Analysis

Learn Geometric Measure Theory

Surfaces built out of measure theory, so that a minimiser exists before anyone knows it is smooth. Covers Hausdorff measure and rectifiability, the area and coarea formulas, sets of finite perimeter, currents and varifolds, and why regularity holds up to dimension eight and fails after it.

Free to start · adaptive placement finds your level · reviews timed to your own forgetting.

What you'll learn

30 lessons in Geometric Measure Theory

Hausdorff measureBox dimension and contentDensities and density pointsThe Besicovitch covering theoremLipschitz maps and Rademacher's theoremThe area formulaThe coarea formulaRectifiable setsPurely unrectifiable setsTangent measuresSets of finite perimeterDe Giorgi's structure theoremIsoperimetry and Brunn-MinkowskiPlateau's problemCurrents, mass and flat normThe Federer-Fleming compactness theoremVarifolds and first variationRegularity and the singular setThe Assouad dimensionBesicovitch sets and KakeyaThe Whitney extension theoremThe analyst's travelling salesman theoremUniform rectifiabilityReifenberg flatnessSets of positive reach and the Steiner formulaThe Wulff shapeThe quantitative isoperimetric inequalityThe double bubble theoremMean curvature flowThe dimension of Brownian paths
How Erudia teaches

Built to be understood — and remembered.

Every idea is taught with motivation and a worked example before the drills, and an FSRS spaced-repetition engine schedules each review for the day its model predicts you would forget it. A short placement check finds what you already know, so you start Geometric Measure Theory exactly where it's useful.

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