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Concept
Geometric measure theory
Summary
In mathematics, geometric measure theory (GMT) is the study of geometric properties of sets (typically in Euclidean space) through measure theory. It allows mathematicians to extend tools from differential geometry to a much larger class of surfaces that are not necessarily smooth.
Geometric measure theory was born out of the desire to solve Plateau's problem (named after Joseph Plateau) which asks if for every smooth closed curve in there exists a surface of least area among all surfaces whose boundary equals the given curve. Such surfaces mimic soap films.
The problem had remained open since it was posed in 1760 by Lagrange. It was solved independently in the 1930s by Jesse Douglas and Tibor Radó under certain topological restrictions. In 1960 Herbert Federer and Wendell Fleming used the theory of currents with which they were able to solve the orientable Plateau's problem analytically without topological restrictions, thus sparking geometric measure theory. Later Jean Taylor after Fred Almgren proved Plateau's laws for the kind of singularities that can occur in these more general soap films and soap bubbles clusters.
The following objects are central in geometric measure theory:
Hausdorff measure and Hausdorff dimension
Rectifiable sets (or Radon measures), which are sets with the least possible regularity required to admit approximate tangent spaces.
Characterization of rectifiability through existence of approximate tangents, densities, projections, etc.
Orthogonal projections, Kakeya sets, Besicovitch sets
Uniform rectifiability
Rectifiability and uniform rectifiability of (subsets of) metric spaces, e.g. SubRiemannian manifolds, Carnot groups, Heisenberg groups, etc.
Connections to singular integrals, Fourier transform, Frostman measures, harmonic measures, etc
Currents, a generalization of the concept of oriented manifolds, possibly with boundary.
Flat chains, an alternative generalization of the concept of manifolds, possibly with boundary.
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Related courses (3)
MATH-731: Topics in geometric analysis I
The subject deals with differential geometry and its relation to global analysis, partial differential equations, geometric measure theory and variational principles to name a few.
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Introduction to classical Calculus of Variations and a selection of modern techniques.
MATH-688: Reading group in applied topology I
The focus of this reading group is to delve into the concept of the "Magnitude of Metric Spaces". This approach offers an alternative approach to persistent homology to describe a metric space across
Related concepts (3)
Hausdorff measure
In mathematics, Hausdorff measure is a generalization of the traditional notions of area and volume to non-integer dimensions, specifically fractals and their Hausdorff dimensions. It is a type of outer measure, named for Felix Hausdorff, that assigns a number in [0,∞] to each set in or, more generally, in any metric space. The zero-dimensional Hausdorff measure is the number of points in the set (if the set is finite) or ∞ if the set is infinite.
Isoperimetric inequality
In mathematics, the isoperimetric inequality is a geometric inequality involving the perimeter of a set and its volume. In -dimensional space the inequality lower bounds the surface area or perimeter of a set by its volume , where is a unit sphere. The equality holds only when is a sphere in . On a plane, i.e. when , the isoperimetric inequality relates the square of the circumference of a closed curve and the area of a plane region it encloses. Isoperimetric literally means "having the same perimeter".
Measure (mathematics)
In mathematics, the concept of a measure is a generalization and formalization of geometrical measures (length, area, volume) and other common notions, such as magnitude, mass, and probability of events. These seemingly distinct concepts have many similarities and can often be treated together in a single mathematical context. Measures are foundational in probability theory, integration theory, and can be generalized to assume negative values, as with electrical charge.