Federer defines the on currents (via Stokes’ theorem), compactness theorems (essential for solving variational problems), and the flat norm , which measures how close two currents are.
Before 1969, analyzing geometric shapes often required them to be "smooth" (differentiable). Federer’s work developed a rigorous framework allowing mathematicians to apply analytical tools (like calculus) to "rough" or discontinuous shapes. The book blends:
—generalized surfaces that allow mathematicians to solve the "Plateau Problem" (finding the surface of least area for a given boundary) in any dimension without restrictive topological assumptions. Key technical highlights from the text include:
If you are looking for a review of the text or a "PDF" version for study, here is the breakdown of what to expect: federer geometric measure theory pdf
) maps. Rectifiable sets serve as the measure-theoretic analog to smooth manifolds. Federer explores their properties, proving that they possess approximate tangent planes almost everywhere, allowing for the application of calculus on highly irregular surfaces. 3. The Theory of Currents
It is the definitive source for the proofs of major theorems in GMT.
Despite its reputation for being difficult to read, Federer's text is essential for several reasons: Federer defines the on currents (via Stokes’ theorem),
Traditional calculus and differential geometry rely heavily on smoothness. They assume that surfaces are differentiable, curves are well-behaved, and manifolds do not have sharp, unpredictable singularities. However, physical reality and optimization problems rarely obey these clean rules. Soap films pop and merge, fractures propagate through materials, and optimal transport paths branch unpredictably.
: Chapters 1 and 2 cover Grassmann algebra (tensor products, exterior algebra) and General measure theory (Borel sets, Radon measures) to establish the necessary formal framework.
Geometry of Sets and Measures in Euclidean Spaces: Fractals and Rectifiability Federer explores their properties, proving that they possess
For centuries, mathematicians sought a general solution to Plateau's Problem: Does every closed curve in space bound a surface of minimal area?
The book provides the analytical tools necessary to understand why soap films take the shapes they do in higher dimensions. Modern Alternatives and Supplements