1. The Square Variation of Rearranged Fourier Series Allison Lewko Mark Lewko Columbia University Institute for Advanced Study TexPoint fonts used in EMF. Read the TexPoint manual before you delete this box.: AA A A AAAA

2. Background on Orthonormal Systems

4. Sensitivity to Ordering Would imply “Yes” above

5. Known Results For Reorderings

6. Variation Operators

7. Comparing Maximal and Variation Operators

8. Variation Results for the Trigonometric System

9. What Tools Do We Have to Analyze Variation?

10. Dyadic Intervals Arbitrary subinterval is contained in dyadic interval of comparable length (approx.) Arbitrary subinterval can be decomposed into dyadic pieces

11. How Do We Reorder?

12. From Selectors to Fixed Size Subsets

13. Structure of the Proof

14. Reducing to a Sub-Level of Intervals

15. Tool for Controlling Smaller Intervals: Orlicz Space Norms

16. Orlicz Space Norms

17. Proof of Decomposition Property

18. Proof of Decomposition Continued

19. Deriving L p, L 2 bounds for Decomposition

20. Deriving L p, L 2 bounds from ¡ K (contd.)

21. Getting from ¡ K Bounds to V 2 Bounds

22. Controlling ¡ K Norms by Probabilistic Estimates

23. Controlling the Supremum of a Random Process

24. Generic Chaining

25. Covering Numbers

26. Strategy for our Base Estimates

27. Further Improving the Bounds

28. High-Level Recap of Proof Lots of details swept under the rug!

29. Remaining Questions

30. Other Implications of Variational Quantities

32. Implications of Variational Quantities (contd.)

33. Thanks! Questions?