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