Preface.- 1 Removable Sets and Analytic Capacity.- 1.1 Removable Sets.- 1.2 Analytic Capacity.- 2 Removable Sets and Hausdor Measure.- 2.1 Hausdor Measure and Dimension.- 2.2 Painleve's Theorem.- 2.3 Frostman's Lemma.- 2.4 Conjecture & Refutation: The Planar Cantor Quarter Set.- 3 Garabedian Duality for Hole-Punch Domains.- 3.1 Statement of the Result and an Initial Reduction.- 3.2 Interlude: Boundary Correspondence for H1(U).- 3.3 Interlude: Some F. & M. Riesz Theorems.- 3.4 Construction of the Boundary Garabedian Function.- 3.5 Construction of the Interior Garabedian Function.- 3.6 A Further Reduction.- 3.7 Interlude: Some Extension and Join Propositions.- 3.8 Analytically Extending the Ahlfors and Garabedian Functions.- 3.9 Interlude: Consequences of the Argument Principle.- 3.10 An Analytic Logarithm of the Garabedian Function.- 4 Melnikov and Verdera's Solution to the Denjoy Conjecture.- 4.1 Menger Curvature of Point Triples.- 4.2 Melnikov's Lower Capacity Estimate.- 4.3 Interlude: A Fourier Transform Review.- 4.4 Melnikov Curvature of Some Measures on Lipschitz Graphs.- 4.5 Arclength & Arclength Measure: Enough to Do the Job.- 4.6 The Denjoy Conjecture Resolved Affirmatively.- 4.7 Conjecture & Refutation: The Joyce-Morters Set.- 5 Some Measure Theory.- 5.1 The Caratheodory Criterion and Metric Outer Measures.- 5.2 Arclength & Arclength Measure: The Rest of the Story.- 5.3 A Vitali Covering Lemma and Planar Lebesgue Measure.- 5.4 Regularity Properties of Hausdor Measures.- 5.5 The Besicovitch Covering Lemma and Lebesgue Points.- 6 A Solution to Vitushkin's Conjecture Modulo Two Difficult Results.- 6.1 Statement of the Conjecture and a Reduction.- 6.2 Cauchy Integral Representation.- 6.3 Estimates of Truncated Cauchy Integrals.- 6.4 Estimates of Truncated Suppressed Cauchy Integrals.- 6.5 Vitushkin's Conjecture Resolved Affirmatively Modulo Two Difficult Results.- 6.6 Postlude: The Original Vitushkin Conjecture.- 7 The T(b) Theorem of Nazarov, Treil, and Volberg.- 7.1 Restatement of the Result.- 7.2 Random Dyadic Lattice Construction.- 7.3 Lip(1)-Functions Attached to Random Dyadic Lattices.- 7.4 Construction of the Lip(1)-Function of the Theorem.- 7.5 The Standard Martingale Decomposition.- 7.6 Interlude: The Dyadic Carleson Imbedding Inequality.- 7.7 The Adapted Martingale Decomposition.- 7.8 Bad Squares and Their Rarity.- 7.9 The Good/Bad-Function Decomposition.- 7.10 Reduction to the Good Function Estimate.- 7.11 A Sticky Point, More Reductions, and Course Setting.- 7.12 Interlude: The Schur Test.- 7.13 G1: The Crudely Handled Terms.- 7.14 G2: The Distantly Interacting Terms.- 7.15 Splitting Up the G3 Terms.- 7.16 Gterm 3 : The Suppressed Kernel Terms.- 7.17 Gtran 3 : The Telescoping Terms.- 8 The Curvature Theorem of David and Leger.- 8.1 Restatement of the Result and an Initial Reduction.- 8.2 Two Lemmas Concerning High Density Balls.- 8.3 The Beta Numbers of Peter Jones.- 8.4 Domination of Beta Numbers by Local Curvature.- 8.5 Domination of Local Curvature by Global Curvature.- 8.6 Selection of Parameters for the Construction.- 8.7 Construction of a Baseline L0.- 8.8 De nition of a Stopping-Time Region S0.- 8.9 De nition of a Lipschitz Set K0 over the Base Line.- 8.10 Construction of Adapted Dyadic Intervals on the Base Line.- 8.11 Assigning Linear Functions to Adapted Dyadic Intervals.- 8.12 Construction of a Lipschitz Graph G Threaded through K0.- 8.13 Veri cation that the Graph is Indeed Lipschitz.- 8.14 A Partition of K n K0 into Three Sets: K1, K2, & K3.- 8.15 The Smallness of the Set K2.- 8.16 The Smallness of a Horrible Set H.- 8.17 Most of K Lies in the Vicinity of the Lipschitz Graph.- 8.18 The Smallness of the Set K1.- 8.19 Gamma Functions of the Lipschitz Graph.- 8.20 A Point Estimate on One of the Gamma Functions.- 8.21 A Global Estimate on the Other Gamma Function.- 8.22 Interlude: