Conformal restriction, Brownian motion, and SLE
Student No.:50
Time:Mon/Wed/Fri 10:10-12:00, 2014-08-18 ~ 2014-08-29
Instructor:Hao Wu  [Department of Mathematics, MIT]
Place:Conference Room 1, Floor 1, Jin Chun Yuan West Building
Starting Date:2014-8-18
Ending Date:2014-8-29





When people tried to understand two-dimensional statistical physics models, it is realized that any conformally invariant process satisfying a certain restriction property has crossing or intersection exponents. Conformal field theory has been extremely successful in predicting the exact values of critical exponents describing the behavior of two-dimensional systems from statistical physics. The main goal of the present course is to investigate the restriction property and the related critical exponents on the mathematical level. The course has six lectures:


1.    Introduction on two-dimensional statistical physics models, conformal invariance, restriction property. Recall the basic properties of Brownian motion and introduce Brownian motion intersection exponents.


2.    Brownian motion, Brownian excursion and Brownian loop.


3.    Introduction on chordal SLE: Loewner Evolution, and basic properties.


4.    Calculation with chordal SLE(8/3). Chordal restriction samples.


5.    Introduction on radial SLE. And calculation with radial SLE(8/3).


6.    Radial restriction samples.






Conditional expectation; Martingale; Ito Formula; Brownian motion.






[1] Random Planar Curves and Schramm Loewner Evolutions, Wendelin Werner, Lecture notes from the 2002-Saint Flour summer school


[2] Conformal Restriction: the Chordal Case, Gregory Lawler, Oded Schramm, Wendlin Werner, J.Amer.Math.Sco., 16(4):917-955(electronic), 2003


[3] Conformal Restriction: the Radial Case, Hao Wu


[4] Conformally invariant processes in the plane, Gregory Lawler, volume 114 of Mathematical Surveys and Monographs. American Mathematical Society, Providence, RI, 2005


[5] Values of Brownian Intersection Exponents I, II, III, Gregory Lawler, Oded Schramm, Wendelin Werner, Acta Math. 187(2):237-273, 2001; Acta Math., 187(2):275-308, 2001; Ann. Inst. H. Poincare Probab. Statist. 38(1):109-123,2002