Related papers: Directed Homotopy in Non-Positively Curved Spaces

Diagrammatic sets are presheaves on a rich category of shapes, whose definition is motivated by combinatorial topology and higher-dimensional diagram rewriting. These shapes include representatives of oriented simplices, cubes, and positive…

Recent research has made impressive progress in large-scale multimodal pre-training. In the context of the rapid growth of model size, it is necessary to seek efficient and flexible methods other than finetuning. In this paper, we propose…

We propose a novel method for motion planning and illustrate its implementation on several canonical examples. The core novel idea underlying the method is to define a metric for which a path of minimal length is an admissible path, that is…

We formulate explicit predictions concerning the symmetry of optimal codes in compact metric spaces. This motivates the study of optimal codes in various spaces where these predictions can be tested.

Let $G/H$ be a closed, simply connected homogeneous manifold. Suppose every stable class of real vector bundles over $G/H$ contains a homogeneous bundle. Then, for any closed, simply connected smooth manifold $M$ homotopy equivalent to…

Prompt tuning has become a new paradigm for model tuning and it has demonstrated success in natural language pretraining and even vision pretraining. In this work, we explore the transfer of prompt tuning to multimodal pretraining, with a…

Deep learning models in medical imaging are susceptible to shortcut learning, relying on confounding metadata (e.g., scanner model) that is often encoded in image embeddings. The crucial question is whether the model actively utilizes this…

Despite the widespread adoption of higher-order mathematical structures such as hypergraphs, methodological tools for their analysis lag behind those for traditional graphs. This work addresses a critical gap in this context by proposing…

We show that the canonical map from the associative operad to the unital associative operad is a homotopy epimorphism for a wide class of symmetric monoidal model categories. As a consequence, the space of unital associative algebra…

Given a metrizable topological vector space, we can also use its von Neumann bornology or its bornology of precompact subsets to do analysis. We show that the bornological and topological approaches are equivalent for many problems. For…

This paper is the third paper of a series devoted to higher dimensional transition systems. The preceding paper proved the existence of a left determined model structure on the category of cubical transition systems. In this sequel, it is…

In condensed matter physics and related areas, topological defects play important roles in phase transitions and critical phenomena. Homotopy theory facilitates the classification of such topological defects. After a pedagogic introduction…

We present a toolchain for solving path planning problems for concentric tube robots through obstacle fields. First, ellipsoidal sets representing the target area and obstacles are constructed from labelled point clouds. Then, the nonlinear…

We present a robust model predictive control (MPC) framework for linear systems facing bounded parametric uncertainty and bounded disturbances. Our approach deviates from standard MPC formulations by integrating multi-step predictors, which…

We study real nonsingular projective cubic fourfolds up to deformation equivalence combined with projective equivalence and prove that they are classified by the conjugacy classes of involutions induced by the complex conjugation in the…

In this note, we answer positively a question by Belegradek and Kapovitch about the relation between rational homotopy theory and a problem in Riemannian geometry which asks that total spaces of which vector bundles over compact nonnegative…

In this paper, we aim to investigate the synchronization problem of dynamical systems, which can be of generic linear or Lipschitz nonlinear type, communicating over directed switching network topologies. A mild connectivity assumption on…

We show that basic homotopical notions such as homotopy sets and groups, connected and truncated maps, cellular constructions and skeleta, etc., extend to the setting of $(\infty,\infty)$-categories, as well as to presentable categories…

This paper is concerned with the study of the geometry of determinant line bundles associated to families of spectral triples parametrized by the moduli space of gauge equivalent classes of Hermitian connections on a Hermitian finite…

We explore the geometry behind the modular bootstrap and its image in the space of Taylor coefficients of the torus partition function. In the first part, we identify the geometry as an intersection of planes with the convex hull of moment…