SLE and two-dimensional statistical physics - syllabus
Note: This syllabus will evolve with time. A calendar indicating
what will be covered each day follows the list of topics.
Notes: Notes are available for sections that have a link.
Following the link will get you a pdf file. The date following the link
is when the pdf file was last changed.
2.1 Definition and properties
2.2 Convergence of random walks to Brownian motion
2.3 Conditional expectation
2.4 Markov properties of Brownian motion
3.1 Percolation
3.2 Loop-erased random walk
3.3 Self-avoiding random walk
3.4 Ising model
3.5 FK percolation and Potts models
3.6 Conformal invariance
3.7 Markov property
4.1 Definition, Riemann mapping theorem
4.2 Univalent functions
4.3 Half plane capacity
4.4 Loewner differential equation
5.1 Definition via Lowener equation
5.2 Derivation of SLE
6.1 Notation and some definitions
6.2 Definition of integration with respect to Brownian motion
6.3 Ito's formula
6.4 Time change of martingales
6.5 Bessel processes
7.1 Phases
7.2 Some computations
7.3 Restriction property
7.4 Locality property
7.5 Discrete SLE
Wednesday, Jan 12 - Class 1
1. Introduction
Friday, Jan 14 - Class 2
2.1 Definition and properties of Brownian motion
Wednesday, Jan 19 - Class 3
2.2 Convergence of random walks to Brownian motion
Friday, Jan 21 - Class 4
2.2 continued
Monday, Jan 24 - Class 5
2.3 Conditional expectation
Wednesday, Jan 26 - Class 6
2.4 Markov property of BM, martingales
Friday, Jan 28 - Class 7
2.4 Stopping times, Strong Markov property
Monday, Jan 31 - Class 8
3.1 Percolation
Wednesday, Feb 2 - Class 9
3.2 Loop-erased random walk
Friday, Feb 4 - Class 10
3.3 Self-avoiding random walk
Monday, Feb 7 - Class 11
3.4 Ising model
Wednesday, Feb 9 - Class 12
3.5 FK percolation and Potts models
Friday, Feb 11 - Class 13
3.6 Conformal invariance
3.7 Markov property
Monday, Feb 14 - Class 14
BEGIN 4. Conformal maps
Wednesday, Feb 16 - Class 15
Friday, Feb 18 - Class 16
Monday, Feb 21 - Class 17
Wednesday, Feb 23 - Class 18
Friday, Feb 25 - No Class (rodeo days)
Monday, Feb 28 - Class 20
4.4 Loewner differential equation - cont
Wednesday, Mar 2 - Class 21
4.4 Loewner differential equation - cont
Friday, Mar 4 - Class 22
5.1 Definition of SLE via Lowener equation
Monday, Mar 7 - Class 23
5.2 "Derivation" of SLE
Wednesday, Mar 9 - Class 24
BEGIN 6. Stochastic differential equations
Friday, Mar 11 - Class 25
Def of stochastic integral
Monday, Mar 21 - Class 26
Def of stochastic integral continued
Wednesday, Mar 23 - Class 27
Def of stochastic integral continued, Ito formula
Friday, Mar 25 - Class 28
Ito formula examples
Monday, Mar 28 - Class 29
Time change of martingles
Wednesday, Mar 30 - Class 30
Bessel processes
Friday, Apr 1 - Class 31
Bessel processes -cont
Monday, Apr 4 - Class 32
7.1 Properties of SLE - phases
Wednesday, Apr 6 - Class 33
7.1 continued
Friday, Apr 8 - Class 34
7.1 continued
7.2 Some computations
Monday, Apr 11 - Class 35
7.2 continued
Wednesday, Apr 13 - Class 36
7.2 continued
Friday, Apr 15 - Class 37
7.3 Locality property
Monday, Apr 18 - Class 38
7.3 Locality property continued
Wednesday, Apr 20 - Class 39
7.4 Restriction property
Friday, Apr 22 - Class 40
Restriction measures
Monday, Apr 25 - Class 41
7.5 Discrete SLE
Wednesday, Apr 27 - Class 42
8. Convergence of LERW to SLE_2
Friday, Apr 29 - Class 43
Convergence of LERW to SLE_2 - continued
Monday, May 2 - Class 44
Convergence of LERW to SLE_2 - continued
Wednesday, May 4 - Class 45