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Part I. Introduction: 1. The isoperimetric problem 2. The isoperimetric inequality in the plane 3. Preliminaries 4. Bibliographic notes Part II. Differential Geometric Methods: 1. The C2 uniqueness theory 2. The C1 isoperimetric inequality 3. Bibliographic notes Part III. Minkowski Area and Perimeter: 1. The Hausdorff metric on compacta 2. Minkowski area and Steiner symmetrization 3. Application: the Faber-Krahn inequality 4. Perimeter 5. Bibliographic notes Part IV. Hausdorff Measure and Perimeter: 1. Hausdorff measure 2. The area formula for Lipschitz maps 3. Bibliographic notes Part V. Isoperimetric Constants: 1. Riemannian geometric preliminaries 2. Isoperimetric constants 3. Discretizations and isoperimetric inequalities 4. Bibliographic notes Part VI. Analytic Isoperimetric Inequalities: 1. L2-Sobolev inequalities 2. The compact case 3. Faber-Kahn inequalities 4. The Federer-Fleming theorem: the discrete case 5. Sobolev inequalities and discretizations 6. Bibliographic notes Part VII. Laplace and Heat Operators: 1. Self-adjoint operators and their semigroups 2. The Laplacian 3. The heat equation and its kernels 4. The action of the heat semigroup 5. Simplest examples 6. Bibliographic notes Part VIII. Large-Time Heat Diffusion: 1. The main problem 2. The Nash approach 3. The Varopoulos approach 4. Coulhon's modified Sobolev inequality 5. The denoument: geometric applications 6. Epilogue: the Faber-Kahn method 7. Bibliographic notes Bibliography.