Mathematics / research
Geometric measure theory
Measure irregular sets and weak geometric objects through Hausdorff dimension, rectifiability, currents, and compactness.
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Module 01 research
Geometric measure theory
Measure irregular sets and weak geometric objects through Hausdorff dimension, rectifiability, currents, and compactness.
- 01
Hausdorff measures and dimension
Fix a geometric normalization and measure sets that occupy noninteger dimensions.
36 min - 02
Coverings, densities, and Vitali theorems
Use disjoint covering selections and compute local measure ratios with explicit exceptional points.
36 min - 03
Lipschitz maps and the area formula
Calculate geometric Jacobians and retain the multiplicity of a folded parametrization.
36 min - 04
The coarea formula and level sets
Disintegrate volume across fibers with the correct dimension and gradient factor.
36 min - 05
Rectifiable sets and approximate tangent planes
Separate almost-everywhere planar structure from singular crossings and measure multiplicity.
36 min - 06
Bounded variation and distributional gradients
Measure jumps and diffuse derivatives in an almost-everywhere function class.
36 min - 07
Sets of finite perimeter and reduced boundary
Choose an outward-normal convention and recover geometric boundary area from a distributional measure.
36 min - 08
Currents, boundaries, and Stokes’ theorem
Represent oriented geometry by functionals and track multiplicity and cancellation explicitly.
36 min - 09
Compactness and lower semicontinuity for currents
State the support and boundary bounds behind integral compactness and measure lost oscillation in a flat limit.
36 min - 10
Minimal surfaces and regularity thresholds
Separate mass minimization from stationarity and track hypersurface dimension at the first singular cone.
36 min