INSTRUCTOR INFORMATION:
Instructor: Boris Solomyak
Office: Padelford C-328, Phone: 685-1307.
E-mail: solomyak@math.washington.edu
Office hours: Monday 4-5, Tuesday 11-12 or by appointment.
Instructor: Tatiana Toro.
Office: Padelford C-343, Phone: 543-1173.
E-mail: toro@math.washington.edu
Office hours: Tuesday 11-12, Friday 1:30-2:30 or by appointment.
RESERVE LIST:
- Geometry of Sets and Measures in Euclidean Spaces, P. Mattila
- Measure Theory and Fine Properties of Functions, L. C. Evans and
R. F. Gariepy
COURSE DESCRIPTION: The course will consist of two parts. In the first part, taught by Boris Solomyak, we will develop the tools of geometric measure
theory and apply them to a wide range of examples, mostly of fractal nature,
such as purely unrectifiable sets. The second part, taught by Tatiana Toro,
will focus on the relationship between the densities of a measure and the
structure of its support. The properties of rectifiable sets, to be discussed
in the second part of the course, will contrast sharply with those of
the purely unrectifiable sets studied in the first part of the course.
PART I TOPICS:
- Review of measure theory
- Covering theorems and differentiation theorems for Radon measures
- Hausdorff measure, Hausdorff and Minkowski dimensions,
examples of fractals
- Potential-theoretic methods, Frostman's Lemma
- Projection theorems
- Extensions, applications, open problems
PART II TOPICS:
- Densities of measures
- Lipschitz maps (Rademacher's theorem)
- Tangent measures (Mastrand density theorem)
- Rectifiable sets
- Rectifiability and tangent measures
- Rectifiability and densities (Preiss's Theorem)
HOMEWORK DUE DATES:
- Monday, February 11 (Solomyak)
- Monday, March 17 (Toro)