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My Apologies

My Apologies. Thank You. Some Philosophy. Some Philosophy -- Continued. Some Philosophy -- Continued. Some Philosophy -- Continued. Some Philosophy -- Continued. The Problem. Some Basic Observations. Some Basic Observations -- Continued. Our problem demands that M be pathological.

sean-sellers
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My Apologies

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  1. My Apologies

  2. Thank You

  3. Some Philosophy

  4. Some Philosophy -- Continued

  5. Some Philosophy -- Continued

  6. Some Philosophy -- Continued

  7. Some Philosophy -- Continued

  8. The Problem

  9. Some Basic Observations

  10. Some Basic Observations -- Continued

  11. Our problem demands that M be pathological

  12. A surface can have finite Lebesgue area and still occupy positive measure

  13. Lebesgue Area of a Surface

  14. What Is Area?

  15. Competing definitions of Area

  16. Definitive Work on Area: “Currents and Area”

  17. Sobolev functions can have “small” supports in the sense topology, but “large” in the sense of analysis

  18. More Pathology

  19. We have a solution for n=3, p>2.(First Proof)

  20. We have a solution for n=3, p>2. (Cont.)

  21. We have a solution for n=3, p>2. (Cont.)

  22. The Result Can Be Improved

  23. The Proof

  24. The Proof Requires the following ideas -- Cont

  25. Bagby-Gauthier Result

  26. Proof of Bagby-Gauthier result

  27. Continuity of Sobolev functions on subspaces

  28. The proof of Bagby-Gauthier is concluded

  29. The Main Result

  30. The proof offers some hope for solving the problem inits greatest generality

  31. Brief Outline of the Proof -- Notation

  32. Recall Two Equivalent Definitions of Sobolev Space

  33. The Precise Representative of a Sobolev Function

  34. Quasi-Continuity & The Coarea Formula

  35. The Fibers of Q are Horizontal (m+1)-planes

  36. A Sobolev Function is a continuous Sobolev function on a.e. Fiber of Q

  37. The Basic Idea

  38. The Basic Idea – Cont

  39. Linked Spheres in Full Generality

  40. Topological Degree

  41. (n-1)-manifolds cannot contain linked spheres

  42. The Main Theorem -- Concluded

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