN-heavy, and retain geometric properties distinct from dense parameter rewriting. The sparse structure is operationally useful, but sparsity-inducing SGD underperforms AdamW due to heterogeneous gradient scales.

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 investigates whether spatial geometry improves language-agent memory recall, demonstrating that geometry must lead recall over recency and importance, and that a ray-tracing visibility predicate is crucial for occlusion handling in 3D voxel worlds.