Mathematicians study many different mathematical objects, and some of these objects are more complicated than others. Combining ideas from different areas of mathematical logic, from computability theory, descriptive set theory, and infinitary model theory, the Scott analysis is a way of measuring and understanding the complexity of mathematical objects. The Scott analysis is robust in the sense that it captures several different kinds of complexity that all coincide: the complexity of describing an object, the complexity of identifying two copies of an object, and the complexity of an object's internal structure. Though there remain many questions, the Scott analysis has now been well-developed for the case of discrete structures such as many structures appearing in algebra. More recently there has been increasing interest in studying continuous structures such as those appearing in analysis, a setting in which we do not have a robust Scott analysis, as there are further complication which do not arise in the discrete setting. This project will develop a robust Scott analysis in this continuous setting, including applications, while also further applying developing the Scott analysis in the more classical discrete setting. The long-term goals are to give a more rigorous and formal understanding of why certain mathematical questions are difficult or even impossible to solve, and what the barriers are to solving them. This project includes the training of undergraduate and gr