One of the hardest challenges faced by the systems community stems from the exponential explosion of data, fueled by recent advances in sensing technology. During the past few years a large research effort has been devoted to developing computationally tractable methods that seek to mitigate the ``curse of dimensionality" by exploiting sparsity. The goals of this course are: 1) provide a quick introduction to the subject for people in the systems community faced with ``big data" and scaling problems, and 2) serve as a ``quick reference" guide for researchers, summarizing the state of the art . Part I of the course covers the issue of handling large data sets and sparsity priors in systems identification, model (in)validation and control, presenting techniques that exploit a connection to semi-algebraic geometry, rank minimization and matrix completion. Several applications of these techniques will be discussed, including control and filter design subject to information flow constraints, subspace clustering and time-series classification, including activity recognition and anomaly detection. Part II of the course focuses on recently developed Generative AI tools. In particular we will discuss how these new techniques can be applied to identification and control problems, what performance guarantees are currently available and open problems where a systems perspective can help. Topics include: • Review of convex optimization and Linear Matrix Inequalities • Promoting sparsity via convex optimization. Convex surrogates for cardinality and rank • Fast algorithms for rank and cardinality minimization • Fast, scalable algorithms for Semi-Definite Programs that exploit sparsity • Sparsity in Systems Identification: • Identification of LTI systems with missing data and outliers • Identification of Switched Linear and Wiener Systems • Identification of sparse networks • Connections to Machine Learning: subspace clustering and manifold embedding • Applications: Time series classification from video data, fault detection, actionable information extraction from large data sets, nonlinear dimensionality reduction • GenAI and Control: • Diffusion models: theory and applications to trajectory planning in uncertain environments • Transformers and applications to filtering • Selective state space models (Mamba)