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# Schramm-Loewner evolution (Introduction to mathematical physics), spring 2016

**Teacher:** Antti Kemppainen

**Scope:** 10 cr

**Type:** Advanced studies in mathematics.

**Teaching:**

**Topics:**

1 Introduction to Brownian motion and stochastic analysis

2 Introduction to conformal invariants and boundary behaviour of conformal maps

3 Conformal maps, growth processes and the Loewner differential equation

4 Stochastic Loewner flows and Schramm—Loewner evolutions

5 Schramm—Loewner evolution in statistical physics

Take also a look at the fall 2011 web page of the course.

**Prerequisites:**

Probability theory (or familiarity with probability and some measure theory) and some complex analysis (including at least conformal maps and Riemann mapping theorem).

Physics students are welcome to the course, but background in physics is not required.

## News

- Most important changes to the webpage (recently):
- (31.3.) Chapter 5 was added
- (12.4.) Problem sheet 10 was posted
- (20.4.) Problem sheet 11 was posted
- (21.4.) Chapter 5 was updated
- (27.4.) Problem sheet 12 was posted
- (28.4.) Chapter 5 was updated
- (3.5.) Chapter 5 was updated
- (11.5.) Chapter 5 was updated and Chapter 6 was added
- (16.5.) Chapters 3, 4 and 5 got small updates

**The course has now ended.**Thank you everybody for attending!

## Teaching schedule

Weeks 3-9 and 11-18, lectures Friday 10-12 in room C123 and exercises Tuesday 10-12 in room B121.

Easter holiday 24.-30.3.

### Lectures

One lecture per week. A rough plan of the next lectures will appear here. The lecture notes will be posted in the Course material section below after each lecture.

**Tue 19.1.**Introduction and practical matters. Introduction to Brownian motion.**Fri****22.1.**Introduction to Brownian motion and stochastic calculus.**Fri 29.1.**Stochastic calculus, continues.**Fri 5.2.**Stochastic calculus, Ito's formula.**Fri 12.2.**End of stochastic calculus, conformal invariance of Brownian motion.**Fri 19.2.**Harmonic functions, Poisson kernels, conformal maps**Fri 26.2.**Loewner theory**Fri 4.3.**Loewner theory continues**Period break during the week 7.-11.3. -> NO LECTURE on Fri 11.3.****Fri 18.3.**Loewner theory continues**Easter break during 24.-30.3. -> NO LECTURE on**Fri 25.3.**Fri 1.4.**SLE definition, motivation and elementary properties**NO LECTURE on****Fri 8.4.****Tue 12.4. (the time and place of the exercise session)**Dimension of SLE continues**Fri 15.4.**Phases of SLE**Fri 22.4.**Phases of SLE continuous, variants of SLE**Fri 29.4.**Further topics in SLE**Fri 6.5.**Further topics in SLE

## Evaluation

The grading is based on the exercises done during the course. The evaluation is discussed more on the first lecture.

Roughly the idea is the following: at the beginning of the exercise session you return your solutions, at the session you present the solution or see somebody else presenting it. There are compulsory and bonus problems. If you get \( P \) compulsory problem points and \( B \) bonus points (active participation in the exercise sessions is included) during the course, \( P_{max} \) is the maximum number of compulsory problem points and

then passing the course is based on the number \( p \) and the grade on the number \( q. \)

## Course material

Typed lecture notes.

### Lecture notes

Main chapters:

- Chapter 1: Introduction
Chapter 2: Introduction to stochastic calculus

- Chapter 3: Introduction to conformal mappings
- Chapter 4: Loewner equation
- Chapter 5: Schramm-Loewner evolution
- Chapter 6: Regularity and convergence of random curves

Appendices:

- Appendix A: Supplementary material on probability theory
- Appendix B: Supplementary material on stochastic calculus
- Appendix C: Supplementary material on complex analysis

### Links to references

- Rick Durrett: Probability: theory and examples
- Timo Seppäläinen: Basics of Stochastic Analysis

Did you forget to register? What to do?

## Exercises

Return your solutions at the exercise session. See Evaluation above. Scanned copies of your solutions you can submit through submit assignment form (insert the solutions as an attachment file).

### Assignments

- Problem sheet 1, Tue 26.1.
- Problem sheet 2, Tue 2.2.
- Problem sheet 3, Tue 9.2.
- Problem sheet 4, Tue 16.2.
- Problem sheet 5, Tue 23.2.
- Problem sheet 6, Tue 1.3.
**Period break during the week 7.-11.3.**- Problem sheet 7, Tue 15.3.
- Problem sheet 8, Tue 22.3.
**Easter break during 24.-30.3.**- Problem sheet 9, Tue 5.4.
**No exercise session on Tue 12.4., a lecture instead.**Problem sheet 10, Mon 18.4.**Exercise session of Tue 19.4. moved to Mon 18.4. 14-16 room CK****111 (due to****department's recreational day).**- Problem sheet 11, Tue 26.4.
- Problem sheet 12, Tue 3.5.

### Exercise classes

Group | Day | Time | Room | Instructor |
---|---|---|---|---|

1. | Tue | 10-12 | B121 | Petri Tuisku |

## Course feedback

Course feedback can be given at any point during the course. Click here.