Series: Penn State Logic Seminar Date: Tuesday, November 4, 2003 Time: 2:30 - 3:45 PM Place: 324 Sackett Building (note unusual location) Speaker: Ksenija Simic, Carnegie Mellon University, Mathematics Title: The Mean Ergodic Theorem in Weak Subsystems of Second Order Arithmetic Abstract: The mean ergodic theorem states that for an appropriately defined measure preserving transformation T on a space X, the sequence S_n=(1/n)sum_{k=0}^{n-1}f(T^k) converges in the L_2 norm for all f in L_2(X). Due to the restrictions second order arithmetic imposes, it is not possible to define T pointwise. Instead, we define it as a norm preserving linear operator on L_2(X). As it transpires, it is more convenient to state and prove the theorem for the more abstract case - that of Hilbert spaces, following the approach of Halmos. A number of results from Hilbert space theory then needs to be established, before proving the actual theorem. I will give a brief overview of some of these results, and focus on the proof of the mean ergodic theorem. Finally, I will show that the mean ergodic theorem is equivalent to arithmetic comprehension over the base theory RCA_0.