Explore the mathematical foundations of well-posedness for Hamilton-Jacobi equations in the context of linearly controlled gradient flows in this lecture by Richard Kraai. Delve into the intricate connections between probability theory, optimization techniques, and network science as part of the NETWORKS program. Gain insights into the applications of these theoretical concepts in fields such as finance, large data set analysis, rare event modeling, and social networks. Examine the structural properties of random graphs, dynamics on networks, interacting particle systems, and percolation phenomena. Discover how this research contributes to the broader goals of the NETWORKS project, which aims to apply theoretical aspects of probability theory, computer science, and statistics to real-life scenarios.