Geometric problems require the passage from very natural energy bounds on curvature to very unnatural pointwise bounds. The long-standing approach has been to use the technology of analysis, yet this relies on the persistence of a geometric-analytic nexus, expressed concisely as the Sobolev constant, that is dicult or impossible to control in nature. In this talk we discuss a more intrinsically geometric way of approaching regularity questions on critical 4-manifolds.

Advisor: Prof. Zhiqin Lu

Classical ergodic averages give good norm approximations, but these averages are not necessarily giving the best norm approximation among all possible averages. We consider
1) what the optimal Cesaro norm approximation can be in terms of the transformation and the function,
2) when these optimal Cesaro norm approximations are comparable to the norm of the usual ergodic average, and
3) oscillatory behavior of these norm approximations.