Universität Wien

250140 VO Schramm-Loewner Evolution (2023S)

6.00 ECTS (4.00 SWS), SPL 25 - Mathematik
Moodle

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Details

Language: English

Examination dates

Lecturers

Classes (iCal) - next class is marked with N

There will be a second time slot (most likely Friday from 13.15 to 14.45).

  • Monday 06.03. 13:15 - 14:45 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Friday 10.03. 13:15 - 14:45 Hörsaal 7 Oskar-Morgenstern-Platz 1 1.Stock
  • Friday 17.03. 13:15 - 14:45 Hörsaal 7 Oskar-Morgenstern-Platz 1 1.Stock
  • Monday 20.03. 13:15 - 14:45 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Friday 24.03. 13:15 - 14:45 Hörsaal 7 Oskar-Morgenstern-Platz 1 1.Stock
  • Monday 27.03. 13:15 - 14:45 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Friday 31.03. 13:15 - 14:45 Hörsaal 7 Oskar-Morgenstern-Platz 1 1.Stock
  • Monday 17.04. 13:15 - 14:45 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Friday 21.04. 13:15 - 14:45 Hörsaal 7 Oskar-Morgenstern-Platz 1 1.Stock
  • Monday 24.04. 13:15 - 14:45 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Friday 28.04. 13:15 - 14:45 Hörsaal 7 Oskar-Morgenstern-Platz 1 1.Stock
  • Friday 05.05. 13:15 - 14:45 Hörsaal 7 Oskar-Morgenstern-Platz 1 1.Stock
  • Monday 08.05. 13:15 - 14:45 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Friday 12.05. 13:15 - 14:45 Hörsaal 7 Oskar-Morgenstern-Platz 1 1.Stock
  • Monday 15.05. 13:15 - 14:45 Seminarraum 1 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Friday 19.05. 13:15 - 14:45 Hörsaal 7 Oskar-Morgenstern-Platz 1 1.Stock
  • Monday 22.05. 13:15 - 14:45 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
    Seminarraum 1 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Friday 26.05. 13:15 - 14:45 Hörsaal 7 Oskar-Morgenstern-Platz 1 1.Stock
  • Friday 02.06. 13:15 - 14:45 Hörsaal 7 Oskar-Morgenstern-Platz 1 1.Stock
  • Monday 05.06. 13:15 - 14:45 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
    Seminarraum 1 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Friday 09.06. 13:15 - 14:45 Hörsaal 7 Oskar-Morgenstern-Platz 1 1.Stock
  • Monday 12.06. 13:15 - 14:45 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
    Seminarraum 1 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Friday 16.06. 13:15 - 14:45 Hörsaal 7 Oskar-Morgenstern-Platz 1 1.Stock
  • Monday 19.06. 13:15 - 14:45 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
    Seminarraum 1 Oskar-Morgenstern-Platz 1 Erdgeschoß
  • Friday 23.06. 13:15 - 14:45 Hörsaal 7 Oskar-Morgenstern-Platz 1 1.Stock
  • Monday 26.06. 13:15 - 14:45 Hörsaal 2 Oskar-Morgenstern-Platz 1 Erdgeschoß
    Seminarraum 1 Oskar-Morgenstern-Platz 1 Erdgeschoß

Information

Aims, contents and method of the course

Schramm--Loewner Evolutions, or SLE for short, are a family of random curves in some simply connected domain of the plane. They were introduced by Oded Schramm in 1999, who proved that SLE is the only possible scaling limit for interfaces in models of statistical mechanics at their critical point. In some cases this has been proved (leading to the Fields medals of Werner in 2006 and Smirnov in 2010). In others the problem remains a tantalising conjecture.

The theory is *very* beautiful, blending stochastic and complex analysis in a remarkable way, and has revolutionised our understanding of two-dimensional probability.

In this course we will learn the basics of SLE theory. We will describe how the SLE curves are constructed, and prove Schramm's theorem that they are characterised by conformal invariance and a certain domain Markov property. We will establish the two phase transitions describing the geometry of these curves as the parameter \(\kappa\) is changed (a.s simple for \(\kappa \le 4\), a.s. not simple but not space-filling for \(4< \kappa < 8\), and a.s. space-filling for \(\kappa \ge 8\)).
We will also explain the connections to various models of statistical mechanics (\(\kappa = 8/3\) for self-avoiding walks, \(\kappa =6\) for percolation, \(\kappa = 2\) for loop-erased random walks, etc.).

The theory will be explained from the point of view of people who have some familiarity with Ito calculus but not so much complex analysis. We will rely on the notes by myself and James Norris.

Assessment and permitted materials

There will be a written, closed book examination at the end of the class.

Minimum requirements and assessment criteria

Students must obtain a passing grade.

Examination topics

Reading list

We will use the notes by myself and James Norris as a main source throughout:
https://www.dropbox.com/s/k3pnm3nuzrstn43/sle.pdf?dl=0

Other sources will be given throughout the course as necessary.

Association in the course directory

MSTV; MANV

Last modified: Th 23.05.2024 09:46