GEOMETRIC MEASURE THEORY

MATH 582D - WINTER 2012

TIME AND PLACE:

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

INSTRUCTOR INFORMATION:

Instructor: Boris Solomyak
Office: Padelford C-328, Phone: 685-1307.
E-mail: solomyak "at" math.washington.edu
Office hours: Monday 10-11, Tuesday 4-5 (a bit later on 1/10, 2/7, 3/6) 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 first part of a 2-quarter course in Geometric Measure Theory. The second part (Spring 2012) will be taught by Tatiana Toro.

The goal of the course is to study the geometric structure of general Borel sets and Borel measures in the Euclidean space. Such sets and measures can be very irregular, like Cantor-type sets, non-rectifiable curves having tangent nowhere, etc., in short, sets to which the descriptive term ``fractal'' applies. An abundance of such sets comes from dynamical systems, e.g. Julia sets for rational functions, and from Probability, e.g. paths of the Brownian motion.

On the other hand, there are very general curve- and surface-like objects, which are called rectifiable sets and measures. These are very important in the modern theory of PDEs. In particular, they appear naturally in the study of free boundaries. Furthermore, in recent years tools from harmonic analysis and geometric measure theory have been combined to solve longstanding conjectures (e.g Vitushkin's conjecture) . Some of these recent developments will be discussed in the class (in part II).

Roughly speaking, the first part of the course will be more ``fractal,'' and the second part, more ``rectifiable.'' Of course, there are many important tools and techniques which are common, so the interested students should plan on taking both parts.

I am planning to use the book by C. Bishop and Y. Peres Fractal Sets in Probability and Analysis (a subset of the first five chapters), which should be published soon. We will also use the book Geometry of sets and measures in Euclidean spaces: fractals and rectifiability by P. Mattila, but the lectures will not follow it very closely.

ADDITIONAL ELECTRONIC RESOURCES:

• Fractals with positive length and zero Buffon needle probability by Y. Peres, K. Simon and B. Solomyak
• Projecting the one-dimensional Sierpinski gasket by R. Kenyon

Measure theory and integration at the level of graduate Real Analysis.

• summary pdf file

There will be several homework assignments. The grades will be based on homework and class participation. There will be no exams.
• Homework 1 (due F, Jan 20)
• Homework 2 (due F, Feb 3)
• Homework 3 (due F, Feb 17)
• Homework 4 (due F, Mar 9)