Wendelin Werner: Random Planar Curves and Schramm-Loewner Evolutions

1 Introduction1.1 General Motivation1.2 Loop-Erased Random Walks1.3 Iterations of Conformal Maps and SLE1.4 The Critical Percolation Exploration Process1.5 Chordal versus Radial1.6 Conclusion2 Loewner Chains2.1 Measuring the Size of Subsets of the Half-Plane2.2 Loewner Chains3 Chordal SLE3.1 Definition3.2 A First Computation3.3 Chordal $<math display='block'> <mrow> <mi>S</mi><mi>L</mi><msub> <mi>E</mi> <mi>&#x03BA;</mi> </msub> </mrow> </math> $$SLE_\kappa $ in Other Domains3.4 Transience4 Chordal SLE and Restriction4.1 Image of SLE under Conformal Maps4.2 Locality for SLE₆4.3 Restriction for SLE_8/35 SLE and the Brownian Frontier5.1 A Reflected Brownian Motion5.2 Brownian Excursions and SLE_8/36 Radial SLE6.1 Definitions6.2 Relation between Radial and Chordal SLE6.3 Radial SLE₆ and Reflected Brownian Motion7 Some Critical Exponents for SLE7.1 Disconnection Exponents7.2 Derivative Exponents7.3 First Consequences8 Brownian Exponents8.1 Introduction8.2 Brownian Crossings8.3 Disconnection Exponent8.4 Other Exponents8.5 Hausdorff Dimensions9 SLE, UST and LERW9.1 Introduction, LERW9.2 Uniform Spanning Trees, Wilson’s Algorithm9.3 Convergence to Chordal SLE₈9.4 The Loop-Erased Random Walk10 SLE and Critical Percolation10.1 Introduction10.2 The Cardy-Smirnov Formula10.3 Convergence to SLE₆ and Consequences11 What Is Missing11.1 A List of Ideas11.2 A List of Open ProblemsReferences