My Apologies
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Transcript of 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
A surface can have finite Lebesgue area and still occupy positive measure
Lebesgue Area of a Surface
What Is Area?
Competing definitions of Area
Definitive Work on Area: “Currents and Area”
Sobolev functions can have “small” supports in the sense topology, but “large” in the sense of analysis
More Pathology
We have a solution for n=3, p>2.(First Proof)
We have a solution for n=3, p>2. (Cont.)
We have a solution for n=3, p>2. (Cont.)
The Result Can Be Improved
The Proof
The Proof Requires the following ideas -- Cont
Bagby-Gauthier Result
Proof of Bagby-Gauthier result
Continuity of Sobolev functions on subspaces
The proof of Bagby-Gauthier is concluded
The Main Result
The proof offers some hope for solving the problem inits greatest generality
Brief Outline of the Proof -- Notation
Recall Two Equivalent Definitions of Sobolev Space
The Precise Representative of a Sobolev Function
Quasi-Continuity & The Coarea Formula
The Fibers of Q are Horizontal (m+1)-planes
A Sobolev Function is a continuous Sobolev function on a.e. Fiber of Q
The Basic Idea
The Basic Idea – Cont
Linked Spheres in Full Generality
Topological Degree
(n-1)-manifolds cannot contain linked spheres
The Main Theorem -- Concluded