GEOMETRIC MEASURE THEORY

MATH 582E - SPRING 2012

TIME AND PLACE:

MWF 12:30-1:20 in PDL C-401

INSTRUCTOR INFORMATION:

Instructor: Tatiana Toro
Office: Padelford C-343, Phone: 543-1173.
E-mail: toro "at" math.washington.edu
Office hours: Monday 11-12, Friday 10-11 or by appointment.

RESERVE LIST (overnight loan at the Math. Research Library):

• Geometry of sets and measures in Euclidean spaces: fractals and rectifiability by P. Mattila. Cambridge, 1995.
  
• Measure theory and fine properties of functions by Lawrence C. Evans and Ronald F. Gariepy. CRC Press, 1991.
  
• The geometry of fractal sets by K. Falconer. Cambridge, 1986.
  
• Fractal geometry: mathematical foundations and applications by K. Falconer. Wiley, 2003.
  

ELECTRONIC RESOURCES:

• Fractal Sets in Probability and Analysis by C. Bishop and Y. Peres
  This is a book soon to be published. The authors allowed downloading the Draft

This is the second part of a 2-quarter course in Geometric Measure Theory. The first part (Winter 2012) was taught taught by Boris Solomyak.

While the first part of the course focused on "fractal" structures, the second will be concerned with "rectifiable" ones. We will start by studying density theorems for Hausdorff and Radon measures. Then we will study Lipschitz maps and in particular Rademacher's theorem. We will introduce sets of locally finite perimeter as our main example of rectifiable set. These sets introduced by Caccioppoli and studied in detail by De Giorgi played a fundamental role in the development of geometric measure theory. More recently they have appeared naturally in the study of free boundary regularity problems and questions concerning boundary regularity of solutions to PDEs on non-smooth domains. We expect to have time to discuss, in some detail, the proof of the Traveling Salesman problem which appears in Bishop and Peres's new book Fractal Sets in Probability and Analysis.

We will use the books Geometry of sets and measures in Euclidean spaces: fractals and rectifiability by P. Mattila and Measure theory and fine properties of functions by L.C. Evans and R.F. Gariepy, as well as Fractal Sets in Probability and Analysis by C. Bishop and Y. Peres (chapter 8), which should be published soon.

Measure theory and integration at the level of graduate Real Analysis, 582D (Winter 2012) or instrctor's approval.

There will be several homework assignments. The grades will be based on homework and class participation. There will be no exams.
• Homework 1 (due Friday, April 20)
• Homework 2 (due Friday, May 11)
• Homework 3 (due Friday, June 1)