# 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. Brownian motion (last changed 1/27)

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. Statistical physics models (last changed 2/11)

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. Conformal maps (last changed 3/10)

4.1 Definition, Riemann mapping theorem
4.2 Univalent functions
4.3 Half plane capacity
4.4 Loewner differential equation

### 5. SLE (last changed 3/10)

5.1 Definition via Lowener equation
5.2 Derivation of SLE

### 6. Stochastic differential equations (last changed 4/1)

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. Properties of SLE (last changed 4/29)

7.1 Phases
7.2 Some computations
7.3 Restriction property
7.4 Locality property
7.5 Discrete SLE

### 8. Convergence of discrete models to SLE (last changed 4/29)

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