Effective exploration is a vital component in the success of robotic applications in ocean and space exploration, environmental monitoring, and search and rescue tasks. This talk presents a novel formulation of exploration that permits optimality criteria and performance guarantees for robotic exploration tasks. We define the problem of exploration as a coverage problem on continuous (infinite-dimensional) spaces based on ergodic theory and derive control methods that satisfy various notions of optimality and robustness such as asymptotic coverage, set-invariance, time-optimality, and reachability in exploration tasks. Last, we demonstrate successful execution of the approach on a range robotic systems and present an outlook on novel directions.

The Langlands program predicts some surprising connections between number theory and representation theory, most of which are still conjectural. In recent years, powerful new tools in non-archimedean geometry have led to dramatic progress on these questions. One fascinating aspect of these new tools is that in a certain precise sense, they allow us to treat a fixed prime number p as a variable, which opens the door to importing ideas from the geometric Langlands program into more arithmetic settings. I will give a historically motivated introduction to this circle of ideas, and give a big-picture view of some of the most exciting recent and ongoing developments.