Geometric Measure Theory treats measure-theoretic properties
of geometrically defined sets of various dimensions. Some of the critical
notions are *Hausdorff**
measure, rectifiable sets,* and* rectifiable currents. * The k
dimensional Hausdorff (outer) measure

H^{ k}(A) gives, for every nonnegative number
k , a precise notion of the k dimensional size of
A for any subset A of a metric space X . In case X= R^{ 3} , C is a
smooth embedded curve, S is a smooth embedded surface, and U
is an open region. Then H^{ 1}(C) is the length of
C , H^{2} (S) is the area of S , and H^{
3}(U) is the volume or Lebesgue measure of
U . Also H^{2} (C)=H^{ 2}
(C)=H^{ 3}(S)=0 while H^{1}(S)=H^{ 1}(U)
=H^{2}(U)= oo. Noninteger
Hausdorff measures are useful for some fractals, e.g.

0 < H^{log2/log3(}Cantor
set) < oo . For integer k
, a subset M of X is called * k rectifiable*
if M is H^{ k }almost the countable union of Lipschitz images of subsets of R^{ k}. Even in a general
metric space X , one may associate to H^{ k} almost every
point a in M a “tangent” k plane Tan(M,a) .

By assigning, for H^{k} almost
every point a in a k
dimensional rectifiable set M, an integer multiplicity m(a)
and an orientation v(a) for Tan(M,a)
, one obtains a k dimensional *rectifiable chain *T*. * One
may integrate smooth k forms of R^{
n} on M . The boundary of T is then given
simply by d T(f) = T(df) for a k-1
form f , and the mass of T is the is J_{ M} m(a)dH^{ k}a . The *Compactness Theorem* of Federer-Fleming for rectifiable currents implies the
existence, for a given k-1 dimensional
rectifiable boundary, a k dimensional rectifiable current of
minimum mass. This provides a solution of the “Plateau Problem” in
general dimensions and in many metric space contexts.

These results have enjoyed new, more accessible proofs in the nineties based on
properties of slices of sets or currents by general affine
n-k planes. Also some of these notions have remarkably been
carried over to more general metric spaces and to chains with coefficients in
other groups.

We will try to present some of these new ideas in the course as well as many of the standard analytic and geometric constructions for rectifiable sets and currents.

Prerequesites for the course include some knowledge of basic analysis and measure theory as in Math 425, 426 and beginning topology as in Math 443.

We will meet initially MWF 10-10:50 in HB453 and discuss possible changes of time then.

BV
Compactness for Maps to a Metric Space

Compactness of
Rectifiable Currents in a Metric Space

F. Morgan, *Geometric Measure Theory*,* A Beginner's Guide,* Academic Press, 1988.

H. Federer, *Geometric Measure Theory*,
Springer-Verlag, 1970.

P. Mattilla, *Geometry of Sets and Measures in
Euclidean Spaces, *, 1995.

L. Simon,

B. White,

B. Kirchheim, *Rectifiable Metric Spaces, Local structure and Regularity of the Hausdorff Measures,* Proc. A.M.S., 121 (1994)113-123.

L. Ambrosio & B. Kirchheim,
*Rectifiable sets in metric and Banach spaces.* Math. Ann.318(2000), 527--555.

L. Ambrosio & B. Kirchheim, *Currents
in metric spaces.* Acta Math. 185 (2000), 1--80.

J. Heinenon, Lectures on analysis in Metric Spaces, Springer, 2000.

Robert Hardt Office: Herman Brown 430; Office hours: 11-12 MWF (and others by appt.),

Email: hardt@rice.edu, Telephone: ext 3280

http://math.rice.edu/~hardt/522S07/

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