In this work, we investigate the geometry of the k-th homology embedding and focus on cases reminiscent of spectral clustering. Namely, we analyze the connected sum of manifolds as a perturbation of the direct sum of their homology embeddings.

Identifying and comparing topological features, particularly cycles, across different topological objects remains a fundamental challenge in persistent homology and topological data analysis. This work introduces a novel framework for constructing cycle communities through two complementary approaches. First, a dendrogram-based methodology leverages merge-tree algorithms to construct hierarchical representations of homology classes from persistence intervals. The Wasserstein distance on merge trees is introduced as a metric for comparing dendrograms, establishing connections to hierarchical clustering frameworks. Through simulation studies, the discriminative power of dendrogram representations for identifying cycle communities is demonstrated. Second, an extension of Stratified Gradient Sampling simultaneously learns multiple filter functions that yield cycle barycenter functions capable of faithfully reconstructing distinct sets of cycles. The set of cycles each filter function can reconstruct constitutes cycle communities that are non-overlapping and partition the space of all cycles. Together, these approaches transform the problem of cycle matching into both a hierarchical clustering and topological optimization framework, providing principled methods to identify similar topological structures both within and across groups of topological objects.

Efficient transfer learning has shown remarkable performance in tuning large-scale vision-language models (VLMs) toward downstream tasks with limited data resources. The key challenge of efficient transfer lies in adjusting image-text alignment to be task-specific while preserving pre-trained general knowledge.

We propose an information-topological framework in which cycle closure is the fundamental mechanism of memory and consciousness. Memory is not a static store but the ability to re-enter latent cycles in neural state space, with invariant cycles serving as carriers of meaning by filtering order-specific noise and preserving what persists across contexts. The dot-cycle dichotomy captures this: transient dots scaffold exploration, while nontrivial cycles encode low-entropy content invariants that stabilize memory. Biologically, polychronous neural groups realize 1-cycles through delay-locked spiking reinforced by STDP, nested within theta-gamma rhythms that enforce boundary cancellation. These micro-cycles compose hierarchically, extending navigation loops into general memory and cognition. The perception-action cycle introduces high-order invariance: closure holds even across sense-act alternations, generalizing ancestral homing behavior. Sheaf-cosheaf duality formalizes this process: sheaves glue perceptual fragments into global sections, cosheaves decompose global plans into actions and closure aligns top-down predictions with bottom-up cycles. Consciousness then arises as the persistence of high-order invariants that integrate (unity) yet differentiate (richness) across contexts. We conclude that cycle is all you need: persistent invariants enable generalization in non-ergodic environments with long-term coherence at minimal energetic cost.

Understanding the structure of the homology embedding can thus disclose geometric or topological information from the data.

We propose a unified framework based on persistent homology (PH) to characterize both local and global structures in disordered systems. It can simultaneously generate local and global descriptors using the same algorithm and data structure, and has shown to be highly effective and interpretable in predicting particle rearrangements and classifying global phases. We also demonstrated that using a single variable enables a linear SVM to achieve nearly perfect three-phase classification. Inspired by this discovery, we define a non-parametric metric, the Separation Index (SI), which not only achieves this classification without sacrificing significant performance but also establishes a connection between particle environments and the global phase structure. Our methods provide an effective framework for understanding and analyzing the properties of disordered materials, with broad potential applications in materials science and even wider studies of complex systems.

We study the problem of path planning with soft homology constraints on a surface topologically equivalent to a disk with punctures. Specifically, we propose an algorithm, named $\Hstar$, for the efficient computation of a path homologous to a user-provided reference path. We show that the algorithm can generate a suite of paths in distinct homology classes, from the overall shortest path to the shortest path homologous to the reference path, ordered both by path length and similarity to the reference path. Rollout is shown to improve the results produced by the algorithm. Experiments demonstrate that $\Hstar$ can be an efficient alternative to optimal methods, especially for configuration spaces with many obstacles.

Topological Data Analysis (TDA) allows us to extract powerful topological and higher-order information on the global shape of a data set or point cloud. Tools like Persistent Homology or the Euler Transform give a single complex description of the global structure of the point cloud. However, common machine learning applications like classification require point-level information and features to be available. In this paper, we bridge this gap and propose a novel method to extract node-level topological features from complex point clouds using discrete variants of concepts from algebraic topology and differential geometry. We verify the effectiveness of these topological point features (TOPF) on both synthetic and real-world data and study their robustness under noise.

Navigating mobile robots in social environments remains a challenging task due to the intricacies of human-robot interactions. Most of the motion planners designed for crowded and dynamic environments focus on choosing the best velocity to reach the goal while avoiding collisions, but do not explicitly consider the high-level navigation behavior (avoiding through the left or right side, letting others pass or passing before others, etc.). In this work, we present a novel motion planner that incorporates topology distinct paths representing diverse navigation strategies around humans. The planner selects the topology class that imitates human behavior the best using a deep neural network model trained on real-world human motion data, ensuring socially intelligent and contextually aware navigation. Our system refines the chosen path through an optimization-based local planner in real time, ensuring seamless adherence to desired social behaviors. In this way, we decouple perception and local planning from the decision-making process. We evaluate the prediction accuracy of the network with real-world data. In addition, we assess the navigation capabilities in both simulation and a real-world platform, comparing it with other state-of-the-art planners. We demonstrate that our planner exhibits socially desirable behaviors and shows a smooth and remarkable performance.