Related papers: TokenBlowUp: Resolving Representational Singularit…
We provide a numerical validation method of blow-up solutions for finite dimensional vector fields admitting asymptotic quasi-homogeneity at infinity. Our methodology is based on quasi-homogeneous compactifications containing a new…
We give a combinatorial algorithm for equivariant embedded resolution of singularities of a toric variety defined over a perfect field. The algorithm is realized by a finite succession of blowings-up with smooth invariant centres that…
Large language models (LLMs) have proven to be very capable, but access to frontier models currently relies on inference providers. This introduces trust challenges: how can we be sure that the provider is using the model configuration they…
The quantum plane $A=\mathbb{C}_\rho[x,y,z]$ with $\rho$ a root of unity has singularities in its representation variety $\mathbf{trep}_nA$ and its center $Z(A)$. Using the technique of a noncommutative blow-up, we prove that this technique…
Multimodal Large Language Models (MLLMs) have demonstrated exceptional capabilities in processing vision-language tasks. One of the crux of MLLMs lies in vision tokenization, which involves efficiently transforming input visual signals into…
While Diffusion Large Language Models (dLLMs) offer structural advantages for global planning, efficiently verifying that they arrive at correct answers via valid reasoning traces remains a critical challenge. In this work, we propose a…
Classical unsupervised learning methods like clustering and linear dimensionality reduction parametrize large-scale geometry when it is discrete or linear, while more modern methods from manifold learning find low dimensional representation…
The rapid emergence of Large Language Models (LLMs) has precipitated a profound paradigm shift in Artificial Intelligence, delivering monumental engineering successes that increasingly impact modern society. However, a critical paradox…
Recent developments in local quantum physics have led to revolutionary conceptual changes in the thinking about a more intrinsic formulation and in particular about unexpected aspects of localized degrees of freedom. This paradigmatic…
Geometry problems are a crucial testbed for AI reasoning capabilities. Most existing geometry solving systems cannot express problems within a unified framework, thus are difficult to integrate with other mathematical fields. Besides, since…
Latent space geometry provides a rigorous and empirically valuable framework for interacting with the latent variables of deep generative models. This approach reinterprets Euclidean latent spaces as Riemannian through a pull-back metric,…
Recent advances in product bundling have leveraged multimodal information through sophisticated encoders, but remain constrained by limited semantic understanding and a narrow scope of knowledge. Therefore, some attempts employ In-context…
The use of geometric and symmetry techniques in quantum and classical information processing has a long tradition across the physical sciences as a means of theoretical discovery and applied problem solving. In the modern era, the emergent…
Processing long visual token sequences poses a significant computational burden on Multimodal Large Language Models (MLLMs). While token pruning offers a path to acceleration, we find that current methods, while adequate for general…
The objective of this paper is to discuss invariants of singularities of algebraic schemes over fields of positive characteristic, and to show how they yield the simplification of singularities. We focus here on invariants which arise in an…
In this paper we investigate new applications of the blow-up desingularization method in the context of singular Riemannian foliations. First, we relate the dynamics of such a foliation, which is governed by the so-called Molino sheaf, with…
Understanding how Large Language Models (LLMs) perform complex reasoning and their failure mechanisms is a challenge in interpretability research. To provide a measurable geometric analysis perspective, we define the concept of the…
In this paper, blow-up solutions of autonomous ordinary differential equations (ODEs) which are unstable under perturbations of initial points, referred to as saddle-type blow-up solutions, are studied. Combining dynamical systems machinery…
Mathematical geometric reasoning is essential for scientific discovery and educational development, requiring precise logic and rigorous formal verification. While recent advances in Multimodal Large Language Models (MLLMs) have improved…
Katz and Vafa showed how charged matter can arise geometrically by the deformation of ADE-type orbifold singularities in type IIa, M-theory, and F-theory compactifications. In this paper we use those same basic ingredients, used there to…