Current frontiers in machine learning, data science, and, more broadly, artificial intelligence reveal the limits of purely predictive approaches and motivate a shift toward decentralized, scalable, and causal systems. Such systems require processing and learning on increasingly complex networks. A promising heterogeneous toolbox, loosely grouped under the name of Topological Deep Learning (TDL), aims to design deep architectures that integrate ideas from algebraic topology, non-Euclidean geometry, and category theory to address this complexity. In Dr. Battiloro’s approach to TDL, the basic units of a network are cells, which generalize graph nodes. A cell may represent, for example, a single agent or a group of agents in an agentic AI system, a neuron or a brain region in a neural circuit, or a sensor or sensor type in an environmental monitoring network. Cells can be organized hierarchically and exhibit rich intra-and inter-cell interaction patterns. In this seminar, Dr. Battiloro will (1) introduce and discuss TDL’s current landscape and explain why developing a modern, coherent language for it matters broadly, (2) argue that this language should be grounded in the theory of poset sheaves, (3) briefly highlight goals TDL has already achieved—such as inferring higher-order, hierarchical goal-driven interactions in data or jointly modeling and relating subjective causal structures across cells—and (4) outline an ambitious sheaf-centric research pathway for TDL.