Geometric Measure Theory

Frank Morgan is the Dennis Meenan '54 Third Century Professor of Mathematics at Williams College. He obtained his B.S. from MIT and his M.S. and Ph.D. from Princeton University. His research interest lies in minimal surfaces, studying the behavior and structure of minimizers in various settings. He has also written Riemannian Geometry: A Beginner's Guide, Calculus Lite, and most recently The Math Chat Book, based on his television program and column on the Mathematical Association of America Web site.

• 1. Geometric Measure Theory2. Measures3. Lipschitz Functions and Rectifiable Sets4. Normal and Rectifiable Currents5. The Compactness Theorem and the Existence of Area-Minimizing Surfaces6. Examples of Area-Minimizing Surfaces7. The Approximation Theorem8. Survey of Regularity Results9. Monotonicity and Oriented Tangent Cones10. The Regularity of Area-Minimizing Hypersurfaces11. Flat Chains Modulo v, Varifolds, and (M,E,)-Minimal Sets12. Miscellaneous Useful Results13. Soap Bubble Clusters14. Proof of Double Bubble Conjecture15. The Hexagonal Honeycomb and Kelvin Conjectures16. Immiscible Fluids and Crystals17. Isoperimetric Theorems in General Codimension18. Manifolds with Density and Perelman's Proof of the Poincaré Conjecture19. Double Bubbles in Spheres, Gauss Space, and Tori20. The Log-Convex Density Conjecture21. Solutions to Exercises

"Morgan’s book offers the best access I know into this difficult subject. It won’t make the reader an expert, but it does open the door to more detailed treatments. An excellent bibliography is provided." --MAA Reviews