Dilators, Reflection, and Forcing: A Proof-Theoretic Analysis of $\Pi^1_1-\mathsf{CA}_0$

In this talk I will sketch a new approach to ordinal analysis of $\Pi^1_1-\mathsf{CA}_0$ that uses the interplay of the methods of functorial proof theory and iterated reflection. An interesting feature of the approach is that it completely sidesteps predicative cut elimination.